Mathematics is the science of measuring things and calculating things in our world. Math is a science or a group of related sciences dealing with the logic of quantity, numbers, structure, patterns, space, change, shape and arrangement. Math is a language using symbols that are use to describe patterns in nature, and used to interpret the behaviors of matter, which in turn helps us to make sense of the world, and also communicate information and abstract ideas. Math helps us to predict the future and understand the past, while measuring the present time. Math is used in engineering, reasoning, decision making, planning and problem solving, just to name a few.

Add - Subtract - Divide - Multiply - Fractions - Algebra - Geometry - Calculus - Trigonometry - Statistics - Symmetry

Mathematics Education is the practice of teaching and learning mathematics, along with the associated scholarly research.

Outline of Mathematics (PDF)

The Map of Mathematics (youtube).

What Math Skills Are Needed to Become an Engineer?

Mathematician is someone who uses an extensive knowledge of mathematics in his/her work, typically to solve mathematical problems. Mathematics is concerned with numbers, data, quantity, structure, space, models, and change.

Mathematical intelligence is being number smart and being good at reasoning using math. Math smart is the ability to determine the number or amount of something and the ability to correctly apply mathematics when needed. Math smart is the capacity to carry out mathematical operations that would help you to analyze problems logically, and investigate issues scientifically.

Number Sense is having an intuitive understanding of numbers, their magnitude, relationships, and how they are affected by operations. A person who knows how to solve mathematical problems that are not bound by traditional algorithms.

Mathematical Sciences is a group of areas of study that includes, in addition to mathematics, those academic disciplines that are primarily mathematical in nature but may not be universally considered subfields of mathematics proper.

Applied Mathematics is a branch of mathematics that deals with mathematical methods that find use in science, engineering, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge.

Discrete Mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous.

Babylonian Mathematics was any mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC. The first positional numerical system was developed in Babylon in the 2nd millennium. Principles of Math.

Dyscalculia is difficulty in learning or comprehending arithmetic, such as difficulty in understanding numbers, learning how to manipulate numbers, and learning facts in mathematics.

Mathematics as a Language is the system used by mathematicians to communicate mathematical ideas among themselves. This language consists of a substrate of some natural language like English, using technical terms and grammatical conventions that are peculiar to mathematical discourse, supplemented by a highly specialized symbolic notation for mathematical formulas. Like natural languages in general, discourse using the Language of Mathematics can employ a scala of registers. Research articles in academic journals are sources for detailed theoretical discussions about ideas concerning mathematics and its implications for society. Mathematical jargon (wiki).

Mathematics and Art are related in a variety of ways. Mathematics has itself been described as an art motivated by beauty. Mathematics can be discerned in arts such as music, dance, painting, architecture, sculpture, and textiles. This article focuses, however, on mathematics in the visual arts. Symmetry.

Music and Mathematics the basis of musical sound can be described mathematically in acoustics and exhibits a remarkable array of number properties. Elements of music such as its form, rhythm and metre, the pitches of its notes and the tempo of its pulse can be related to the measurement of time and frequency, offering ready analogies in geometry. The attempt to structure and communicate new ways of composing and hearing music has led to musical applications of set theory, abstract algebra and number theory. Some composers have incorporated the golden ratio and Fibonacci numbers into their work.

Principia Mathematica had three aims: (1) to analyze to the greatest possible extent the ideas and methods of mathematical logic and to minimize the number of primitive notions and axioms, and inference rules; (2) to precisely express mathematical propositions in symbolic logic using the most convenient notation that precise expression allows; (3) to solve the paradoxes that plagued logic and set theory at the turn of the 20th century, like Russell's paradox. The Principia covered only set theory, cardinal numbers, ordinal numbers, and real numbers. Deeper theorems from real analysis were not included, but by the end of the third volume it was clear to experts that a large amount of known mathematics could in principle be developed in the adopted formalism. It was also clear how lengthy such a development would be.A fourth volume on the foundations of geometry had been planned, but the authors admitted to intellectual exhaustion upon completion of the third.

Foundations of Mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics.

Mathematical Notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations include relatively simple symbolic representations, such as the numbers 0, 1 and 2, function symbols sin and +; conceptual symbols, such as lim, dy/dx, equations and variables; and complex diagrammatic notations such as Penrose graphical notation and Coxeter–Dynkin diagrams. Mathematical notations are used in mathematics, the physical sciences, engineering, and economics. Math Mnemonics (PDF).

Mathematical Symbols (PDF) - Glossary - Logic Symbols - Mathematical Symbols by Subject (PDF) - Basic Math Symbols

Order of Operations is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression. The order of operations, which is used throughout mathematics, science, technology and many computer programming languages, is multiplication and division, addition and subtraction, exponentiation and root extraction. Exponentiation is the operation of raising one quantity to the power of another.

Sign in mathematics originates from the property of every real number being either positive or negative or zero. Depending on local conventions, zero is either considered as being neither a positive, nor a negative number (having no sign, or a specific sign of its own), or as belonging to both, negative and positive numbers (having both signs).

Plus Sign (+) is a binary operator that indicates addition, as in 2 + 3 = 5.

Minus Sign (−) has three main uses in mathematics. The subtraction operator: A binary operator to indicate the operation of subtraction, as in 5 − 3 = 2. Subtraction is the inverse of addition. Directly in front of a number (numeric literal) and when it is not a subtraction operator it means a negative number. For instance −5 is negative 5. A unary operator that acts as an instruction to replace the operand by its additive inverse. For example, if x is 3, then −x is −3, but if x is −3, then −x is 3. Similarly, −(−2) is equal to 2. The above is a special case of this. Calculations.

Multiplication Sign is the symbol X. While similar to the lowercase letter x, the form is properly a rotationally symmetric saltire also known as the times sign or the dimension sign. * symbol on a computer keyboard can be used to Multiply.

Square is the result of multiplying a number by itself. 3 may be written as 3

Square Root of a number

Obelus symbol commonly represents the mathematical operation of division. it's a symbol consisting of a short horizontal line with a dot above and another dot below, commonly called the division sign. Forward Slash symbol on a computer keyboard can be used to divide.

Equals Sign equality sign (=) is a mathematical symbol used to indicate equality when the symbol is placed between two things. A sign indicating what the quantities add up to. Two parallel horizontal lines.

Less-than Sign is a mathematical symbol that denotes an inequality between two values. Examples of typical usage include 1/2 < 1 and -2 < 0. (when the symbol points to the left it's less than).

Greater-than Sign is a mathematical symbol that denotes an inequality between two values. Examples of typical usage include 1.5 > 1 and 1 > -2.

Two other comparison symbols are ≥ (greater than or equal to) and ≤ (less than or equal to).

Approximation ≈ tilde (U+2248, almost equal to) - Key Board Symbols.

Decimal is the standard system for denoting integer and non-integer numbers.(3.141). Decimal Separator is a symbol used to separate the integer part from the fractional part of a number written in decimal form.

Function is a relation such that one thing is dependent on another. A set sequence of steps, part of larger computer program. Serve as a purpose and what something is used for. Algebra.

Function in mathematics is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. A mathematical relation such that each element of a given set (the domain of the function) is associated with an element of another set (the range of the function). List of Mathematical Functions (wiki).

Implicit Function is a function that is defined implicitly by an implicit equation, by associating one of the variables (the value) with the others (the arguments).

Injective Function is a function that maps distinct elements of its domain to distinct elements of its codomain. In other words, every element of the function's codomain is the image of at most one element of its domain. The term one-to-one function must not be confused with one-to-one correspondence that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain.

Inverse Function is a function that "reverses" another function.

Exponential Function is a function of the form. Utility - Performance.

Equation is a statement containing one or more variables that are either added, subtracted, divided or multiplied in order to get an answer or to determine the values of numbers and what they equate to. A statement of an equality containing one or more variables. Solving the equation consists of determining which values of the variables make the equality true. Equation is a statement that the values of two mathematical expressions are equal (indicated by the sign =) the process of equating one thing with another. Variables are also called unknowns and the values of the unknowns which satisfy the equality are called solutions of the equation. There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variable. A conditional equation is true for only particular values of the variables. (1+3=4, one plus three equals four, one plus three is equal to four). Knowledge.

Equation Solving finding an answer to a set of variables using a mathematical function like adding or subtraction.

Mathematical Operation is a calculation from zero or more input values (called operands) to an output value. The number of operands is the arity of the operation.

Mathematical Expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations, and other aspects of logical syntax.

Interpretation logic is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation. The general study of interpretations of formal languages is called formal semantics.

Interpretation function in mathematical logic is a function that assigns functions and relations to the symbols of a signature.

Interpretability in mathematical logic is a relation between formal theories that expresses the possibility of interpreting or translating one into the other.

Formula is a rule expressed in symbols or a concise way of expressing information symbolically as in a mathematical or chemical formula. A conventionalized statement expressing some fundamental principle. Directions for making something. A standard procedure for solving a class of mathematical problems. Science Formula.

Well-formed Formula is a finite sequence of symbols from a given alphabet that is part of a formal language. A formal language can be identified with the set of formulas in the language. A formula is a syntactic object that can be given a semantic meaning by means of an interpretation. Two key uses of formulas are in propositional logic and predicate logic.

Mathematical Notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations are used in mathematics, the physical sciences, engineering, and economics. Mathematical notations include relatively simple symbolic representations, such as the numbers 0, 1 and 2; variables such as x, y and z; delimiters such as "(" and "|"; function symbols such as sin; operator symbols such as "+"; relational symbols such as "<"; conceptual symbols such as lim and dy/dx; equations and complex diagrammatic notations such as Penrose graphical notation and Coxeter–Dynkin diagrams.

Factorization is to reduce something to “basic building blocks”, such as numbers to prime numbers, or polynomials to irreducible polynomials. Factorization consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. For example, 3 × 5 is a factorization of the integer 15, and (x – 2)(x + 2) is a factorization of the polynomial x2 – 4. Factorization is not usually considered meaningful within number systems possessing division, such as the real or complex numbers. However, a meaningful factorization for a rational number or a rational function can be obtained by writing it in lowest terms and separately factoring its numerator and denominator. Factorization was first considered by ancient Greek mathematicians in the case of integers. They proved the fundamental theorem of arithmetic, which asserts that every positive integer may be factored into a product of prime numbers, which cannot be further factored into integers greater than 1. Moreover, this factorization is unique up to the order of the factors. Although integer factorization is a sort of inverse to multiplication, it is much more difficult algorithmically, a fact which is exploited in the RSA cryptosystem to implement public-key cryptography. Polynomial factorization has also been studied for centuries. In elementary algebra, factoring a polynomial reduces the problem of finding its roots to finding the roots of the factors. Polynomials with coefficients in the integers or in a field possess the unique factorization property, a version of the fundamental theorem of arithmetic with prime numbers replaced by irreducible polynomials. In particular, a univariate polynomial with complex coefficients admits a unique (up to ordering) factorization into linear polynomials: this is a version of the fundamental theorem of algebra. In this case, the factorization can be done with root-finding algorithms. The case of polynomials with integer coefficients is fundamental for computer algebra. There are efficient computer algorithms for computing (complete) factorizations within the ring of polynomials with rational number coefficients (see factorization of polynomials). A commutative ring possessing the unique factorization property is called a unique factorization domain. There are number systems, such as certain rings of algebraic integers, which are not unique factorization domains. However, rings of algebraic integers satisfy the weaker property of Dedekind domains: ideals factor uniquely into prime ideals. Factorization may also refer to more general decompositions of a mathematical object into the product of smaller or simpler objects. For example, every function may be factored into the composition of a surjective function with an injective function. Matrices possess many kinds of matrix factorizations. For example, every matrix has a unique LUP factorization as a product of a lower triangular matrix L with all diagonal entries equal to one, an upper triangular matrix U, and a permutation matrix P; this is a matrix formulation of Gaussian elimination.

Integer Factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are sufficiently large, no efficient, non-quantum integer factorization algorithm is known. An effort by several researchers, concluded in 2009, to factor a 232-digit number (RSA-768) utilizing hundreds of machines took two years and the researchers estimated that a 1024-bit RSA modulus would take about a thousand times as long. However, it has not been proven that no efficient algorithm exists. The presumed difficulty of this problem is at the heart of widely used algorithms in cryptography such as RSA. Many areas of mathematics and computer science have been brought to bear on the problem, including elliptic curves, algebraic number theory, and quantum computing. Not all numbers of a given length are equally hard to factor. The hardest instances of these problems (for currently known techniques) are semiprimes, the product of two prime numbers. When they are both large, for instance more than two thousand bits long, randomly chosen, and about the same size (but not too close, for example, to avoid efficient factorization by Fermat's factorization method), even the fastest prime factorization algorithms on the fastest computers can take enough time to make the search impractical; that is, as the number of digits of the primes being factored increases, the number of operations required to perform the factorization on any computer increases drastically. Many cryptographic protocols are based on the difficulty of factoring large composite integers or a related problem—for example, the RSA problem. An algorithm that efficiently factors an arbitrary integer would render RSA-based public-key cryptography insecure.

Factor Analysis is a statistical method used to describe variability among observed, correlated variables in terms of a potentially lower number of unobserved variables called factors.

Evaluation (testing)

Axiom is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.

Mathematical Proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms, along with accepted rules of inference. Axioms may be treated as conditions that must be met before the statement applies. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed to be true is known as a conjecture.

Logarithm is the inverse operation to exponentiation. That means the logarithm of a number is the exponent to which another fixed number, the base, must be raised to produce that number.

Combination is a way of selecting items from a collection.

Parameters is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when identifying the system, or when evaluating its performance, status, condition, etc.

Frame of Reference consists of an abstract coordinate system and the set of physical reference points that uniquely fix (locate and orient) the coordinate system and standardize measurements. Matrix.

Mathematical Induction is a mathematical proof technique used to prove a given statement about any well-ordered set. Most commonly, it is used to establish statements for the set of all natural numbers.

Physics Math - Math Games

Combinatorial Game Theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information.

Mathematical Analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.

Mathematical Model is a description of a system using mathematical concepts and language.

Mathematical Models are systems of differential equations used to describe biological mechanisms, such as a cell: Irina Kareva (video and text) - Asking the right question and translating it to the right equation, and then back. I formulate assumptions about how these elements interact with each other and with their environment. Then, I translate these assumptions into equations. Finally, I analyze my equations and translate the results back into the language of biology.

Graphical Model is a probabilistic model for which a graph expresses the conditional dependence structure between random variables. They are commonly used in probability theory, statistics—particularly Bayesian statistics—and machine learning.

Probabilistic Model is a class of mathematical model, which embodies a set of assumptions concerning the generation of some sample data, and similar data from a larger population. A statistical model represents, often in considerably idealized form, the data-generating process.

Mathematical Visualization is an aspect of geometry which allows one to understand and explore mathematical phenomena via visualization.

Probabilistic Graphical Models (coursera)

Anomaly - Pattern Recognition - Ai

Similarity Geometry if two objects both have the same shape, or one has the same shape as the mirror image of the other.

Films about Math - Math Videos

Éléments de mathématique is a treatise on mathematics by the collective Nicolas Bourbaki, composed of twelve books (each divided into one or more chapters).

Pure Mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but pure mathematicians are not primarily motivated by such applications. Instead, the appeal is attributed to the intellectual challenge and aesthetic beauty of working out the logical consequences of basic principles.

Arithmetic is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations on them—addition, subtraction, multiplication and division. The branch of Pure Mathematics dealing with the theory of numerical Calculations.

Elementary Arithmetic is the simplified portion of arithmetic that includes the operations of addition, subtraction, multiplication, and division. It should not be confused with elementary function arithmetic.

Calculate is to determine the amount or number of something using mathematical methods. To add things up. To determine something by reasoning, common sense, or practical experience; to estimate; evaluate; gauge. Recalculate is the act of calculating something again to include additional data or to eliminate possible errors.

Calculation is to judge something to be probable. To predict something in advance using mathematical methods. To have a certain value or carry a certain weight. Calculation is a deliberate process that transforms one or more inputs into one or more results, with variable change. Calculation is a set of formulas and equations that produce answers that are know to be accurate and valid, which allows the user to be confident that the information that is being produced is accurate and can also be used to make important decisions or make more calculations. Analog Computer.

Mental Calculation comprises arithmetical calculations using only the human brain, with no help from calculators, computers, or pen and paper.

Computation is any type of calculation that includes both arithmetical and non-arithmetical steps and which follows a well-defined model (e.g. an algorithm). Mechanical or electronic devices (or, historically, people) that perform computations are known as computers. An especially well-known discipline of the study of computation is computer science.

Counting is the action of finding the number of elements of a finite set of objects.

Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures.

Running the Numbers means to do the calculations and to do the math in order to come up with an answer. "I ran the numbers and determined that we only have one option."

Computer Model is a program that runs on a computer that creates a model or a simulation of a real-world feature, phenomenon or event. A computer model is an abstract mathematic representations of a real-world event, system, behavior, or natural phenomenon. A computer model is designed to behave just like the real-life system. A computer model is a translation of objects or phenomena from the real world into mathematical equations. A computer program witha set of instructions that a computer uses to perform some analysis or computation.

Quantify is to express something as a number or as a measure or quantity, which shows how many or how little there are of something.

Factor is one of two or more integers that can be exactly divided into another integer. Any of the numbers (or symbols) that form a product when multiplied together. An independent variable in statistics. Factor can also mean to consider something as relevant when making a decision. Be a contributing factor. Anything that contributes causally to a result. An abstract part of something.

Abacus is a calculating tool constructed as a bamboo frame with beads sliding on wires, but originally they were beans or stones moved in grooves in sand or on tablets of wood, stone, or metal. The abacus system of mental calculation is a system where users mentally visualize an abacus to do calculations. The standard abacus can be used to perform addition, subtraction, division, and multiplication; the abacus can also be used to extract square-roots and cubic roots. Mental Abacus (wiki). Visualization Abacus (youtube).

Computation is the procedure of calculating and determining something by mathematical or logical methods. Problem solving that involves numbers or quantities.

Calculator is typically a portable electronic device used to perform calculations, ranging from basic arithmetic to complex mathematics. Scientific calculators include trigonometric and statistical calculations. Graphing calculators can be used to graph functions defined on the real line, or higher-dimensional Euclidean space. Some calculators even have the ability to do computer algebra.

Mechanical Calculator is a mechanical device used to perform automatically the basic operations of arithmetic. Most mechanical calculators were comparable in size to small desktop computers and have been rendered obsolete by the advent of the electronic calculator. Old Mechanical Calculators (youtube)

Curta is a small mechanical calculator accumulating values on cogs, which are added or complemented by a stepped drum mechanism.

Formula Calculator is a software calculator that can perform a calculation in two steps: Enter the calculation by typing it in from the keyboard. Press a single button or key to see the final result.

Software Calculator is a calculator that has been implemented as a computer program, rather than as a physical hardware device. they are among the simpler interactive software tools, and, as such, they: Provide operations for the user to select one at a time. Can be used to perform any process that consists of a sequence of steps each of which applies one of these operations. Have no purpose other than these processes, because the operations are the sole, or at least the primary, features of the calculator, rather than being secondary features that support other functionality that is not normally known simply as calculation. As a calculator, rather than a computer, they usually: Have a small set of relatively simple operations. Perform short processes that are not compute intensive. Do not accept large amounts of input data or produce many results.

Adder in electronics is a digital circuit that performs addition of numbers. In many computers and other kinds of processors adders are used in the arithmetic logic units or ALU. They are also utilized in other parts of the processor, where they are used to calculate addresses, table indices, increment and decrement operators, and similar operations. Although adders can be constructed for many number representations, such as binary-coded decimal or excess-3, the most common adders operate on binary numbers. In cases where two's complement or ones' complement is being used to represent negative numbers, it is trivial to modify an adder into an adder–subtractor. Other signed number representations require more logic around the basic adder.

Tally Marks is a unary numeral system. They are a form of numeral used for counting. They are most useful in counting or tallying ongoing results, such as the score in a game or sport, as no intermediate results need to be erased or discarded.

Tally Stick was an ancient memory aid device used to record and document numbers, quantities, or even messages. 18,000 to 20,000 BC. Principally, there are two different kinds of tally sticks: the single tally and the split tally. A common form of the same kind of primitive counting device is seen in various kinds of prayer beads.

Unary Language is a formal language (a set of strings) where all strings have the form 1k, where "1" can be any fixed symbol. For example, the language {1, 111, 1111} is unary, as is the language {1k | k is prime}. The complexity class of all such languages is sometimes called TALLY.

Download Calculators for PC - Mathematical Integrals Calculator

Wolfram Alpha - Calculators - Conversions - Translations

Method of Exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area between the n-th polygon and the containing shape will become arbitrarily small as n becomes large. As this difference becomes arbitrarily small, the possible values for the area of the shape are systematically "exhausted" by the lower bound areas successively established by the sequence members. The method of exhaustion typically required a form of proof by contradiction, known as reductio ad absurdum. This amounts to finding an area of a region by first comparing it to the area of a second region (which can be “exhausted” so that its area becomes arbitrarily close to the true area). The proof involves assuming that the true area is greater than the second area, and then proving that assertion false, and then assuming that it is less than the second area, and proving that assertion false, too.

Nerve Cells in the Human Brain can Count. In humans, the neurons activated in response to a "2" are for instance not the same as the neurons activated for a "5." We are born with the ability to count: Shortly after birth, babies can estimate the number of events and even perform simple calculations. We learn digits differently from characters.

Number is a mathematical object used to count, measure, and label. Symbol.

Prime Number - Natural Number

Composite Number is a positive integer that can be formed by multiplying together two smaller Positive Integers.

Complex Number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a solution of the equation x2 = −1, which is called an imaginary number because there is no real number that satisfies this equation.

Integer is a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5 1⁄2, and √2 are not. Integer is any of the Natural Number (positive or negative) or zero. Whole Number.

Square Number is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it can be written as 3×3.

Square Root is the result of multiplying the number by itself. For example, 4 and −4 are square roots of 16 because 4

Number Theory is a branch of pure mathematics devoted primarily to the study of the integers.

Numeral System is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. It can be seen as the context that allows the symbols "11" to be interpreted as the binary symbol for three, the decimal symbol for eleven, or a symbol for other numbers in different bases.

Roman Numerals originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. The history of Roman numerals follows the history of Ancient Rome itself, from its beginnings at the Latin Palatine Hill in 8th and 9th century B.C.. Numbers in this system are represented by combinations of letters from the Latin alphabet. Roman numerals, as used today, employ seven symbols, each with a fixed integer value, as follows: Symbol (I=1) (V=5) (X=10) (L=50) (C=100) (D=500) (M=1,000).

Arabic Numerals are the ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9; or numerals written using them in the Hindu–Arabic numeral system (where the position of a digit indicates the power of 10 to multiply it by). It is the most common system for the symbolic representation of numbers in the world today.

Tally Marks are a unary numeral system. They are a form of numeral used for counting. They are most useful in counting or tallying ongoing results, such as the score in a game or sport, as no intermediate results need to be erased or discarded. However, because of the length of large numbers, tallies are not commonly used for static text. Notched sticks, known as tally sticks, were also historically used for this purpose. Unary numeral system is the simplest numeral system to represent natural numbers, to represent a number N, a symbol representing 1 is repeated N times. Empty string or empty word is the unique string of length zero.

Positional Notation denotes usually the extension to any base of the Hindu–Arabic Numeral System (or decimal system). More generally, a positional system is a numeral system in which the contribution of a digit to the value of a number is the product of the value of the digit by a factor determined by the position of the digit. In early numeral systems, such as Roman numerals, a digit has only one value: I means one, X means ten and C a hundred (however, the value may be negated if placed before another digit). In modern positional systems, such as the decimal system, the position of the digit means that its value must be multiplied by some value: in 555, the three identical symbols represent five hundreds, five tens, and five units, respectively, due to their different positions in the digit string.

Decimal is a proper fraction whose denominator is a power of 10. Numbered or proceeding by tens or based on ten. A numeral system is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral system. The way of denoting numbers in the decimal system is often referred to as Decimal notation. Decimal is also called base-ten positional numeral system, and occasionally called denary. Pi (circle).

Decimal Point is a dot placed after the figure representing units in a decimal fraction. A point or dot we use to separate the whole number part from the fractional part of a decimal number. (0.1 or 2.3).

Decimal Separator is a symbol used to separate the integer part from the fractional part of a number written in decimal form. (e.g., "." in 12.45). Any such symbol can be called a decimal mark, decimal marker, or decimal sign. Period is a punctuation mark (.) placed at the end of a declarative sentence to indicate a full stop or after abbreviations.

Decimal Place is the position of a digit to the right of a decimal point.

Decimal Data Type provide a built-in library decimal data type to represent non-repeating decimal fractions like 0.3 and -1.17 without rounding, and to do arithmetic on them.

Approximate Number System is a cognitive system that supports the estimation of the magnitude of a group without relying on language or symbols.

List of Numbers - Large Numbers - More Numbers - "Crunching the Numbers"

Palindromic Number is a number that remains the same when its digits are reversed.

If you start counting from one and spell out the numbers as you go, you won't use the letter "A" until you reach 1,000.

Addition is determine the sum of. The act of adding one thing to another. A quantity that is added. Something added to what you already have. The arithmetic operation of summing; calculating the sum of two or more numbers. A component that is added to something to improve it.

Add is to make an addition to something or join or combine or unite something with other things. To increase the quality, quantity, size or scope of something. Make an addition by combining numbers.

Figure how many things we have by adding things together. Figure how much there is of something by adding things up. Figure the size of something by measuring and adding the numbers up. Figure how many things I will have in the future by adding things up. Predict the future by calculating actions over a period of time.

Counting is the action of finding the number of elements of a finite set of objects. Work Sheets.

Subitizing is the rapid, accurate, and confident judgments of numbers performed for small numbers of items.

List of Numbers

Enumeration is a complete, ordered listing of all the items in a collection.

Sum is a quantity obtained by the addition of a group of numbers. The whole amount. A set containing all and only the members of two or more given sets.

Quantity is how much there is or how many there are of something that you can quantify or add up or measure.

Summation is the addition of a sequence of numbers; the result is their sum or total. Summary.

Extra is something added. Something additional of the same kind. Beyond or more than is needed, desired, or required.

Polynomial is a mathematical function that is the sum of a number of terms. Polynomial is an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s). A Latin name with more than two parts. Polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. An example of a polynomial of a single indeterminate, x, is x2 - 4x + 7. An example in three variables is x3 + 2xyz2 - yz + 1.

Subtraction is a mathematical operation that represents the operation of removing objects from a collection. It is signified by the minus sign (−). When we have less. Predicting shortages when we have less of something. Decay.

Division is an arithmetic operation that is the inverse of multiplication; if a × b = c, then a = c ÷ b, as long as b is not zero. Division by zero is undefined for the real numbers and most other contexts, because if b = 0, then a cannot be deduced from b and c, as then c will always equal zero regardless of a. In some contexts, division by zero can be defined although to a limited extent, and limits involving division of a real number as it approaches zero are defined. Division is the act of equally partitioning numbers or things into parts, pieces, or sections that are separated by a boundary that divides them or keeps them apart or separate. Division is the quotient of two numbers when computed. Quotient is the ratio of two quantities to be divided. Division is one of the four basic operations of arithmetic, the others being addition, repeated subtraction, and multiplication. The division of two natural numbers is the process of calculating the number of times one number is contained within one another. For example, in the picture on the right, the 20 apples are divided into groups of five apples, and there exist four groups, meaning that five can be contained within 20 four times, or 20 ÷ 5 = 4.

Dividing is about Sharing. How much will each of us have if we equally divide? How much will each of us need if we all use the same amount?

Divide is to separate into parts or portions. Make a division or separation.

Share is to use jointly or in common. Give, or receive a share of. Sharing

Equal is having the same quantity, value, or measure as another. Be identical or equivalent to. Make equal, uniform, corresponding, or matching.

Square-Free Integer is an integer which is divisible by no perfect square other than 1. For example, 10 is square-free but 18 is not, as 18 is divisible by 9 = 32. The smallest positive square-free numbers are 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, ... (sequence A005117 in the OEIS).

Proportion is the balance among the parts of something. Harmonious arrangement or relation of parts or elements within a whole (as in a design). The relation between things (or parts of things) with respect to their comparative quantity, magnitude, or degree. The quotient obtained when the magnitude of a part is divided by the magnitude of the whole. Magnitude or extent. Adjust in size relative to other things. A part, share, or number considered in comparative relation to a whole.

Whole is including all components without exception; being one unit or constituting the full amount or extent or duration; complete.

Part is something determined in relation to something that includes it. A portion of a natural object. One of the portions into which something is regarded as divided and which together constitute a whole.

Graduated is something marked with or divided into degrees. To make fine adjustments or divide into marked intervals for optimal measuring.

Proportionality in mathematics states that two variables are proportional if there is always a constant ratio between them. The constant is called the coefficient of proportionality or proportionality constant. If one variable is always the product of the other variable and a constant, the two are said to be directly proportional. x and y are directly proportional if the ratio y/x is constant. If the product of the two variables is always a constant, the two are said to be inversely proportional. x and y are inversely proportional if the product xy is constant.

Fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, vulgar, or simple fraction (examples: 1 2 {\displaystyle {\tfrac {1}{2}}} {\tfrac {1}{2}} and 17/3) consists of an integer numerator displayed above a line (or before a slash), and a non-zero integer denominator, displayed below (or after) that line. Numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals.

Fractions Poster (image)

Visual Fractions - Fractions

Percentage: 10% is equal to 1/10 fraction. 20% is equivalent to ⅕ fraction. 25% is equivalent to ¼ fraction. 50% is equivalent to ½ fraction. 75% is equivalent to ¾ fraction. 90% is equivalent to 9/10 fraction.

Lowest Common Denominator is the lowest common multiple of the denominators of a set of fractions. It simplifies adding, subtracting, and comparing fractions. The lowest common denominator of a set of fractions is the lowest number that is a multiple of all the denominators: their lowest common multiple. The product of the denominators is always a common denominator.

Least Common Multiple of two integers a and b, usually denoted by LCM(a, b), is the smallest positive integer that is divisible by both a and b. Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero. However, some authors define lcm(a,0) as 0 for all a, which is the result of taking the lcm to be the least upper bound in the lattice of divisibility. The LCM is the "lowest common denominator" (LCD) that can be used before fractions can be added, subtracted or compared. The LCM of more than two integers is also well-defined: it is the smallest positive integer that is divisible by each of them.

Multiplication of whole numbers may be thought as a repeated addition; that is, the multiplication of two numbers is equivalent to adding as many copies of one of them, the multiplicand, as the value of the other one, the multiplier. Normally, the multiplier is written first and multiplicand second, though this can vary, as the distinction is not very meaningful. (Times Symbol is X).

Multiplication Table (image)

Discovering the power of many. Predicting Growth based on many different inputs. Predicting consumption amounts and production amounts based how many people. Percentage.

Exponentiation is a mathematical operation, written as bn, involving two numbers, the base b and the exponent n. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, bn is the product of multiplying n bases.

Fold Change is a measure describing how much a quantity changes going from an initial to a final value. For example, an initial value of 30 and a final value of 60 corresponds to a fold change of 1 (or equivalently, a change to 2 times), or in common terms, a one-fold increase. Fold change is calculated simply as the ratio of the difference between final value and the initial value over the original value. Thus, if the initial value is A and final value is B, the fold change is (B - A)/A or equivalently B/A - 1. As another example, a change from 80 to 20 would be a fold change of -0.75, while a change from 20 to 80 would be a fold change of 3 (a change of 3 to 4 times the original).

Square in algebra is the result of multiplying a number by itself. For example, 9 is a square number, since it can be written as 3 times 3. 3 squared = 9. (3

Trachtenberg System is a system of rapid mental calculation. The system consists of a number of readily memorized operations that allow one to perform arithmetic computations very quickly. (add a zero before the number being multiplied).

Multiplication Algorithm is an algorithm or method to multiply two numbers. Depending on the size of the numbers, different algorithms are in use. Efficient multiplication algorithms have existed since the advent of the decimal system. The most important algorithms are the ones for general multiplication, division and addition.

Complex Multiplication is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties.

Algebra is when several of the factors of a problem are known and one or more are unknown. Algebra uses alphabetic characters representing a number which is either arbitrary or not fully specified or unknown. Algorithm - Correlation.

Abstract Algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras.

Elementary Algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are either unknown or allowed to take on many values. For example, in x + 2 = 5, the letter x is unknown, but the law of inverses can be used to discover its value: x = 3. In E = mc2, the letters E and m are variables, and the letter c is a constant, the speed of light in a vacuum. Algebra gives methods for solving equations and expressing formulas that are much easier (for those who know how to use them) than the older method of writing everything out in words. Functions.

Linear Algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces.

Boolean Algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0 respectively.

Square in algebra is the result of multiplying a number by itself.

Polynomial is an expression consisting of variables (or indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. Separable Polynomial.

Quadratic Equation is any equation having the form where x represents an unknown, and a, b, and c represent known numbers such that a is not equal to 0. If a = 0, then the equation is linear, not quadratic.

Symbols (letters) - Mathematical Symbols - Logic Symbols

Logic Alphabet also called the X-stem Logic Alphabet (XLA), constitutes an iconic set of symbols that systematically represents the sixteen possible binary truth functions of logic.

Lattice - Deductive Reasoning - Percentage

See all the layers of information underneath and not just see the surface.

Converse in logic "If I am a bachelor, then I am an unmarried man" is logically equivalent to "If I am an unmarried man, then I am a bachelor."

Interpolation in mathematics is the calculation of the value of a function between the values already known. Interpolation is a method of constructing new data points within the range of a discrete set of known data points. Interpolate is to estimate the value of that function for an intermediate value of the independent variable. Interpolation can also mean the insertion of something of a different nature into something else. A message that is spoken or written that is introduced or inserted.

Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Mathematics of points, lines, curves, circles, angles, surfaces and planes.

Euclidean geometry the study of plane and solid figures on the basis of axioms and theorems employed by the Greek mathematician Euclid around 323–283 BC. In its rough outline, Euclidean geometry is the plane and solid geometry commonly taught in secondary schools.

Congruence in geometry two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other. Optical Illusions.

Computational Geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Sacred Geometry.

Proportion is a central principle of architectural theory and an important connection between mathematics and art. It is the visual effect of the relationships of the various objects and spaces that make up a structure to one another and to the whole. These relationships are often governed by multiples of a standard unit of length known as a "module". Arithmetic-Geometric Mean (wiki).

Line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation, but in a more abstract setting, such as incidence geometry, a line may be an independent object, distinct from the set of points which lie on it.

Axis is a straight line through a body or figure that satisfies certain conditions. The center around which something rotates.

Linearity is something that can be graphically represented as a straight line. Linear is relating to a line; involving a single dimension. ________________ Angles.

Point is the precise location of something; a spatially limited location. A geometric element that has position but no extension. Point can also be a symbol.

Plane in geometry is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as subspaces of some higher-dimensional space, as with a room's walls extended infinitely far, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry.

Planar is something involving two dimensions. Navigation - Orbital Plane.

Grade-school students teach a Robot to help themselves Learn Geometry.

Amorphous is something that has no definite form or distinct shape, something lacking in system or structure that is characteristic of living bodies, something without real or apparent crystalline form.

Shapes is the form of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material composition. Symbols - Visual Language.

Geometric Shape is the geometric information which remains when location, scale, orientation and reflection are removed from the description of a geometric object. That is, the result of moving a shape around, enlarging it, rotating it, or reflecting it in a mirror is the same shape as the original, and not a distinct shape. Objects that have the same shape as each other are said to be similar. If they also have the same scale as each other, they are said to be congruent. Many two-dimensional geometric shapes can be defined by a set of points or vertices and lines connecting the points in a closed chain, as well as the resulting interior points. Such shapes are called polygons and include triangles, squares, and pentagons. Other shapes may be bounded by curves such as the circle or the ellipse. Many three-dimensional geometric shapes can be defined by a set of vertices, lines connecting the vertices, and two-dimensional faces enclosed by those lines, as well as the resulting interior points. Such shapes are called polyhedrons and include cubes as well as pyramids such as tetrahedrons. Other three-dimensional shapes may be bounded by curved surfaces, such as the ellipsoid and the sphere. A shape is said to be convex if all of the points on a line segment between any two of its points are also part of the shape. Geometric Modeling.

Mathematica Object is an abstract object arising in mathematics.

Structures - Patterns - Symmetry - Dimensions (space) - Polyhedron

Tetris is a tile-matching puzzle video game. Tetromino is a geometric shape composed of four squares, connected orthogonally. This, like dominoes and pentominoes, is a particular type of polyomino. The corresponding polycube, called a tetracube, is a geometric shape composed of four cubes connected orthogonally. Tetris Effect occurs when people devote so much time and attention to an activity that it begins to pattern their thoughts, mental images, and dreams.

Edge in geometry is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope.

Vertex is a point where two or more line segments meet or a point where two or more curves, lines, or edges meet to form a corner. The point where two lines meet to form an angle and the corners of polygons and polyhedra are called vertices. Vertices is plural for vertex.

Apex also know as summit, peak, tip, top, or extreme end, is the vertex which is in some sense the "highest" of the figure to which it belongs. The term is typically used to refer to the vertex opposite from some "base."

Side is a surface forming part of the outside of an object. An extended outer surface of an object. A line segment forming part of the perimeter of a plane figure. A place within a region identified relative to a center or reference location.

Segment is one of several parts or pieces that fit with others to constitute a whole object. One of the parts into which something naturally divides.

Closed Plane figure is formed by three or more line segments. It is closed and has no openings between line segments and has no curved sides. Where the two sides of a polygon intersect is called a vertex of the polygon.

Face in geometry is a flat surface that forms part of the boundary of a solid object. A three-dimensional solid bounded exclusively by flat faces is a polyhedron. In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, the term is also used to mean an element of any dimension of a more general polytope, in any number of dimensions. Boundary is the line or plane indicating the limit or extent of something. A line determining the limits of an area. The greatest possible degree of something. Wall.

Rectangle is a quadrilateral with four right angles.

Triangle is a polygon with three edges and three vertices. Trigonometry.

Equilateral Triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon, so it is also referred to as a regular triangle.

Isosceles Triangle is a triangle that has two sides of equal length.

Golden Triangle in mathematics is an isosceles triangle in which the duplicated side is in the golden ratio to the distinct side: Golden triangles are found in the nets of several stellations of dodecahedrons and icosahedrons. (also known as the sublime triangle).

Square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or right angles). It can also be defined as a rectangle in which two adjacent sides have equal length. Cube.

Square–Cube Law describes the relationship between the volume and the area as a shape's size increases or decreases.

Inverse-square law is any physical law stating that a specified physical quantity or intensity is inversely proportional to the square of the distance from the source of that physical quantity. The fundamental cause for this can be understood as geometric dilution corresponding to point-source radiation into three-dimensional space.

Cube is a hexahedron with six equal squares as faces. A three-dimensional shape with six square or rectangular sides.

Hypercubes is an n-dimensional analogue of a square (n = 2) and a cube (n = 3).

Tesseract is the four-dimensional analog of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eight cubical cells. The tesseract is one of the six convex regular 4-polytopes.

Rhombus is a simple (non-self-intersecting) quadrilateral whose four sides all have the same length.

Flower of Life - Spatial Awareness

Scutoid is a geometric solid between two parallel surfaces. The boundary of each of the surfaces (and of all the other parallel surfaces between them) is a polygon, and the vertices of the two end polygons are joined by either by a curve or a Y-shaped connection. Scutoids present at least one vertex between these two planes. The faces of the scutoids are not necessarily convex, so several scutoids can pack together to fill all the space between the two parallel surfaces. The object was first described in Nature Communications in July 2018, and the name scutoid was coined because of its resemblance to the shape of the scutum and scutellum in some insects, such as beetles in the Cetoniidae subfamily.

Circle is a simple closed shape in Euclidean geometry. It is the set of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves so that its distance from a given point is constant. The distance between any of the points and the center is called the radius.

Round is something having a circular shape. Wind around; move along a circular course. On all sides; so as to encircle.

Orbit - Pi - Angle of Rotation - Polygons - Bubbles

Arc is a continuous portion of a circle. Something curved in shape. Form an arch or curve.

Semicircle is a one-dimensional locus of points that forms half of a circle. The full arc of a semicircle always measures 180°. It has only one line of symmetry.

Radius is a straight line from the center to the circumference of a circle or sphere. Of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is the length of any of them.

Diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. 180 Degrees.

Circumference of a closed curve or circular object is the linear distance around its edge.

Pi is the ratio of a circle's circumference to its diameter. (3.14159). Dividing the circumference by its diameter will equal 3.141. The circumference is a little over 3 times the size of the diameter. Calculating Pi involves taking any circle and dividing its circumference by its diameter. The circumference of a circle is found with the formula C= π*d = 2*π*r. Thus, pi equals a circle's circumference divided by its diameter. Circumference of Earth is 24,901 miles, divided by Pi or 3.1415 = Diameter 7,926 miles, then divided by 2.002 = Radius of Earth is 3,959 miles. Multiply the diameter by π, or 3.14. The result is the circle's circumference. For any circle, if you divide the circumference by the diameter you get pi, which is an irregular number usually rounded to 3.14. A physicist or an engineer who uses pi in numerical calculations may need to have access to 5 or 15 decimal place approximations. How are the decimal numbers calculated in pi? The oldest method used is to use the power series expansion of atan(x) = x - x^3/3 + x^5/5 - ... together with formulas like pi = 16*atan(1/5) - 4*atan(1/239). This gives about 1.4 decimals per term. A second method used is to use formulas coming from Arithmetic-Geometric mean computations. The third method comes from the theory of complex multiplication of elliptic curves. Infinity - Decimals.

Cylinder is the surface formed by the points at a fixed distance from a given straight line called the axis of the cylinder. It is one of the most basic curvilinear geometric shapes.

Curve is an object similar to a line that is not straight or flat. There are no straight lines in the universe, everything eventually curves.

Curvature is the amount by which a geometric object such as a surface deviates from being a flat plane, or a curve from being straight as in the case of a line, but this is defined in different ways depending on the context.

Convex is something curving or bulging outward. Three Dementional.

Curves we (mostly) don't learn in high school (and applications) (youtube)

Elliptic Curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O.

Spiral is a curve which emanates from a point, moving farther away as it revolves around the point. Golden Spiral.

Ulam Spiral is a graphical depiction of the set of prime numbers.

Sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball, (viz., analogous to a circular object in two dimensions. (12 around 1).

Sphere within a Sphere is a bronze sculpture by Italian sculptor Arnaldo Pomodoro.

Spheres in the Vacuum of Space.

Spherical Polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhedra is most conveniently derived in this way. The most familiar spherical polyhedron is the soccer ball, thought of as a spherical truncated icosahedron. The next most popular spherical polyhedron is the beach ball, thought of as a hosohedron.

Stereographic Projection is a particular mapping (function) that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point: the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it preserves angles at which curves meet. It is neither isometric nor area-preserving: that is, it preserves neither distances nor the areas of figures. Intuitively, then, the stereographic projection is a way of picturing the sphere as the plane, with some inevitable compromises. Because the sphere and the plane appear in many areas of mathematics and its applications, so does the stereographic projection; it finds use in diverse fields including complex analysis, cartography, geology, and photography. In practice, the projection is carried out by computer or by hand using a special kind of graph paper called a stereographic net, shortened to stereonet, or Wulff net.

Train Wheels have a conical geometry, which is the primary means of keeping the train's motion aligned with the track. Train wheels have a flange on one side to keep the wheels, and hence the train, running on the rails, when the limits of the geometry-based alignment are reached, e.g. due to some emergency or defect. Some wheels have a cylindrical geometry, where flanges are essential to keep the train on the rail track.

Conical Surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the apex or vertex — and any point of some fixed space curve — the directrix — that does not contain the apex. Each of those lines is called a generatrix of the surface. Every conic surface is ruled and developable. In general, a conical surface consists of two congruent unbounded halves joined by the apex. Each half is called a nappe, and is the union of all the rays that start at the apex and pass through a point of some fixed space curve. (In some cases, however, the two nappes may intersect, or even coincide with the full surface.) Sometimes the term "conical surface" is used to mean just one nappe.

Cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. Light Cone.

Platonic Solid is a regular, convex polyhedron in three-dimensional space. It is constructed by congruent regular polygonal faces with the same number of faces meeting at each vertex. Five solids meet those criteria, and each is named after its number of faces.

Symmetry - Bucky Ball - Domes - Bubbles

Polyhedron is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices. In several protolanguages the word poly comes from words that means many, much, more, and great. The word hedron means base or seat. Pyramid Triangles.

Goldberg Polyhedron is a convex polyhedron made from hexagons and pentagons.

Polygon is a closed plane figure bounded by straight sides. A polygon is a flat, two-dimensional shape with straight sides that is fully closed and all the sides are joined up. The sides must be straight. Polygons may have any number of sides. A shape with curved sides is not a polygon. a polygon is a plane figure that is described by a finite number of straight line segments connected to form a closed polygonal chain or polygonal circuit. The bounded plane region, the bounding circuit, or the two together, may be called a polygon. The segments of a polygonal circuit are called its edges or sides. The points where two edges meet are the polygon's vertices (singular: vertex) or corners. The interior of a solid polygon is sometimes called its body. An n-gon is a polygon with n sides; for example, a triangle is a 3-gon. A simple polygon is one which does not intersect itself. Mathematicians are often concerned only with the bounding polygonal chains of simple polygons and they often define a polygon accordingly. A polygonal boundary may be allowed to cross over itself, creating star polygons and other self-intersecting polygons. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. There are many more generalizations of polygons defined for different purposes.

Convex Polygon is a simple polygon (not self-intersecting) in which no line segment between two points on the boundary ever goes outside the polygon.

Tetrahedron is a polyhedron composed of 4 triangular faces, 6 straight edges, and 4 vertex corners. The tetrahedron, also known as a triangular pyramid, is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces. Volume: (√2)/12 × a³ - Surface area: √3 × a² - Base shape: Triangle. Shapes with similar faces: Octahedron, Icosahedron, Triangular prism, Square pyramid, Hexagonal pyramid, Pentagonal pyramid.

Tetrahedral Number is a figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron. The nth tetrahedral number is the sum of the first n triangular numbers. The first ten tetrahedral numbers are: 1, 4, 10, 20, 35, 56, 84, 120, 165, 220.

Cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

Octahedron is a polyhedron with eight faces, twelve edges, and six vertices. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. A regular octahedron is the dual polyhedron of a cube. It is a rectified tetrahedron. It is a square bipyramid in any of three orthogonal orientations. It is also a triangular antiprism in any of four orientations. An octahedron is the three-dimensional case of the more general concept of a cross polytope.

Dodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120. Regular Dodecahedron is composed of twelve regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges, and 160 diagonals (60 face diagonals, 100 space diagonals). It is represented by the Schläfli symbol {5,3}.

Pentagon is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simple or self-intersecting. A self-intersecting regular pentagon (or star pentagon) is called a pentagram. The 15th kind of Pentagon that can Tile a Plane.

Pentagon Tiling is a tiling of the plane where each individual piece is in the shape of a pentagon.

Pentagram is the shape of a five-pointed star drawn with five straight strokes. Triangles.

Hexagram is a hallow six-pointed geometric star with two overlapping equilateral triangles. Equilateral triangle is a triangle in which all three sides are equal. The hexagram is part of an infinite series of shapes which are compounds of two n-dimensional simplices. In three dimensions, the analogous compound is the stellated octahedron, and in four dimensions the compound of two 5-cells is obtained. N is a vector space having n vectors as its basis.

Hexagon is a polygon with six sides and six angles, or a six-sided polygon or 6-gon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. A regular hexagon is defined as a hexagon that is both equilateral and equiangular. It is bicentric, meaning that it is both cyclic (has a circumscribed circle) and tangential (has an inscribed circle). Honeycomb.

Hexagonal Prism is a prism with hexagonal base. This polyhedron has 8 faces, 18 edges, and 12 vertices. Since it has 8 faces, it is an octahedron. However, the term octahedron is primarily used to refer to the regular octahedron, which has eight triangular faces. Because of the ambiguity of the term octahedron and tilarity of the various eight-sided figures, the term is rarely used without clarification. Before sharpening, many pencils take the shape of a long hexagonal prism.

Prism in geometry is a polyhedron comprising an n-sided polygonal base, a second base which is a translated copy (rigidly moved without rotation) of the first, and n other faces (necessarily all parallelograms) joining corresponding sides of the two bases. All cross-sections parallel to the bases are translations of the bases. Prisms are named for their bases, so a prism with a pentagonal base is called a pentagonal prism. The prisms are a subclass of the prismatoids. Crystals.

Icosahedron is a polyhedron with 20 faces. There are infinitely many non-similar shapes of icosahedra, some of them being more symmetrical than others. The best known is the (convex, non-stellated) regular icosahedron—one of the Platonic solids—whose faces are 20 equilateral triangles.

Octagon is an eight-sided polygon or 8-gon.

Pythagorean Theorem (wiki) - Truncated Octahedron (wiki)

Cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square.

Tetradecahedron is a polyhedron with 14 faces. There are numerous topologically distinct forms of a tetradecahedron, with many constructible entirely with regular polygon faces. A tetradecahedron is sometimes called a tetrakaidecahedron. Skin.

Trapezoid is a quadrilateral with two parallel sides. The wrist bone between the trapezium and the capitate bones. Trapezoidal is something resembling a trapezoid.

Quadrilateral is a polygon in Euclidean plane geometry with four edges (sides) and four vertices (corners). Other names for quadrilateral include quadrangle (in analogy to triangle), tetragon (in analogy to pentagon, 5-sided polygon, and hexagon, 6-sided polygon), and 4-gon (in analogy to k-gons for arbitrary values of k). Optical Illusions.

Three-Dimensional Space - Dimensions

Weaire-Phelan Structure is a complex 3-dimensional structure representing an idealised foam of equal-sized bubbles.

Tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps.

Parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure.

Vector is a variable quantity that can be resolved into components. A straight line segment whose length is magnitude and whose orientation in space is direction.

Vector in mathematics and physics is an element of a vector space. In physics and geometry, a Euclidean vector, used to represent physical quantities that have both magnitude and direction.

Complex Plane or z-plane, is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis. It can be thought of as a modified Cartesian plane, with the real part of a complex number represented by a displacement along the x-axis, and the imaginary part by a displacement along the y-axis.

Polygonal Chain is a connected series of line segments. More formally, a polygonal chain P is a curve specified by a sequence of points ( A 1 , A 2 , … , A n ) called its vertices. The curve itself consists of the line segments connecting the consecutive vertices. A polygonal chain may also be called a polygonal curve, polygonal path, polyline, piecewise linear curve, or, in geographic information systems, a linestring or linear ring.

Origami is the art of paper folding, which is often associated with Japanese culture. Mathematics of Paper Folding (wiki) - Box Pleat - Programmable Matter - Creativity.

Topology is the properties of space that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing. Important topological properties include connectedness and compactness. Spatial intelligence

Topological Space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighborhoods.

Topography - Geography - Euler

Problem Solving - Management

Polynomial is an expression consisting of variables (or indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

Geometric Progression is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common Ratio.

Scale - Sizes - Mind Maps

Graphs and Charts using shapes, symbols and images to communicate.

Graphing Calculator is a handheld computer that is capable of plotting graphs, solving simultaneous equations, and performing other tasks with variables. Most popular graphing calculators are also programmable, allowing the user to create customized programs, typically for scientific/engineering and education applications. Because they have large displays in comparison to standard 4-operation handheld calculators, graphing calculators also typically display several lines of text and calculations at the same time.

Computer Algebra System is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists.

Trigonometry is the science of measuring triangles. A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C. Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. Navigation.

Trigonometric Functions (wiki)

Polygon is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain or circuit.

Pyramid is a structure whose outer surfaces are triangular and converge to a single point at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilateral, or any polygon shape. A pyramid has at least three outer triangular surfaces (at least four faces including the base). The square pyramid, with square base and four triangular outer surfaces, is a common version. Polyhedron

Slope or gradient of a line is a number that describes both the direction and the steepness of the line. Slope is calculated by finding the ratio of the "vertical change" to the "horizontal change" between (any) two distinct points on a line.

Frustum is the portion of a solid (normally a cone or pyramid) that lies between one or two parallel planes cutting it. A right frustum is a parallel truncation of a right pyramid or right cone.

Harmonic Mathematics terminology to that of music is not accidental: the equations of motion of vibrating strings, drums and columns of air are given by formulas involving Laplacians; the solutions to which are given by eigenvalues corresponding to their modes of vibration. Thus, the term "harmonic" is applied when one is considering functions with sinusoidal variations, or solutions of Laplace's equation and related concepts.

Sine of an angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle (that is not the hypotenuse) to the length of the longest side of the triangle (i.e., the hypotenuse).

Cosine is the ratio of the adjacent side to the hypotenuse of a right-angled triangle.

Pythagorean Theorem is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides. This theorem can be written as an equation relating the lengths of the sides a, b and c, often called the "Pythagorean equation" A squared plus B squared equals C squared, where c represents the length of the hypotenuse and a and b the lengths of the triangle's other two sides.

Hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides. For example, if one of the other sides has a length of 3 (when squared, 9) and the other has a length of 4 (when squared, 16), then their squares add up to 25. The length of the hypotenuse is the square root of 25, that is, 5.

Special Right Triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45°–45°–90°. This is called an "angle-based" right triangle. A "side-based" right triangle is one in which the lengths of the sides form ratios of whole numbers, such as 3 : 4 : 5, or of other special numbers such as the golden ratio. Knowing the relationships of the angles or ratios of sides of these special right triangles allows one to quickly calculate various lengths in geometric problems without resorting to more advanced methods.

Angle is the space between two lines or planes that intersect; the inclination of one line to another; measured in degrees or radians, which is the standard unit of angular measure. Degree is a measurement of a plane angle, defined so that a full rotation is 360 degrees. Triangulation - Navigation.

Angle in planar geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane. Angles are also formed by the intersection of two planes in Euclidean and other spaces. These are called dihedral angles. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the spherical angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles. Golden Angle.

Angular of an object, outline, or shape having angles or sharp corners. Denoting physical properties or quantities measured with reference to or by means of an angle, especially those associated with rotation. Something measured by an angle or by the rate of change of an angle. Having angles or an angular shape. Speed Square.

Protractor is a measuring instrument for measuring angles, typically made of transparent plastic or glass.

Incline is to be at an angle. A non-flat area of ground that tends upwards or downwards. Inclined Plane or ramp is a flat supporting surface tilted at an angle, with one end higher than the other, used as an aid for raising or lowering a load. Stairs is a construction designed to bridge a large vertical distance by dividing it into smaller vertical distances, called steps. Stairs may be straight, round, or may consist of two or more straight pieces connected at angles. Recliner is an armchair or sofa that reclines when the occupant lowers the chair's back and raises its front. It has a backrest that can be tilted back, and often a footrest that may be extended by means of a lever on the side of the chair, or may extend automatically when the back is reclined. Recline is to move the upper body backwards and down and lean in a comfortable resting position.

Pitch is to be at an angle. Degree of deviation from a horizontal plane. Slope downward. Pitched Roof - Music Pitch.

Slope is a non-flat area of ground that tends upwards or downwards. The property possessed by a line or surface that departs from the horizontal.

Gradient is the property possessed by a line or surface that departs from the horizontal. A graded change in the magnitude of some physical quantity or dimension.

Grade refers to the tangent of the angle of that surface to the horizontal of a physical feature, landform or constructed line. It is a special case of the slope, where zero indicates horizontality. A larger number indicates higher or steeper degree of "tilt". Often slope is calculated as a ratio of "rise" to "run", or as a fraction ("rise over run") in which run is the horizontal distance (not the distance along the slope) and rise is the vertical distance. The grades or slopes of existing physical features such as canyons and hillsides, stream and river banks and beds are often described. Grades are typically specified for new linear constructions (such as roads, landscape grading, roof pitches, railroads, aqueducts, and pedestrian or bicycle circulation routes). The grade may refer to the longitudinal slope or the perpendicular cross slope. (also called slope, incline, gradient, mainfall, pitch or rise). Grade is the gradient of a slope or road or other surface. The height of the ground on which something stands. A position on a scale of intensity or amount or quality. One-hundredth of a right angle. Grade as a Score.

Gable is the triangular part of a house's exterior wall that supports a pointed or peaked roof. Gothic-style houses are well known for their many gables. Gable is the generally triangular portion of a wall between the edges of intersecting roof pitches. The shape of the gable and how it is detailed depends on the structural system used, which reflects climate, material availability, and aesthetic concerns. A gable wall or gable end more commonly refers to the entire wall, including the gable and the wall below it. Some types of roofs do not have a gable (for example hip roofs do not). One common type of roof with gables, the gable roof, is named after its prominent gables. Shed Roof.

Angle of Rotation is a measurement of the amount, namely the angle, that a figure is rotated about a fixed point, often the center of a circle. A clockwise rotation is considered a negative rotation, so that, for instance, a rotation of 310° (counterclockwise) can also be called a rotation of –50° (since 310° + 50° = 360°, a full rotation (turn)). A counterclockwise rotation of more than one complete turn is normally measured modulo 360°, meaning that 360° is subtracted off as many times as possible to leave a non-negative measurement less than 360°. For example, the carts on a Ferris wheel move along a circle around the center point of that circle. If a cart moves around the wheel once, the angle of rotation is 360 degrees. If the cart was stuck halfway, at the top of the wheel, at that point its angle of rotation was only 180 degrees. This is also referred to as the "order of symmetry." Angles are commonly measured in degrees, radians, gons (gradians) and turns, sometimes also in angular mils and binary radians. They are central to polar coordinates and trigonometry.

Plane of Rotation is an abstract object used to describe or visualize rotations in space. In three dimensions it is an alternative to the axis of rotation, but unlike the axis of rotation it can be used in other dimensions, such as two, four or more dimensions. Plane.

Rotational Symmetry also known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which it looks exactly the same for each rotation.

Hinge is a mechanical bearing that connects two solid objects, typically allowing only a limited angle of rotation between them. Two objects connected by an ideal hinge rotate relative to each other about a fixed axis of rotation: all other translations or rotations being prevented, and thus a hinge has one degree of freedom. Hinges may be made of flexible material or of moving components. In biology, many joints function as hinges like the elbow joint.

Degree in Relation to Angles, usually denoted by the degree symbol (°), is a measurement of a plane angle, defined so that a full rotation is 360 degrees. It is not an SI unit, as the SI unit of angular measure is the radian, but it is mentioned in the SI brochure as an accepted unit. Because a full rotation equals 2π radians, one degree is equivalent to π/180 radians. (in full, a degree of arc, arc degree, or arc-degree). Temperature.

Degree is a measure for arcs and angles. A position on a scale of intensity or amount or quality. A specific identifiable position in a continuum or series or especially in a process. A unit of temperature on a specified scale.

Horizontal is parallel to the ground or flat on the water or in the plane of the horizon or a base line at right angles to the vertical. Latitude - Parallel Wiring.

Vertical is straight up and down or in an upright position or posture. At right angles to the plane of the horizon or a base line. Longitude - Horizontal and Vertical (wiki).

Altitude is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The reference datum also often varies according to the context. Although the term altitude is commonly used to mean the height above sea level of a location, in geography the term elevation is often preferred for this usage.

Spirit Level or bubble level or simply a level is an instrument designed to indicate whether a surface is horizontal (level) or vertical (plumb). Different types of spirit levels may be used by carpenters, stonemasons, bricklayers, other building trades workers, surveyors, millwrights and other metalworkers, and in some photographic or videographic work.

Level Instrument is an optical instrument used to establish or verify points in the same horizontal plane in a process known as levelling, and is used in conjunction with a levelling staff to establish the relative heights levels of objects or marks. It is widely used in surveying and construction to measure height differences and to transfer, measure, and set heights of known objects or marks.

Levelling is a branch of surveying, the object of which is to establish or verify or measure the height of specified points relative to a datum. It is widely used in cartography to measure geodetic height, and in construction to measure height differences of construction artifacts.

Vanishing Point is a point on the image plane of a perspective drawing where the two-dimensional perspective projections (or drawings) of mutually parallel lines in three-dimensional space appear to converge. When the set of parallel lines is perpendicular to a picture plane, the construction is known as one-point perspective, and their vanishing point corresponds to the oculus, or "eye point", from which the image should be viewed for correct perspective geometry. Traditional linear drawings use objects with one to three sets of parallels, defining one to three vanishing points. Spatial Intelligence.

Sideways is moving left to right parallel to a plane or moving from the side at an angle. With one side forward or to the front.

Parallel in geometry are lines that are side by side and having the same distance continuously between them, like an equal sign. Two lines in a plane that do not intersect or touch each other at any point are said to be parallel. An imaginary line around the Earth parallel to the equator. Parallel in computing is the simultaneous performance of multiple operations. Parallel Wiring.

Perpendicular is the relationship between two lines which meet at a right angle (90 degrees). The property extends to other related geometric objects.

Right Angle is an angle of exactly 90° (degrees), corresponding to a quarter turn.

Diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. In matrix algebra, a diagonal of a square matrix is a set of entries extending from one corner to the farthest corner. From angle to angle.

Radian is the standard unit of angular measure, used in many areas of mathematics. The length of an arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends; one radian is just under 57.3 degrees.

Turn in geometry is a unit of plane angle measurement equal to 2π radians, 360 degrees or 400 gradians. A turn is also referred to as a revolution or complete rotation or full circle or cycle or rev or rot.A turn can be subdivided in many different ways: into half turns, quarter turns, centiturns, milliturns, binary angles, points etc.

Truss is a structure that "consists of two-force members only, where the members are organized so that the assemblage as a whole behaves as a single object". A "two-force member" is a structural component where force is applied to only two points. Although this rigorous definition allows the members to have any shape connected in any stable configuration, trusses typically comprise five or more triangular units constructed with straight members whose ends are connected at joints referred to as nodes.

Triangle Center is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles, i.e. a point that is in the middle of the figure by some measure.

Ancient Babylonian Tablet - World's First Trig Table (youtube)

Trigonometry to calculate the elevation of a mountain, scientists would measure the distance between two points on the ground and then measure the angles between the top of the mountain and each point. "If you have two angles, you know the third, because the sum of the angles is 180 [degrees].

You can calculate the height of an object using the distance and angle. distance * cos (angle), where distance is the horizontal distance to the object, and angle is the angle above horizontal of the top of the object (from the viewer). The result will be the height above the viewer. Surveying.

Tree Height Measurement is the vertical distance between the base of the tree and the tip of the highest branch on the tree, and is difficult to measure accurately. How to measure a tree (using your thumb or looking between your legs).

Topographic Prominence measures the height of a mountain or hill's summit relative to the lowest contour line encircling it but containing no higher summit within it. It is a measure of the independence of a summit. A peak's key col (highest gap between two mountains) is a unique point on this contour line and the parent peak is some higher mountain, selected according to various objective criteria.

Deductive Reasoning

Eratosthenes was the first person to calculate the circumference of the Earth more than 2000 years ago, which he did by comparing angles of the mid-day Sun at two places a known North-South distance apart. His calculation was remarkably accurate. He was also the first to calculate the tilt of the Earth's axis, again with remarkable accuracy.

Calculus is the mathematical study of change. Calculating changes and calculating the actions needed to correctly adjust to these changes. The same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. It has two major branches, differential calculus (concerning rates of change and slopes of curves), and integral calculus (concerning accumulation of quantities and the areas under and between curves); these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. Generally, modern calculus is considered to have been developed in the 17th century by Isaac Newton and Gottfried Leibniz. Today, calculus has widespread uses in science, engineering and economics and can solve many problems that elementary algebra alone cannot.

Calculus 1 - limits and basic Differentiation and integration.

Calculus 2 - more sophisticated Integration techniques, and infinite series.

Calculus 3 - Multivariable calculus a.k.a. vector calculus.

Limit is the value that a function or sequence "approaches" as the input or index approaches some value. Limits are essential to calculus (and mathematical analysis in general) and are used to define continuity, derivatives, and integrals.

MIT 2006 Integration Bee Competitive Calculus (youtube)

Deductive Reasoning

Operation in mathematics is a calculation from zero or more input values (called operands) to an output value.

Operand is the object of a mathematical operation, a quantity on which an operation is performed. There are two common types of operations: unary and binary. Unary operations involve only one value, such as negation and trigonometric functions. Binary operations, on the other hand, take two values, and include addition, subtraction, multiplication, division, and exponentiation.

Unary Operation is an operation with only one operand, i.e. a single input. An example is the function f : A - A, where A is a set. The function f is a unary operation on A.

Binary Operation on a set is a calculation that combines two elements of the set (called operands) to produce another element of the set (more formally, an operation whose arity is two, and whose two domains and one codomain are (subsets of) the same set). Examples include the familiar elementary arithmetic operations of addition, subtraction, multiplication and division. Other examples are readily found in different areas of mathematics, such as vector addition, matrix multiplication and conjugation in groups.

Differentials is the instantaneous change of one quantity relative to another; df(x)/dx. A quality that differentiates between similar things. The result of mathematical differentiation. Differentiation is the process of finding a derivative. The derivative of a function is the rate of change of the output value with respect to its input value, whereas differential is the actual change of function.

Differentiates is something marked as different with a distinctive feature, attribute, or trait.

Differential Equations is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

Predictions - Causality

Stochastic Differential Equation is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as unstable stock prices or physical systems subject to thermal fluctuations. Typically, SDEs contain a variable which represents random white noise calculated as the derivative of Brownian motion or the Wiener process. However, other types of random behaviour are possible, such as jump processes.

Derivative of a function of a real variable measures the sensitivity to change of a quantity (a function value or dependent variable) which is determined by another quantity (the independent variable). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time is advanced.

Quadratic Formula is the solution of the quadratic equation. There are other ways to solve the quadratic equation instead of using the quadratic formula, such as factoring, completing the square, or graphing. Using the quadratic formula is often the most convenient way.

Differential also means a bevel gear that permits rotation of two shafts at different speeds; used on the rear axle of automobiles to allow wheels to rotate at different speeds on curves.

Statistics is the study of the collection, analysis, interpretation, presentation, and organization of data. In applying statistics to, e.g., a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model process to be studied. Populations can be diverse topics such as "all people living in a country" or "every atom composing a crystal". Statistics deals with all aspects of data including the planning of data collection in terms of the design of surveys and experiments. Factoring the possibilities, knowing the odds. Errors.

Mathematical Statistics is the application of mathematics to statistics, which was originally conceived as the science of the state — the collection and analysis of facts about a country: its economy, land, military, population, and so forth. Mathematical techniques which are used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure-theoretic probability theory.

Computational Statistics is the interface between statistics and computer science.

Statistical Mechanics is a branch of theoretical physics using probability theory to study the average behaviour of a mechanical system, where the state of the system is uncertain. A common use of statistical mechanics is in explaining the thermodynamic behavior of large systems.

Statistical Significance is attained whenever the observed p-value of a test statistic is less than the significance level defined for the study.

Statistical Model is a class of mathematical model, which embodies a set of assumptions concerning the generation of some sample data, and similar data from a larger population. A statistical model represents, often in considerably idealized form, the data-generating process. The assumptions embodied by a statistical model describe a set of probability distributions, some of which are assumed to adequately approximate the distribution from which a particular data set is sampled. The probability distributions inherent in statistical models are what distinguishes statistical models from other, non-statistical, mathematical models. A statistical model is usually specified by mathematical equations that relate one or more random variables and possibly other non-random variables. As such, a statistical model is "a formal representation of a theory" (Herman Adèr quoting Kenneth Bollen). All statistical hypothesis tests and all statistical estimators are derived from statistical models. More generally, statistical models are part of the foundation of statistical inference.

Multivariate Statistics is a subdivision of statistics encompassing the simultaneous observation and analysis of more than one outcome variable. Multivariate statistics concerns understanding the different aims and background of each of the different forms of multivariate analysis, and how they relate to each other. The practical application of multivariate statistics to a particular problem may involve several types of univariate and multivariate analyses in order to understand the relationships between variables and their relevance to the problem being studied.

In addition, multivariate statistics is concerned with multivariate probability distributions, in terms of both. How these can be used to represent the distributions of observed data; how they can be used as part of statistical inference, particularly where several different quantities are of interest to the same analysis. Certain types of problems involving multivariate data, for example simple linear regression and multiple regression, are not usually considered to be special cases of multivariate statistics because the analysis is dealt with by considering the (univariate) conditional distribution of a single outcome variable given the other variables.

Sample in statistics or a data sample, is a set of data collected and/or selected from a statistical population by a defined procedure. The elements of a sample are known as sample points, sampling units or observations. Cherry Picking Data.

Sampling in statistics is concerned with the selection of a subset of individuals from within a statistical population to estimate characteristics of the whole population. Two advantages of sampling are that the cost is lower and data collection is faster than measuring the entire population. Survey.

Sample Size determination is the act of choosing the number of observations or replicates to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a population from a sample. In practice, the sample size used in a study is usually determined based on the cost, time, or convenience of collecting the data, and the need for it to offer sufficient statistical power. In complicated studies there may be several different sample sizes: for example, in a stratified survey there would be different sizes for each stratum. In a census, data is sought for an entire population, hence the intended sample size is equal to the population. In experimental design, where a study may be divided into different treatment groups, there may be different sample sizes for each group. Sample sizes may be chosen in several ways: using experience – small samples, though sometimes unavoidable, can result in wide confidence intervals and risk of errors in statistical hypothesis testing. Using a target variance for an estimate to be derived from the sample eventually obtained, i.e. if a high precision is required (narrow confidence interval) this translates to a low target variance of the estimator. Using a target for the power of a statistical test to be applied once the sample is collected. Using a confidence level, i.e. the larger the required confidence level, the larger the sample size (given a constant precision requirement).

Overfitting is the production of an analysis that corresponds too closely or exactly to a particular set of data, and may therefore fail to fit additional data or predict future observations reliably. An overfitted model is a statistical model that contains more parameters than can be justified by the data. The essence of overfitting is to have unknowingly extracted some of the residual variation (i.e. the noise) as if that variation represented underlying model structure. Underfitting occurs when a statistical model cannot adequately capture the underlying structure of the data. An underfitted model is a model where some parameters or terms that would appear in a correctly specified model are missing. Underfitting would occur, for example, when fitting a linear model to non-linear data. Such a model will tend to have poor predictive performance. Cherry Picking Data.

Per Capita is for each person or in relation to people taken individually. Phrase means "by heads" or "for each head" or per individual/person. in place of saying "Per Person". Latin for "Per Head".

Per Capita Income measures the average income earned per person in a given area, city, region, country in a specified year. It is calculated by dividing the area's total income by its total population.

Stratified Sampling is a method of sampling from a population. Stratification is the process of dividing members of the population into homogeneous subgroups before sampling. Bias.

Tool for non-statisticians automatically generates models that glean insights from complex datasets and automatically generates models for analyzing raw data. The tool currently lives on Jupyter Notebook, an open-source web framework that allows users to run programs interactively in their browsers. Users need only write a few lines of code to uncover insights.

Survey Methodology studies the sampling of individual units from a population and the associated survey data collection techniques, such as questionnaire construction and methods for improving the number and accuracy of responses to surveys. Survey methodology includes instruments or procedures that ask one or more questions that may, or may not, be answered.

Standard Deviation is a measure that is used to quantify the amount of variation or dispersion of a set of data values. A low standard deviation indicates that the data points tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values.

Statistical Inference is the process of deducing properties of an underlying distribution by analysis of data. Inferential statistical analysis infers properties about a population: this includes testing hypotheses and deriving estimates. The population is assumed to be larger than the observed data set; in other words, the observed data is assumed to be sampled from a larger population.

Statistical Interference is when two probability distributions overlap, statistical interference exists. Knowledge of the distributions can be used to determine the likelihood that one parameter exceeds another, and by how much.

Outlier in statistics is a data point that differs significantly from other observations. An outlier may be due to variability in the measurement or it may indicate experimental error; the latter are sometimes excluded from the data set. An outlier can cause serious problems in statistical analyses.

Confidence Interval is a type of interval estimate of a population parameter. It is an observed interval (i.e., it is calculated from the observations), in principle different from sample to sample, that potentially includes the unobservable true parameter of interest. How frequently the observed interval contains the true parameter if the experiment is repeated is called the confidence level. In other words, if confidence intervals are constructed in separate experiments on the same population following the same process, the proportion of such intervals that contain the true value of the parameter will match the given confidence level. Whereas two-sided confidence limits form a confidence interval, and one-sided limits are referred to as lower/upper confidence bounds (or limits).

5 Sigma is a measure of how confident scientists feel their results are. If experiments show results to a 5 sigma confidence level, that means if the results were due to chance and the experiment was repeated 3.5 million times then it would be expected to see the strength of conclusion in the result no more than once.

Meta-analysis is a statistical analysis that combines the results of multiple scientific studies.

Geometric distribution (wiki)

Statistical Hypothesis Testing is a hypothesis that is testable on the basis of observing a process that is modeled via a set of random variables.

Parametric Statistics is a branch of statistics which assumes that sample data comes from a population that follows a probability distribution based on a fixed set of parameters. Statistics.

Statistical Process Control is a method of quality control which uses statistical methods. SPC is applied in order to monitor and control a process. Monitoring and controlling the process ensures that it operates at its full potential. At its full potential, the process can make as much conforming product as possible with a minimum (if not an elimination) of waste (rework or scrap). SPC can be applied to any process where the "conforming product" (product meeting specifications) output can be measured. Key tools used in SPC include control charts; a focus on continuous improvement; and the design of experiments. An example of a process where SPC is applied is manufacturing lines.

Ordination Statistics is a method complementary to data clustering, and used mainly in exploratory data analysis (rather than in hypothesis testing). Ordination orders objects that are characterized by values on multiple variables (i.e., multivariate objects) so that similar objects are near each other and dissimilar objects are farther from each other. These relationships between the objects, on each of several axes (one for each variable), are then characterized numerically and/or graphically. Stats Direct.

Geo-Statistics is a branch of statistics focusing on spatial or spatiotemporal datasets. Developed originally to predict probability distributions of ore grades for mining operations, it is currently applied in diverse disciplines including petroleum geology, hydrogeology, hydrology, meteorology, oceanography, geochemistry, geometallurgy, geography, forestry, environmental control, landscape ecology, soil science, and agriculture (esp. in precision farming). Geostatistics is applied in varied branches of geography, particularly those involving the spread of diseases (epidemiology), the practice of commerce and military planning (logistics), and the development of efficient spatial networks. Geostatistical algorithms are incorporated in many places, including geographic information systems (GIS) and the R statistical environment.

Linear Trend Estimation is a statistical technique to aid interpretation of data. When a series of measurements of a process are treated as a time series, trend estimation can be used to make and justify statements about tendencies in the data, by relating the measurements to the times at which they occurred. This model can then be used to describe the behaviour of the observed data.

Google Trends - Google Hot Trends Visualize

Patterns - Mind Maps - Comparisons

Correlation and Dependence is any statistical relationship, whether causal or not, between two random variables or two sets of data. Correlation is any of a broad class of statistical relationships involving dependence, though in common usage it most often refers to the extent to which two variables have a linear relationship with each other. Familiar examples of dependent phenomena include the correlation between the physical statures of parents and their offspring, and the correlation between the demand for a product and its price.

Predicate in logic is any statistical relationship, whether causal or not, between two random variables or two sets of data. Correlation is any of a broad class of statistical relationships involving dependence, though in common usage it most often refers to the extent to which two variables have a linear relationship with each other. Familiar examples of dependent phenomena include the correlation between the physical statures of parents and their offspring, and the correlation between the demand for a product and its price.

Extrapolation is the process of estimating, beyond the original observation range, the value of a variable on the basis of its relationship with another variable. It is similar to interpolation, which produces estimates between known observations, but extrapolation is subject to greater uncertainty and a higher risk of producing meaningless results. Linear.

Linear Equation is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable (however, different Variables may occur in different terms). A simple example of a linear equation with only one variable, x, may be written in the form: ax + b = 0, where a and b are constants and a ≠ 0. The constants may be numbers, parameters, or even non-linear functions of parameters, and the distinction between variables and parameters may depend on the problem (for an example, see linear regression).

Uniform Distribution is a symmetric probability distribution whereby a finite number of values are equally likely to be observed; every one of n values has equal probability 1/n. Another way of saying "discrete uniform distribution" would be "a known, finite number of outcomes equally likely to happen".

Sensitivity and Specificity are statistical measures of the performance of a binary classification test, also known in statistics as classification function.

Effect Size (number needed to treat)

Second-Order Logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory.

Procedural Generation is a method of creating data algorithmically as opposed to manually. In computer graphics, it is also called random generation and is commonly used to create textures and 3D models. In video games it is used to automatically create large amounts of content in a game. Advantages of procedural generation include smaller file sizes, larger amounts of content, and randomness for less predictable gameplay.

Mode in statistics is the value that appears most often in a set data. The mode of a discrete probability distribution is the value x at which its probability mass function takes its maximum value. In other words, it is the value that is most likely to be sampled. The mode of a continuous probability distribution is the value x at which its probability density function has its maximum value, so the mode is at the peak.

Arthur Benjamin: Teach Statistics before Calculus (video)

Analytics is the discovery, interpretation, and communication of meaningful patterns in data.

Fads and Trends is any form of collective behavior that develops within a culture, a generation or social group and which impulse is followed enthusiastically by a group of people for a finite period of time.

Formulating - Validity

Peter Donnelly: How Stats Fool Juries (video)

Actuarial Science is the discipline that applies mathematical and statistical methods to assess risk in insurance, finance and other industries and professions.

Statistical Survey - Scenarios

Mediocrity Principle is the philosophical notion that "if an item is drawn at random from one of several sets or categories, it's likelier to come from the most numerous category than from any one of the less numerous categories".

Correspondence Mathematics is a term with several related but distinct meanings. Ratings.

Statistical Syllogism is a non-deductive syllogism. It argues, using inductive reasoning, from a generalization true for the most part to a particular case.

Statistical Power of a binary hypothesis test is the probability that the test correctly rejects the null hypothesis (H0) when the alternative hypothesis (H1) is true. It can be equivalently thought of as the probability of accepting the alternative hypothesis (H1) when it is true—that is, the ability of a test to detect an effect, if the effect actually exists.

Information Sources

Stats is the collection and interpretation of quantitative data and the use of probability theory to estimate parameters.

Empirical Statistical Laws or a law of statistics represents a type of behavior that has been found across a number of datasets and, indeed, across a range of types of data sets. Many of these observances have been formulated and proved as statistical or probabilistic theorems and the term "law" has been carried over to these theorems. There are other statistical and probabilistic theorems that also have "law" as a part of their names that have not obviously derived from empirical observations. However, both types of "law" may be considered instances of a scientific law in the field of statistics. What distinguishes an empirical statistical law from a formal statistical theorem is the way these patterns simply appear in natural distributions, without a prior theoretical reasoning about the data.

Probabilities is a measure of how likely it is that some event will occur; a number expressing the ratio of favorable cases to the whole number of cases possible. The quality of being probable; a probable event or the most probable event. Errors.

Probability is the measure of the likelihood that an event will occur. A number expressing the ratio of favorable cases to the whole number of cases possible. Probable is something having a high chance to be true or real.

Probabilistic is based on a theory of probability and subject to or involving chance variation.

Potential - Cherry Picking Data - Risk - Deterministic - Randomness

Possibilities is the capability of existing or happening or being true. A possible alternative.

Probabilistic Logic is to combine the capacity of probability theory to handle uncertainty with the capacity of deductive logic to exploit structure of formal argument. The result is a richer and more expressive formalism with a broad range of possible application areas. Probabilistic logics attempt to find a natural extension of traditional logic truth tables: the results they define are derived through probabilistic expressions instead. A difficulty with probabilistic logics is that they tend to multiply the computational complexities of their probabilistic and logical components. Other difficulties include the possibility of counter-intuitive results, such as those of Dempster-Shafer theory in evidence-based subjective logic. The need to deal with a broad variety of contexts and issues has led to many different proposals.

Odds is calculating the likelihood that the event will happen or not happen, using a numerical expression usually expressed as a pair of numbers. A prediction. A guess.

Chance is a possibility due to a favorable combination of circumstances. An unknown and unpredictable phenomenon that causes an event to result one way rather than another.

Relative - Hypothesis - Ratio - Intuition.

May expresses a possibility that something might happen.

Guess is a message expressing an opinion based on incomplete evidence. An estimate based on little or no information.

Educated Guess is a guess based on knowledge, reasoning and experience and factors that you take into account which might affect the outcome. A well-informed guess or estimate based on experience or theoretical knowledge.

BEST Guess Who Strategy- 96% WIN record using MATH (youtube)

Opportunity is a possibility due to a favorable combination of circumstances. A chance or the possibility of future success. A measure of how likely it is that some event will occur or a number expressing the ratio of favorable cases to the whole number of cases possible.

Scientists discover what happens in our brains when we make educated guesses. Researchers have identified how cells in our brains work together to join up memories of separate experiences, allowing us to make educated guesses in everyday life. By studying both human and mouse brain activity, they report that this process happens in a region of the brain called the hippocampus.

Uncertainty is something dependent on chance. To be in doubt or being unsure of something. Lacking confidence.

Uncertainty refers to epistemic situations involving imperfect or unknown information. It applies to predictions of future events, to physical measurements that are already made, or to the unknown. Uncertainty arises in partially observable and/or stochastic environments, as well as due to ignorance, indolence, or both. It arises in any number of fields, including insurance, philosophy, physics, statistics, economics, finance, psychology, sociology, engineering, metrology, meteorology, ecology and information science.

Certain is something definitely known and destined to happen or inevitable to happen. Feeling no doubt or uncertainty and being confident and assured. To established something beyond doubt or question. Something definite but not specified or identified or established irrevocably. Reliable in operation or effect. Exercising or taking care great enough to bring assurance.

"Once you lower your expectations, the sky's the limit."

Given is when you believe something as being true or when you believe something to be a sure thing that you don't need to prove.

Estimate is an approximate calculation of quantity or degree. Judge tentatively or form an estimate of quantities or time.

Estimation is the process of finding an approximation, a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. Calculate.

Estimation Statistics is a data analysis framework that uses a combination of effect sizes, confidence intervals, precision planning and meta-analysis to plan experiments, analyze data and interpret results.

Reliability in statistics is the consistency that produces similar results under consistent conditions.

Value Bet is a bet where the probability of a given outcome is greater than the bookmakers odds reflect. Simply put, when value betting you will be placing bets that have a larger chance of winning than implied by the bookmakers odds. This means you will have an edge over the bookmaker in the long run.

Approximation is anything that is similar but not exactly equal to something else. Extrapolation.

Approximation Error in when some data is the discrepancy between an exact value and some approximation to it. An approximation error can occur because the measurement of the data is not precise due to the instruments. (e.g., the accurate reading of a piece of paper is 4.5 cm but since the ruler does not use decimals, you round it to 5 cm.) or approximations are used instead of the real data (e.g., 3.14 instead of π).

Order of Approximation refers to formal or informal terms for how precise an approximation is, and to indicate progressively more refined approximations: in increasing order of precision, a zeroth-order approximation, a first-order approximation, a second-order approximation, and so forth. Informally, it is simply the level of precision used to represent quantities which are not perfectly known.

Approximate Number System an adult could distinguish 100 items versus 115 items without counting.

Inconceivable is something unlikely to happen or is unimaginable, but not totally impossible.

Conceivable is something capable of being imagined. Conceive is to have an idea.

Impossible is something not capable of happening or occurring or being accomplished or dealt with. Something that cannot be done or totally unlikely.

Limit is the greatest possible degree of something. The greatest amount of something that is possible or allowed. A final or latest limiting point. As far as something can go. Restrict in quantity or scope. The boundary of a specific area. The mathematical value toward which a function goes as the independent variable approaches infinity.

Fold Change is a measure describing how much a quantity changes between an original and a subsequent measurement. It is defined as the ratio between the two quantities; for quantities A and B, then the fold change of B with respect to A is B/A. Fold change is often used when analyzing multiple measurements of a biological system taken at different times as the change described by the ratio between the time points is easier to interpret than the difference.

Variables is an alphabetic character representing a number, called the value of the variable, which is either arbitrary or not fully specified or unknown. Iterations.

Probability Distribution is a mathematical description of a random phenomenon in terms of the probabilities of events.

Propensity Probability is the tendency of a given type of physical situation to yield an outcome of a certain kind, or to yield a long run relative frequency of such an outcome. (physical propensity, disposition or to behave in a certain way).

Probability Density Function is a function, whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample.

Bean Machine or Galton Board is a device that demonstrates the central limit theorem in particular that with sufficient sample size the binomial distribution approximates a normal distribution. Among its applications, it afforded insight into regression to the mean or "regression to mediocrity".

Independence probability theory is when the occurrence of one does not affect the probability of occurrence of the other (equivalently, does not affect the odds). Similarly, two random variables are independent if the realization of one does not affect the probability distribution of the other. Probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment.

Bayesian Probability represents a level of certainty relating to a potential outcome or idea. This is in contrast to a frequentist probability that represents the frequency with which a particular outcome will occur over any number of trials. An event with Bayesian probability of .6 (or 60%) should be interpreted as stating "With confidence 60%, this event contains the true outcome", whereas a frequentist interpretation would view it as stating "Over 100 trials, we should observe event X approximately 60 times." The difference is more apparent when discussing ideas. A frequentist will not assign probability to an idea; either it is true or false and it cannot be true 6 times out of 10.

Bayesian is relating to statistical methods based on Bayes' theorem.

Bayes' Theorem describes the probability of an event, based on prior knowledge of conditions that might be related to the event. Decisions.

Bayesian Inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available.

Statistical Inference is the process of deducing properties of an underlying probability distribution by analysis of data. Inferential statistical analysis infers properties about a population: this includes testing hypotheses and deriving estimates. The population is assumed to be larger than the observed data set; in other words, the observed data is assumed to be sampled from a larger population. Inferential statistics can be contrasted with descriptive statistics. Descriptive statistics is solely concerned with properties of the observed data, and does not assume that the data came from a larger population.

If Function Algorithms - Ratings - Truth

Bellman Equation is a necessary condition for optimality associated with the mathematical optimization method known as dynamic programming. It writes the value of a decision problem at a certain point in time in terms of the payoff from some initial choices and the value of the remaining decision problem that results from those initial choices. This breaks a dynamic optimization problem into simpler subproblems, as Bellman's "Principle of Optimality" prescribes.

Average is around the middle of a scale. The sum of a list of numbers divided by the number of numbers in the list. In mathematics and statistics, this would be called the arithmetic mean. In statistics, mean, median, and mode are all known as measures of central tendency. Balanced - Moderation.

Mean is the sum of a collection of numbers divided by the number of numbers in the collection. The collection is often a set of results of an experiment, or a set of results from a survey.

Regression toward the Mean is the phenomenon that arises if a random variable is extreme on its first measurement but closer to the mean or average on its second measurement and if it is extreme on its second measurement but closer to the average on its first. To avoid making incorrect inferences, regression toward the mean must be considered when designing scientific experiments and interpreting data. Historically, what is now called regression toward the mean has also been called reversion to the mean and reversion to mediocrity. Cherry Picking Data.

Middle is an area that is approximately central within some larger region. Being neither at the beginning nor at the end in a series. Equally distant from the extremes. Time between the beginning and the end of a temporal period.

Percentage is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%", or the abbreviations "pct.", "pct"; sometimes the abbreviation "pc" is also used. A percentage is a dimensionless number (pure number). (16% of 25 = 4 or 25% of 16 = 4. Lets say that lunch was $45.00, tip of 20%, multiply 45x2=90, the tip is 9 dollars.

Lunch was $45.00, tip of 15%, 45x15=675, the tip is 6.75 dollars).

Algebra - Multiplication - Fractions

Ratio is a relationship between two numbers indicating how many times the first number contains the second. The relative magnitudes of two quantities usually expressed as a quotient or the ratio of two quantities to be divided. For example, if a bowl of fruit contains eight oranges and six lemons, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ratio 4:3). Similarly, the ratio of lemons to oranges is 6:8 (or 3:4) and the ratio of oranges to the total amount of fruit is 8:14 (or 4:7). The numbers in a ratio may be quantities of any kind, such as counts of persons or objects, or such as measurements of lengths, weights, time, etc. Scale - Circles - Symmetry.

Quotient is the ratio of two quantities to be divided. The quantity produced by the division of two numbers. IQ.

Aspect Ratio of an image describes the proportional relationship between its width and its height. It is commonly expressed as two numbers separated by a colon, as in 16:9. For an x:y aspect ratio, no matter how big or small the image is, if the width is divided into x units of equal length and the height is measured using this same length unit, the height will be measured to be y units.

Risk Ratio is the ratio of the probability of an outcome in an exposed group to the probability of an outcome in an unexposed group. Risk.

Parameter is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when identifying the system, or when evaluating its performance, status, condition, etc. Parameter has more specific meanings within various disciplines, including mathematics, computing and computer programming, engineering, statistics, logic and linguistics. Within and across these fields, careful distinction must be maintained of the different usages of the term parameter and of other terms often associated with it, such as argument, property, axiom, variable, function, attribute, etc.

Luck - Comparisons

Parallel Individuation System is a non-symbolic cognitive system that supports the representation of numerical values from zero to three (in infants) or four (in adults and non-human animals). It is one of the two cognitive systems responsible for the

representation of number, the other one being the approximate number system. Unlike the approximate number system, which is not precise and provides only an estimation of the number, the parallel individuation system is an exact system and encodes the exact numerical identity of the individual items. The parallel individuation system has been attested in human adults, non-human animals, such as fish and human infants, although performance of infants is dependent on their age and task

Intraparietal Sulcus is processing symbolic numerical information, visuospatial working memory and interpreting the intent of others.

Operationally Impossible is considered to be 1 in 10 to the 70th Power

Power of 10 is any of the integer powers of the number ten; in other words, ten multiplied by itself a certain number of times (when the power is a positive integer). By definition, the number one is a power (the zeroth power) of ten. Size

Margin of Error is a statistic expressing the amount of random sampling error in a survey's results. It asserts a likelihood (not a certainty) that the result from a sample is close to the number one would get if the whole population had been queried. The likelihood of a result being "within the margin of error" is itself a probability, commonly 95%, though other values are sometimes used. The larger the margin of error, the less confidence one should have that the poll's reported results are close to the true figures; that is, the figures for the whole population. Margin of error applies whenever a population is incompletely sampled.

Accuracy and Precision. Precision is a description of random errors, a measure of statistical variability. Accuracy has two definitions: More commonly, it is a description of systematic errors, a measure of statistical bias; as these cause a difference between a result and a "true" value, ISO calls this trueness. Alternatively, ISO defines accuracy as describing a combination of both types of observational error above (random and systematic), so high accuracy requires both high precision and high trueness. In simplest terms, given a set of data points from a series of measurements, the set can be said to be precise if the values are close to the average value of the quantity being measured, while the set can be said to be accurate if the values are close to the true value of the quantity being measured. The two concepts are independent of each other, so a particular set of data can be said to be either accurate, or precise, or both, or neither.

Precision in statistics is when the precision is the reciprocal of the variance, and the precision matrix, also known as concentration matrix, is the matrix inverse of the covariance matrix. Some particular statistical models define the term precision differently.

Markov Chain is a process satisfies the Markov property if one can make predictions for the future of the process based solely on its present state just as well as one could knowing the process's full history. i.e., conditional on the present state of the system, its future and past are independent. Markov Property refers to the memoryless property of a stochastic process.

Memorylessness is a property of certain probability distributions: the exponential distributions of non-negative real numbers and the geometric distributions of non-negative integers.

Stochastic Process is a probability model used to describe phenomena that evolve over time or space. More specifically, in probability theory, a stochastic process is a time sequence representing the evolution of some system represented by a variable whose change is subject to a random variation.

Time Management - Virtual Reality

Relative Change and Difference are used to compare two quantities while taking into account the "sizes" of the things being compared. The comparison is expressed as a ratio and is a unitless number. By multiplying these ratios by 100 they can be expressed as percentages so the terms percentage change, percent(age) difference, or relative percentage difference are also commonly used. The distinction between "change" and "difference" depends on whether or not one of the quantities being compared is considered a standard or reference or starting value. When this occurs, the term relative change (with respect to the reference value) is used and otherwise the term relative difference is preferred. Relative difference is often used as a quantitative indicator of quality assurance and quality control for repeated measurements where the outcomes are expected to be the same. A special case of percent change (relative change expressed as a percentage) called percent error occurs in measuring situations where the reference value is the accepted or actual value (perhaps theoretically determined) and the value being compared to it is experimentally determined (by measurement).

Observation Errors - Error's

Piwik Analytics Software

Web Analytics Software List (wiki)

Saplumira

Research (science) - Mind Maps

Analytics is the discovery, interpretation, and communication of meaningful patterns in data. Especially valuable in areas rich with recorded information, analytics relies on the simultaneous application of statistics, computer programming and operations research to quantify performance. Google Analytics - Watson Analytics.

Learning Not Just on Paper. If you are teaching math then you should use real life examples that relate and are relative to the students immediate needs. You should also use calculations that students will need to preform in order to solve a problem that they will most likely face in the immediate future or far future. The main reason why you use real life situations or scenarios when learning math is the associations. When you associate knowledge with other knowledge that is used often, you remember it more often, so the knowledge stays with you longer. That is why you can easily remember things when you associate them with other things, which is one of the key techniques in having a good memory. When you have nothing to associate something with, you forget it, to a point where you will not even remember why you even learned this knowledge in the first place. This is what education is today, fragmented and incoherent. Kids have to learn how to use math in their everyday life, if they don't, they will eventually forget it and never use it effectively or efficiently.

Example Choice is when students see a connection between physics and the real world. They learn easier because the subject is more interesting and more relevant to their daily life. Children Learning by Example.

Analogy - Precedent - Proof of Concept - Outdoors - Experience Learning - Read to Learn

Example is a representative form or pattern. Something to be imitated. A task performed or problem solved in order to develop skill or understanding. Clarify something by showing something similar or equivalent.

Exemplary is something serving as a desirable model and representing the best of its kind. Exemplary also means punishment serving as a warning or deterrent.

Instance is to clarify something by giving an example of it. An item of information that is typical of a class or group. An occurrence of something.

Demonstration Teaching involves showing by reason or proof, explaining or making clear by use of examples or experiments. Making something less vague and abstract.

Demonstration is a show or display and the act of presenting something to sight or view. Proof by a process of argument or a series of proposition proving an asserted conclusion. Repeatable (reproducibility).

Ostensive Definition conveys the meaning of a term by pointing out examples. This type of definition is often used where the term is difficult to define verbally, either because the words will not be understood (as with children and new speakers of a language) or because of the nature of the term (such as colors or sensations). It is usually accompanied with a gesture pointing to the object serving as an example, and for this reason is also often referred to as "definition by pointing".

Action Learning is an approach to solving real problems that involves taking action and reflecting upon the results, which helps improve the problem-solving process, as well as the solutions developed by the team. The action learning process includes: a real problem that is important, critical, and usually complex, a diverse problem-solving team or "set", a process that promotes curiosity, inquiry, and reflection, a requirement that talk be converted into action and, ultimately, a solution, and a commitment to learning.

Learning by Doing is when productivity is achieved through practice, self-perfection and minor innovations. An example is a factory that increases output by learning how to use equipment better without adding workers or investing significant amounts of capital. Learning refers to understanding through thinking ahead and solving backward, one of the main problem solving strategies. PDF - Math for America Classroom Lessons.

Authentic Learning is an instructional approach that allows students to explore, discuss, and meaningfully construct concepts and relationships in contexts that involve real-world problems and projects that are relevant to the learner. It refers to a wide variety of educational and instructional techniques focused on connecting what students are taught in school to real-world issues, problems, and applications. The basic idea is that students are more likely to be interested in what they are learning, more motivated to learn new concepts and skills, and better prepared to succeed in college, careers, and adulthood if what they are learning mirrors real-life contexts, equips them with practical and useful skills, and addresses topics that are relevant and applicable to their lives outside of school. DIY Learning.

Public Sphere Pedagogy represents an approach to educational engagement that connects classroom activities with real world civic engagement. The focus of PSP programs is to connect class assignments, content, and readings with contemporary public issues. Students are then asked to participate with members of the community in various forms of public sphere discourse and democratic participation such as town hall meetings and public debate events. Through these events, students are challenged to practice civic engagement and civil discourse.

Content-Based Instruction encourages learners to learn a language by using it as a real means of communication from the very first day in class. The idea is to make students become independent learners so they can continue the learning process even outside the class. Historically, the word content has changed its meaning in second language teaching. Content used to refer to the methods of grammar-translation, audio-lingual methodology, and vocabulary or sound patterns in dialog form. Recently, content is interpreted as the use of subject matter as a vehicle for second or foreign language teaching/learning and linguistic immersion. CBI is designed to provide second-language learners instruction in content and language, hence it is also called content-based language teaching or CBLT.

Knowing how to count doesn't matter if you don't count the things that matter. Count the things that Matter. Real life preparation has to be the goal and the purpose in all educational courses.

Don't force students to figure something out if they can't use that knowledge in real life. Because they will just forget it, which is why school testing is a failure and a disservice. It's been well documented that students forget almost everything they saw on a test, so what's the point? If you want to use a math formula or use a problem solving technique using numbers, then give a clear example of how those numbers can be used to symbolize real things in their life, things that they should know because they are part of a valuable skill set. Relate - Learning Methods.

If someone is going to show you how to use a hammer, then they should also show you how to build a house using a hammer. Learning how to use a hammer is not interesting or fun, but when you learn what a hammer can do, then it becomes an incredible tool. Like math, learning how to do math is boring, but once you learn what math can do, then you can use math to build your own house, or maybe even use math to build a spacecraft and fly to the moon. When learning does not have that bigger goal in mind, then learning becomes pointless and boring, and then people don't learn enough or keep progressing.

Where ever students are, use that students needs in the present moment as a teaching format. What ever a person is struggling with, use that particular struggle to teach them how to over come their struggle using reading, writing, math, science, biology, or any other useful subject or skill. This way you increase their understanding of important subjects and also help solve their problems that they are experiencing now, or may experience in the future. Help them with life, and help prepare them for the future. As you are walking towards a goal, teach them along the way, and most important, show them the power of learning, and make every student understand that they need to be able to learn on their own, because that is the most important skill that they will ever have in life. And if they never learn to learn, or never learn how important it is to be able to learn on their own, then they will struggle with life, and they will most likely never acquire true success or true happiness.

A lesson should have a beginning, a middle and an end. It should explain the procedure used, if one was used. It should explain why particular problem solving skills where used? It should explain the things to be aware of and why? It should explain the learning path that was chosen and that it was not a blind mindless reaction. As history has taught us, just because something was done in a particular way for a long period of time, it does not mean that it can't be improved.

Video Samples: This video is one example, but it needs to be even more reality based. Math Shorts Episode 15 - Applying the Pythagorean Theorem (youtube).

Real World Math Examples This video did not go far enough to teach all the variables. And you could have showed more examples of how to estimate the altitude, like holding the drone over a yard stick, if the drone can see the entire yardstick at 2 feet off the ground, then you could estimate the altitude needed in order to see 100 yards if the drone was in the middle straight up from the 50 yard line. In the video they said the altitude needed was 89.7 feet to have a full view of a 100 yard long area. So the lens of the camera definitely influences field of view like with a wide angle camera lens. It would been even more accurate if you added an Ariel photographers expertise to explain important factors of Ariel photography, and also teach safe Drone Operation.

Education improves decision-making ability and economic rationality, study finds. Using a randomized controlled trial of education support and laboratory experiments that mimic real-life examples, we established causal evidence that an education intervention increases not only educational outcomes but also economic rationality in terms of measuring how consistently people make decisions to seek their economic goals.

Knowing the math behind a problem, or knowing the math behind a solution or goal, helps to clarify its true significance and also helps explain what decisions and choices are available. This is when math reveals its true power. But even knowing that there’s a mathematical equation in almost everything in our lives, math does not explain everything. Especially when knowing that some people can’t do the math, or worse, some people leave out very important factors, that when calculated, clearly paints a different picture to what the real facts are. Math is not the only factor when solving a problem, or the only factor that clarifies true meaning. There are also other factors that could help solve a problem, or reach an understanding.

Some people can understand math a lot sooner than other people can. Some people can understand math at the age of 12 and some people at the age of 20. The only difference is the options that a person will have at that particular time in their life. Once you reach a certain level of knowledge, you have more possibilities to choose from and more options concerning a particular educational direction, like being a doctor, a lawyer, a farmer, a leader, or a representative of the people. The 20 year old will still have the same potential, but only at a later time, but only if they keep learning.

Is what you're doing making a difference? Are you fully conscious of all the causes and effects that you have on the world? Are you aware of the damage that you are afflicting on yourself or on anyone else? Do you know what choices you have? Do you have enough math knowledge in order to correctly calculate your causes and effects? Can you translate these numbers into a language that even a laymen could understand? In order to fully understand yourself and the world around you, you need Knowledge, information and the tools that help explain it. If things need to be calculated, then you must calculate them. Math is a universal language. Math explains why some words are undeniably true. A truth that can be proven and witnessed. If you can confirm something to be true, and it has relevancy, then it is most likely very important. And ignoring this importance is dangerous, the consequences can be catastrophic. Math is a good guide that you can trust and a really good friend that you can count on, literally. And this is fully knowing that even though some things can be counted does not necessary mean that they actually count. In other words, you have to know how to count if you want to count the things that count. The importance of math is constantly revealing itself. In order to educate people about this importance you must show people real life examples of how powerful math knowledge truly is. Teaching math, or learning math, is one thing, knowing how to use math correctly and effectively in real life situations is another. That has to be the ultimate goal of math, otherwise you are just wasting time, people and resources. People need to stop cherry picking data and stop pretending to know things. Stop counting the money and start counting the things that happened because of the money. Calculate the value of your actions and calculate the full cost of your actions. Stop counting the things that don't matter, and stop being vague about the things you think matter.

"If you don't count the things that matter, then knowing how to count won't matter." This goes for reading too.

It counts to count. Count is to determine the number or the amount of something. But Count also means something that has truth, or validity or Value. Like providing a service that counts, or doing something important that counts as a benefit to you and for others. Loss Aversion.

We need to learn how to count more accurately. Numbers should have specific values assigned to them, so that they are not just numbers, they are detailed records of a transaction of what was taken from the Earth and what we gave back to earth in return in order to sustain life. We need to calculate all the factors that are needed for life on earth. A side by side comparison, the pros and cons, the pluses and minuses, the choices, and so on. Knowing the difference between Value and Cost and Hidden Costs. Productivity is measured by work rate, output and yield, and also how much resources were used, what pollution it caused, the effects of that pollution, how much the pollution cost peoples health and the environment. And if the time, people, resources and energy could have been better used more effectively and efficiently that would have been more productive. The truth is in the numbers.

Capstone Project # 1 (problem solving)

How much does food really cost? (time, people, resources, environmental impacts, options, solutions, Et cetera...).

How much does clean water really cost? (time, people, resources, environmental impacts, options, solutions, Et cetera..).

Why does tap water cost 10,000 times less than bottle water?

How much does good health really cost? (time, people, resources, environmental impacts, options, solutions, Et cetera).

How much do clothes really cost? (time, people, resources, environmental impacts, options, solutions, Et cetera).

How much does a home really cost? (time, people, resources, environmental impacts, options, solutions, Et cetera).

How much does energy really cost? (time, people, resources, environmental impacts, options, solutions, Et cetera).

How much does a particular cell phone really cost? (time, people, resources, environmental impacts, options, solutions).

How much does a particular computer really cost? (time, people, resources, environmental impacts, options, solutions).

How much does a good education really cost? (time, people, resources, environmental impacts, options, solutions).

How much does ignorance cost? (time, people, resources, environmental impacts, options, solutions) - Greendex - Ratings.

How much would one of these things cost you if you didn't have it? (lost time, poor health, impacts, Et cetera...).

Sustainable Calculator

Mathematical Optimization is the selection of a best element (with regard to some criterion) from some set of available alternatives.

Mathematical Proof demonstrates that a statement is always true (listing possible cases and showing that it holds in each).

Axiom well-established, that it is accepted without controversy or question. Valid.

"You can’t manage what you don’t measure accurately"

Optimum is the most favorable conditions or greatest degree or amount possible under given circumstances.

Let students see this information and let them challenge these calculations so they can confirm this knowledge for themselves, and also be able to repeat these processes on other subjects of great importance and on other problems that need to be solved.

Measuring Value -

Critical Thinking and Technology - Cause and Effect - Problem Solving

Investigative Dashboard - Alaveteli

Hidden Costs (youtube) - Opportunity Cost

Zipf's Law is an empirical law formulated using mathematical statistics that refers to the fact that many types of data studied in the physical and social sciences can be approximated with a Zipfian distribution, one of a family of related discrete power law probability distributions. Management Tools.

Note:

Of course anyone can do the above capstone project. I'm sure someday there will soon be an App for this, of course!

Food for Thought App tallies the nutritional data and carbon footprint associated with each food item and with the overall meal, such as the amount of calories in a salad and the amount of water that would be used in growing the lettuce. Apps.

Pattie Maes demos the Sixth Sense (youtube)

Rating System - Responsibly Produced Rating

"Criminals know how to use a calculator better than the general public does. That's one of the reasons why education fails to prepare students effectively. If you educate students to be smarter than criminals, then you will have no more criminals."

I wouldn't say that "The unexamined life is not worth living." I would say that "An examined life definitely makes life worth living."

"If you teach students how math is used in the real world, and how math has many benefits, when they graduate, they will know what math is used for, and they also know what it's not used for."

When you're learning math, everyone starts out not understanding math. But with time and practice, you will eventually understand math. If it takes you longer to learn than other people, that's ok, because you will eventually learn math. And you will see the benefits that come from math. But you need to see how math is used in your every day life. So as you're learning math, you are also learning about the world, and learning about yourself. If you can't connect the world with math, then math will seem unimportant to you, so the motivation to learn math will be very low. If you're learning to count, then count the things that matter to you. Then you will eventually see the potential of using math. The most important factor is what the numbers represent. If the numbers represent something arbitrary, then they lose their meaning and their effectiveness.

Teleology - Cause and Effect - Structure

Stop Teaching Calculating, Start Learning Maths! - Conrad Wolfram (youtube)

Factor is anything that contributes causally to a result. Consider as relevant when making a decision. An abstract part of something. Any of the numbers (or symbols) that form a product when multiplied together. Odds.

Count is to show consideration for; take into account. Allow or plan for a certain possibility; concede the truth or validity of something. Have a certain value or carry a certain weight. Determine the number or amount of. Include as if by counting. Have faith or confidence in.

Consideration is the process of giving careful thought to something. Information that should be kept in mind when making a decision. Kind and considerate regard for others. A considerate and thoughtful act.

Instruction is a message describing how something is to be done.

Mathematical Statement

Statement is a message that is stated or declared; a communication (oral or written) setting forth particulars or facts etc. A fact or assertion offered as evidence that something is true.

Calculations or Computations is problem solving that involves numbers or quantities. Planning something carefully and intentionally. The procedure of calculating; determining something by mathematical or logical methods. Time Management.

Procedure is a particular course of action intended to achieve a result. A process or series of acts especially of a practical or mechanical nature involved in a particular form of work. A set sequence of steps, part of larger computer program. Procedure (science).

Process is to perform mathematical and logical operations on data according to programmed instructions in order to obtain the required information. A particular course of action intended to achieve a result. Shape, form, or improve a material. Subject to a process or treatment, with the aim of readying for some purpose, improving, or remedying a condition. Process (science).

Operations is a process or series of acts, especially of a practical or mechanical nature involved in a particular form of work. Operation in psychology is the performance of some composite cognitive activity; an operation that affects mental contents. Operation in mathematics is calculation by mathematical methods. Operation is a calculation from zero or more input values (called operands) to an output value.

Function is a mathematical relation such that each element of a given set (the domain of the function) is associated with an element of another set (the range of the function). The actions and activities assigned to or required or expected of a person or group. A relation such that one thing is dependent on another. What something is used for. Perform as expected when applied. Function in mathematics.

Measure is the assignment of a number or values to a characteristic of an object or event, which can be compared with other objects or events. To determine the measurements of something or somebody, take measurements of. Express as a number or measure or quantity. Have certain dimensions. Evaluate or estimate the nature, quality, ability, extent, or significance of. Any maneuver made as part of progress toward a goal. How much there is or how many there are of something that you can quantify. The act or process of assigning numbers to phenomena according to a rule. A basis for comparison; a reference point against which other things can be evaluated. Measuring instrument having a sequence of marks at regular intervals; used as a reference in making measurements. A container of some standard capacity that is used to obtain fixed amounts of a substance - Psychological Measurement.

Measuring Instrument is a device for measuring a physical quantity. In the physical sciences, quality assurance, and engineering, measurement is the activity of obtaining and comparing physical quantities of real-world objects and events. Established standard objects and events are used as units, and the process of measurement gives a number relating the item under study and the referenced unit of measurement. Measuring instruments, and formal test methods which define the instrument's use, are the means by which these relations of numbers are obtained. All measuring instruments are subject to varying degrees of instrument error and measurement uncertainty. Measuring Tools.

System of Measurement is a collection of units of measurement and rules relating them to each other. Systems of measurement have historically been important, regulated and defined for the purposes of science and commerce. Systems of measurement in modern use include the metric system, the imperial system, and United States customary units, which uses the inch, foot, yard, and mile, which are the only four customary length measurements in everyday use. 94.7 % of the world uses the metric system. Most countries use the Metric System, which uses the measuring units such as meters and grams and adds prefixes like kilo, milli and centi to count orders of magnitude. Scientists often use meters for length, kilograms for mass, and seconds for time. Metric System by Country (image)

Metrology is the science of measurement. It establishes a common understanding of units, crucial in linking human activities.

Units of Measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same quantity. Any other value of that quantity can be expressed as a simple multiple of the unit of measurement. For example, a length is a physical quantity. The metre is a unit of length that represents a definite predetermined length. When we say 10 metres (or 10 m), we actually mean 10 times the definite predetermined length called "metre". Measurement is a process of determining how large or small a physical quantity is as compared to a basic reference quantity of the same kind. In physics and metrology, units are standards for measurement of physical quantities that need clear definitions to be useful. Reproducibility of experimental results is central to the scientific method. A standard system of units facilitates this. Scientific systems of units are a refinement of the concept of weights and measures historically developed for commercial purposes. Science, medicine, and engineering often use larger and smaller units of measurement than those used in everyday life. The judicious selection of the units of measurement can aid researchers in problem solving. In the social sciences, there are no standard units of measurement and the theory and practice of measurement is studied in psychometrics and the theory of conjoint measurement. History of Measurement (wiki).

Imperial Units are units of measurement of the British Imperial System, the traditional system of weights and measures used officially in Great Britain from 1824 until the adoption of the metric system beginning in 1965. (also called British Imperial System).

English Units are the units of measurement used in England up to 1826 (when they were replaced by Imperial units), which evolved as a combination of the Anglo-Saxon and Roman systems of units. Various standards have applied to English units at different times, in different places, and for different applications.

Weights and Measures Acts are acts of the British Parliament determining the regulation of weights and measures.

United States Customary Units are a system of measurements commonly used in the United States and U.S. territories since it was formalized in 1832. The imperial system was officially adopted in 1826, changing the definitions of some of its units. Subsequently, while many U.S. units are essentially similar to their imperial counterparts, there are significant differences between the systems.

Calibration - Standards - Statistics

CGS is a system of measurement based on centimeters and grams and seconds.

Level of Measurement is a classification that describes the nature of information within the numbers assigned to variables. Classification with four levels, or scales, of measurement: nominal, ordinal, interval, and ratio.

Isotropic is having a physical property which has the same value when measured in different directions. (of a property or phenomenon) not varying in magnitude according to the direction of measurement. Isotropy is uniformity in all orientations.

Ruler is an instrument used in geometry, technical drawing, printing, engineering and building to measure distances or to rule straight lines. The ruler is a straightedge which may also contain calibrated lines to measure distance.

Tools for Measuring (engineering)

Slide Rule is a mechanical analog computer. The slide rule is used primarily for multiplication and division, and also for functions such as exponents, roots, logarithms and trigonometry, but is not normally used for addition or subtraction. Though similar in name and appearance to a standard ruler, the slide rule is not ordinarily used for measuring length or drawing straight lines. How to Use a Slide Rule: Multiplication/Division, Squaring/Square Roots (youtube).

Analog Computer is a form of computer that uses the continuously changeable aspects of physical phenomena such as electrical, mechanical, or hydraulic quantities to model the problem being solved.

Logarithm is the inverse operation (a function that "reverses" another function) to exponentiation. That means the logarithm of a number is the exponent to which another fixed number, the base, must be raised to produce that number.

Pros and Cons - Side by Side Comparisons - Value

Our ability to measure is extremely powerful. Measuring gives us the ability to predict the future. So that means we can literally control our own destiny. We can even measure ourselves, to measure the measurer. Learn to measure, measure as much as you can, and measure the things that are the most important. If you can't measure something yourself, then find someone who can measure it for you. Measuring encompasses many different skills, but the skills to accurately decipher your measurements will always be the most important. Why, when, where, who, how, value, priority?"

Quantify is to express as a number or measure or quantity. Calculate.

Quantification in science is the act of counting and measuring that maps human sense observations and experiences into quantities. Quantification in this sense is fundamental to the scientific method.

Quantifier in linguistics is a type of determiner, such as all, some, many, few, a lot, and no, (but not numerals) that indicates quantity. Quantifier in logic is a construct that specifies the quantity of specimens in the domain of discourse that satisfy an open formula.

Quantities is how much there is or how many there are of something that you can quantify. The concept that something has a magnitude and can be represented in mathematical expressions by a constant or a variable.

Quantity is a property that can exist as a magnitude or multitude. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in terms of a unit of measurement. Quantity is among the basic classes of things along with quality, substance, change, and relation. Some quantities are such by their inner nature (as number), while others are functioning as states (properties, dimensions, attributes) of things such as heavy and light, long and short, broad and narrow, small and great, or much and little. A small quantity is sometimes referred to as a quantulum.

Physical Quantity is a physical property of a phenomenon, body, or substance, that can be quantified by measurement. A physical quantity can be expressed as the combination of a magnitude expressed by a number – usually a real number – and a unit: n u where n is the magnitude and u is the unit.

Volume is the amount of 3-dimensional space occupied by an object. The property of something that is great in magnitude. Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains. Volume is often quantified numerically using the SI derived unit, the cubic metre. The volume of a container is generally understood to be the capacity of the container, i. e. the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces. Sound Volume.

Capacity is the capability to perform or produce. The maximum production possible. The power to learn or retain knowledge; in law, the ability to understand the facts and significance of your behavior. Capacity (wiki) - Limits (engineering).

Load is a quantity that can be processed or transported at one time. The power output of a generator or power plant.

Structural Load are forces, deformations, or accelerations applied to a structure or its components. Loads cause stresses, deformations, and displacements in structures. Assessment of their effects is carried out by the methods of structural analysis. Excess load or overloading may cause structural failure, and hence such possibility should be either considered in the design or strictly controlled. Mechanical structures, such as aircraft, satellites, rockets, space stations, ships, and submarines, have their own particular structural loads and actions. Engineers often evaluate structural loads based upon published regulations, contracts, or specifications. Accepted technical standards are used for acceptance testing and inspection.

Mass is the property of a body that causes it to have weight in a gravitational field. A body of matter without definite shape. The property of something that is great in magnitude. Mass (matter).

Weight is the vertical force exerted by a mass as a result of gravity. An artifact that is heavy. The relative importance granted to something. A system of units used to express the weight of something. (statistics) a coefficient assigned to elements of a frequency distribution in order to represent their relative importance. Weight of an object is usually taken to be the force on the object due to gravity. A unit used to measure weight.

Weigh is to determine the weight of something. Show consideration for something and take into account. Have weight; have import, carry weight.

Heavy is of comparatively great physical weight or density.

Density is the spatial property of being crowded together or the amount per unit size.

Dimension is one of three cartesian coordinates that determine a position in space. A construct whereby objects or individuals can be distinguished. Dimension in physics is the physical units of a quantity, expressed in terms of fundamental quantities like time, mass and length. Dimension of a mathematical space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces

Shapes - Geometry - Dimensions in Space

Cartesian Coordinates is one of the coordinates in a system of coordinates that locates a point on a plane or in space by its distance from two lines or three planes respectively; the two lines or the intersections of the three planes are the coordinate axes. Cartesian Coordinate System.

Coordinate is a number that identifies a position relative to an axis, which is a straight line through a body or figure that satisfies certain conditions.

Size is the physical magnitude of something or how big or large it is. Size is the magnitude or dimensions of a thing, or how big something is. Size can be measured as length, width, height, diameter, perimeter, area, volume, or mass.

Sizes (nano) - Atoms - Universe - Spatial Intelligence - Shoe Size

Height is the vertical dimension of extension; distance from the base of something to the top. Height is the measure of vertical distance, either how "tall" something is, or how "high up" it is. Human Body Height.

Geometry (shapes)

Length is the linear extent in space from one end to the other; the longest dimension of something that is fixed in place. Size of the gap between two places. Continuance in time. Length is the most extended dimension of an object, any quantity with dimension distance. Orders of Magnitude (length) (wiki).

Distance is the property created by the space between two objects or points. A remote point in time. The interval between two times. Distance is a numerical description of how far apart objects are.

Furlong is a measure of distance in imperial units and U.S. customary units equal to one eighth of a mile, equivalent to 660 feet, 220 yards, 40 rods, or 10 chains. Using the international definition of the inch as exactly 25.4 millimeters, one furlong is 201.168 meters. Furlong was the distance a team of oxen could plough without resting. This was standardized to be exactly 40 rods or 10 chains.

Acre equals 1⁄640 (0.0015625) square mile, 4,840 square yards, 43,560 square feet. Traditionally defined as the area of one chain by one furlong (66 by 660 feet), which is exactly equal to 10 square chains, 1⁄640 of a square mile, or 43,560 square feet, and approximately 4,047 m2, or about 40% of a hectare. The international symbol of the acre is ac. The most common use of the acre is to measure tracts of land. The acre, based upon the International yard and pound agreement of 1959, is defined as exactly 4,046.8564224 square meters.

Duration is the period of time during which something continues. Duration is the amount of elapsed time between two events.

Frequency (HZ) - Action Physics

Interval is a definite length of time marked off by two instants. A set containing all points (or all real numbers) between two given endpoints. The distance between things. Interval in mathematics is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set.

Planning - Predictions - Patterns

Cycle is an interval during which a recurring sequence of events occurs. A periodically repeated sequence of events. A single complete execution of a periodically repeated phenomenon. Cause to go through a recurring sequence.

Seasons (earth) - Life-Cycle Assessment (development)

Sequence is a serial arrangement in which things follow in logical order or a recurrent pattern. A following of one thing after another in time. The action of following in order. Sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms). Stages.

Measuring Value - Assessments (errors)

Evaluate is to evaluate or estimate the nature, quality, ability, extent, or significance of. Evaluation is a systematic determination of a subject's merit, worth and significance, using criteria governed by a set of standards. It can assist an organization, program, project or any other intervention or initiative to assess any aim, realisable concept/proposal, or any alternative, to help in decision-making; or to ascertain the degree of achievement or value in regard to the aim and objectives and results of any such action that has been completed. The primary purpose of evaluation, in addition to gaining insight into prior or existing initiatives, is to enable reflection and assist in the identification of future change.

Khan Academy: Math Tutorial Videos

Math Videos (youtube)

Math Help

Online Resources for Learning Math

Paul Dirac

Carnegie Learning

The Math Page

Math is Fun

Cool Math 4 Kids

IXL

Math Words

Math World

Art of Problem Solving

Conrad Wolfram: Teaching Kids Real Math with Computers (youtube)

Math Games and Puzzles

The Story of "1" (film)

The Colors Of Infinity (youtube)

Fermat's Last Theorem (youtube)

Agape Satori - Mathematics is The Language of Nature (youtube)

Math Using Lines (youtube)

Number-Phile (youtube channel) Numberphile (website)

4.669 - Numberphile (youtube)

Solving Multivariable Equations (youtube)

How to Use Rectangular Arrays to Teach Multiplication, Factors, Primes, Composites, Squares

Technical Math Courses

Mathologer

Math Symbols

Math Trick: Multiply Numbers Close To Each Other In Your Head (youtube)

Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The Fields Medal is sometimes viewed as the highest honor a mathematician can receive. The Fields Medal and the Abel Prize have often been described as the mathematician's "Nobel Prize". The Fields Medal differs from the Abel in view of the age restriction mentioned above.

Mathematical Olympiad

The International Mathematical Olympiad

MOSP (wiki)

USA Math Camp Advanced Mathematics ("cool math")

Nobel Prize is a set of annual international awards bestowed in a number of categories by Swedish and Norwegian institutions in recognition of academic, cultural, or scientific advances.

Math Trick

Choose a number 1 through 10.

Lets say that you choose the number 8.

Now double that number, which would now be 16.

Now add 6 to 16, which is now 22.

Now dived 22 by 2, which is now 11.

Now minus the original number, which is 8 from 11.

Your answer is 3.

No mater which number you choose from 1 to 10, or 1 to a million, you will always get the same answer, "3"

Kind of like Voting in Politics, no matter how you add it up you always end up with the same old sh*t.

Sum of three cubes for 42 finally solved -- using real life planetary computer. Mathematicians have solved the final piece of the famous 65-year-old math puzzle with an answer for the most elusive number of all - 42.

Singapore Math is teaching students to learn and master fewer mathematical concepts at greater detail as well as having them learn these concepts using a three-step learning process. The three steps are: concrete, pictorial, and abstract. In the concrete step, students engage in hands-on learning experiences using concrete objects such as chips, dice, or paper clips. This is followed by drawing pictorial representations of mathematical concepts. Students then solve mathematical problems in an abstract way by using numbers and symbols.

Metric System is a decimal system of weights and measures based on the meter and the kilogram and the second, multipliers that have positive powers of ten.

International System of Units or SI is the modern form of the metric system, and is the most widely used system of measurement. It comprises a coherent system of units of measurement built on seven base units. The system also establishes a set of twenty prefixes to the unit names and unit symbols that may be used when specifying multiples and fractions of the units.

Cubit is an ancient unit based on the forearm length from the middle finger tip to the elbow bottom.

Roman Numerals is a system represented by combinations of letters from the Latin alphabet. Roman numerals, as used today, are based on seven symbols: I = 1, V + 5, X =10, L= 50, C = 100, D= 500, M= 1,000.

The Principles of Mathematics (wiki)

Mathematics Teaching Standards

Secrets of Mental Math (Book)

Math Forum

Math Lab

The Math League

National Council of Teacher of Mathematics

National Council of Teachers of Mathematics (wiki)

Euclid's Elements is a collection of definitions, postulates, propositions (theorems and constructions), and mathematical proofs of the propositions. A mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c. 300 BC.

Pi is 3.14159, which is the ratio of a circle's circumference to its diameter. Pi is an irrational number, meaning it cannot be referenced exactly as a fraction, but only as a decimal. Pi is not expressible as the quotient of two integers or can't be equally divided. Pi is also a Mathematical constant. A computer calculated PI out to 2.7 trillion decimal places. A man calculated PI in his head out to 70,000 decimal places. 15 decimal places is what most Engineers need to use (3.141592653589793). Pi Day is an annual celebration of the mathematical constant π (pi). Pi Day is observed on March 14 (3/14 in the month/day format) since 3, 1, and 4 are the first three significant digits of π.

Approximations of Pi (wiki) - Tau or Pi (youtube) - TD - Tau (wiki)

Symmetry - Fractals - Mandelbrot Set

Phi also used as a symbol for the golden ratio, a mathematical constant.

Basel Problem is a problem in mathematical analysis with relevance to number theory. The sum of the series is approximately equal to 1.644934. Euler's original derivation of the value π2/6 or Pi squared divided by 6, essentially extended observations about finite polynomials and assumed that these same properties hold true for infinite series.

Planck Units are a set of units of measurement defined exclusively in terms of five universal physical constants, in such a manner that these five physical constants take on the numerical value of 1 when expressed in terms of these units.

Matryoshka Doll or Russian doll, is a set of wooden dolls of decreasing size placed one inside another. The name "matryoshka" (матрёшка), literally "little matron", is a diminutive form of Russian female first name "Matryona" (Матрёна) or "Matriosha".

Borromean Rings consist of three topological circles which are linked and form a Brunnian link (i.e., removing any ring results in two unlinked rings). In other words, no two of the three rings are linked with each other as a Hopf link, but nonetheless all three are linked.

Efimov State is an effect in the quantum mechanics of few-body systems. Efimov’s effect is where three identical bosons interact, with the prediction of an infinite series of excited three-body energy levels when a two-body state is exactly at the dissociation threshold. One corollary is that there exist bound states (called Efimov states) of three bosons even if the two-particle attraction is too weak to allow two bosons to form a pair. A (three-particle) Efimov state, where the (two-body) sub-systems are unbound, are often depicted symbolically by the Borromean rings. This means that if one of the particles is removed, the remaining two fall apart. In this case, the Efimov state is also called a Borromean state.

Elliptic Curves is a plane algebraic curve defined by an equation of the form. Shapes.

Infinity is an abstract concept describing something without any bound or larger than any number.

Myriad is something being too numerous to be counted. A large indefinite number. Uncountable is something that is not able to be counted because there are too many to count. Constant.

Proof by Infinite Descent shows that a given equation has no solutions.

Infinity Plus One are representations of sizes (cardinalities) of abstract sets, which may be infinite. Addition of cardinal numbers is defined as the cardinality of the disjoint union of sets of given cardinalities. Power Set (wiki)

Finite Topological Space is a topological space for which the underlying point set is finite. That is, it is a topological space for which there are only finitely many points.

Finite describes something that is bounded or limited in magnitude or spatial or temporal extent. Having an end or limit; constrained by bounds.

Finite Set is a set that has a finite number of elements. Element of a set is any one of the distinct objects that make up that set. Set is a well-defined collection of distinct objects, considered as an object in its own right. Mathematical object is an abstract object arising in mathematics.

Set Theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects. (odd number + even number = Odd number).

"Infinity in an finite world with finite time. Infinity shows us endless possibilities, a math phenomenon that seemingly goes on forever. But nothing lasts forever. Our planet will die someday, our sun will die someday, and every person on earth will die someday. But new stars will form and new planets will be born, and over 350,000 new humans are born everyday. And the universes has 100's of trillions of years left, if not more, which is not forever, but it may as well be forever. Matter can not be destroyed or created, matter can only be transformed. But matter was created because matter exists, so matter can be created again, but only if it has to be created again. So let's focus on the finites because finites is our Reality. Let infinity be a symbol for endless possibilities, which is a lot easier on the mind then trying to wrap your head around an idea that has no limits, and it also helps us live more in our reality instead of thinking about a perceived reality that can literally blow your mind."

Permutation relates to the act of arranging all the members of a set into some sequence or order, or if the set is already ordered, rearranging (reordering) its elements, a process called permuting.

Recursion occurs when a thing is defined in terms of itself or of its type.

Recursion is a method where the solution to a problem depends on solutions to smaller instances of the same problem (as opposed to iteration).

Iteration in computer science, is a single execution of a set of instructions that are to be repeated. Executing the same set of instructions a given number of times or until a specified result is obtained. Doing or saying again; a repeated performance. Repeating a process.

Googol is the large number 10100. In decimal notation, it is written as the digit 1 followed by one hundred 0s:

Large Numbers are numbers that are significantly larger than those ordinarily used in everyday life. Names of Large Numbers (wiki) - Law of Large Numbers (wiki)

1,000 or one thousand - 1,000,000 or one million - 1,000,000,000 or one billion - 1,000,000,000,000 or one trillion - 1,000,000,000,000,000 or one quadrillion - 1,000,000,000,000,000,000 or one quintillion - 1,000,000,000,000,000,000,000 or one sextillion - 1,000,000,000,000,000,000,000,000 or one septillion - 1,000,000,000,000,000,000,000,000,000 or one octillion, which is a one with 27 zeros after it. Numbers.

Bytes (zeros and ones) - Sizes (big to small)

Natural Number are those used for counting. Numbers - Integer.

Whole Number or Natural Number is a number without fractions. An integer. Cardinal Number is also called whole number or natural number, which are those used to count physical objects in the real world. They are integers that can be zero or positive.

Real Number is a value that represents a measurable quantity. A value of a continuous quantity that can represent a distance along a line.

Definable Real Number is a real number that can be uniquely specified by its description.

Cardinal Number is a number denoting quantity (one, two, three, etc.), as opposed to an ordinal number (first, second, third, etc.). Cardinal Number is the number of elements in a mathematical set; denotes a quantity but not the order. A cardinal number are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set.

Ordinal Number is a number defining a thing's position in a series, such as “first,” “second,” or “third.” Ordinal numbers are used as adjectives, nouns, and pronouns. An ordinal number is one generalization of the concept of a natural number that is used to describe a way to arrange a possibly infinite collection of objects in order, one after another.

Trinity is the cardinal number that is the sum of one and one and one. Three people considered as a unit. Holy Trinity - Polygons.

Hyperreal Number is a way of treating infinite and infinitesimal quantities.

Surreal Number system is a totally ordered class containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number.

0 as the number Zero fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems.

Rational Number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q.

Irrational Number are all the real numbers which are not rational numbers. Numbers constructed from ratios or fractions of integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself. Among irrational numbers are the ratio π of a circle's circumference to its diameter, Euler's number e, the golden ratio φ, and the square root of two; in fact all square roots of natural numbers, other than of perfect squares, are irrational.

Irrational is something that is not consistent or precisely and clearly expressed.

Prime Number is a natural number or whole number greater than 1 whose only factors are 1 and itself and has no positive divisors other than 1 and itself. For example, 5 is prime because 1 and 5 are its only positive integer factors, whereas 6 is composite because it has the divisors 2 and 3 in addition to 1 and 6. A factor is a whole numbers that can be divided evenly into another number. greater than 1 that (5 is a Prime). The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. A natural number (1, 2, 3, 4, 5, 6, etc.) is called a prime number (or a prime) if it is greater than 1 and cannot be written as a product of two natural numbers that are both smaller than it. The numbers greater than 1 that are not prime are called composite numbers. In other words, n is prime if n items cannot be divided up into smaller equal-size groups of more than one item, or if it is not possible to arrange n dots into a rectangular grid that is more than one dot wide and more than one dot high. For example, among the numbers 1 through 6, the numbers 2, 3, and 5 are the prime numbers, as there are no other numbers that divide them evenly (without a remainder). 1 is not prime, as it is specifically excluded in the definition. 4 = 2 × 2 and 6 = 2 × 3 are both composite. There are infinitely many primes, as demonstrated by Euclid around 300 BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically modelled. The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability of a randomly chosen number being prime is inversely proportional to its number of digits, that is, to its logarithm.

Largest known Prime Number a number with 17,425,170 digits.

Great Internet Mersenne Prime Search

Twin Prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime twin or prime pair. Twin primes become increasingly rare as one examines larger ranges, in keeping with the general tendency of gaps between adjacent primes to become larger as the numbers themselves get larger. However, it is unknown whether there are infinitely many twin primes (the so-called twin prime conjecture) or if there is a largest pair. The work of Yitang Zhang in 2013, as well as work by James Maynard, Terence Tao and others, has made substantial progress towards proving that there are infinitely many twin primes, but at present this remains unsolved. Incompleteness.

Strobogrammatic Prime is a prime number that, given a base and given a set of glyphs, appears the same whether viewed normally or upside down.

Prime Quadruplet is a set of four primes of the form {p, p+2, p+6, p+8}. This represents the closest possible grouping of four primes larger than 3.

Composite Number is a positive integer or natural number that can be formed by multiplying together two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, or the unit 1, so the composite numbers are exactly the numbers that are not prime and not a unit.

Palindromic Number is a number that remains the same when its digits are reversed. Like 16461, for example, it is "symmetrical". 101

Transcendental Number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best-known transcendental numbers are π and e. Almost all real and complex numbers are transcendental, since the algebraic numbers are countable while the sets of real and complex numbers are both uncountable. All real transcendental numbers are irrational, since all rational numbers are algebraic. The converse is not true: not all irrational numbers are transcendental.

Imaginary Numbers (Lateral) (youtube) - Fundamental Theorem of Algebra - Square Root of Negative One.

Plato's Number - 216 - Wolfram

5040 is a factorial (7!), a superior highly composite number, a colossally abundant number, and the number of permutations of 4 items out of 10 choices (10 × 9 × 8 × 7 = 5040).

Pseudorandom Number Generator is an algorithm for generating a sequence of numbers whose properties approximate the properties of sequences of random numbers

Square-Free Integer is an integer which is divisible by no other perfect square than 1. For example, 10 is square-free but 18 is not, as 18 is divisible by 9 = 32.

Countable Set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers. A countable set is either a finite set or a countably infinite set. Whether finite or infinite, the elements of a countable set can always be counted one at a time and, although the counting may never finish, every element of the set is associated with a unique natural number.

Digital Root of a non-negative integer is the (single digit) value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit sum. The process continues until a single-digit number is reached. For example, the digital root of 65,536 is 7, because 6 + 5 + 5 + 3 + 6 = 25 and 2 + 5 = 7.

Additive identity of a set which is equipped with the operation of addition is an element which, when added to any element x in the set, yields x. One of the most familiar additive identities is the number 0 from elementary mathematics, but additive identities occur in other mathematical structures where addition is defined, such as in groups and rings.

Numerical Digit is a numeric symbol (such as "2" or "5") used in combinations (such as "25") to represent numbers (such as the number 25) in positional numeral systems.

Numeral System is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. It can be seen as the context that allows the symbols "11" to be interpreted as the binary symbol for three, the decimal symbol for eleven, or a symbol for other numbers in different bases.

TWL #7: This Number is Illegal Prime Numbers and Encryption (youtube)

Logarithmic integral is a special function. It is relevant in problems of physics and has number theoretic significance, occurring in the prime number theorem as an estimate of the number of prime numbers less than a given value.

Logarithm is the inverse operation to exponentiation. That means the logarithm of a number is the exponent to which another fixed number, the base, must be raised to produce that number. In simple cases the logarithm counts factors in multiplication. For example, the base 10 logarithm of 1000 is 3, as 10 to the power 3 is 1000 (1000 = 10 × 10 × 10 = 103); 10 is used as a factor three times. More generally, exponentiation allows any positive real number to be raised to any real power, always producing a positive result, so the logarithm can be calculated for any two positive real numbers b and x where b is not equal to 1.

Logarithmic Scale is a nonlinear scale used when there is a large range of quantities. Common uses include the earthquake strength, sound loudness, light intensity, and pH of solutions. It is based on orders of magnitude, rather than a standard linear scale, so each mark on the scale is the previous mark multiplied by a value.

Logarithmic Growth describes a phenomenon whose size or cost can be described as a logarithm function of some input. e.g. y = C log (x). Note that any logarithm base can be used, since one can be converted to another by multiplying by a fixed constant. Logarithmic growth is the inverse of exponential growth and is very slow. A familiar example of logarithmic growth is the number of digits needed to represent a number, N, in positional notation, which grows as logb (N), where b is the base of the number system used, e.g. 10 for decimal arithmetic

Fermat's Last Theorem states that no three positive integers a, b, and c satisfy the equation

Riemann Hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2.

Goldbach Conjecture states that every even integer greater than 2 can be expressed as the sum of two primes.

Probability Theory is the branch of mathematics concerned with probability, the analysis of random phenomena.

Chaos Theory studies the behavior of dynamical systems that are highly sensitive to initial conditions—a response popularly referred to as the butterfly effect. Small differences in initial conditions (such as those due to rounding Errors in numerical computation) yield widely diverging outcomes for such dynamical systems, rendering long-term prediction of their behavior impossible in general. This happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as deterministic chaos, or simply chaos.

Polynomial is an expression consisting of variables (or indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

Theta Function are special functions of several complex variables. They are important in many areas, including the theories of abelian varieties and moduli spaces, and of quadratic forms. They have also been applied to soliton theory. When generalized to a Grassmann algebra, they also appear in quantum field theory.

Modular Forms is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition.

Constructivism asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists. When one assumes that an object does not exist and derives a contradiction from that assumption, one still has not found the object and therefore not proved its existence, according to constructivism. This viewpoint involves a verificational interpretation of the existence quantifier, which is at odds with its classical interpretation.

Duodecimal is a positional notation numeral system using twelve as its base.

12 Dozenal Society of America

Action Physics - Physics - Magnetics

Euler Identity - The number 0 is the additive identity. The number 1 is the multiplicative identity. The number π or pi is the ratio of the circumference of a circle to its diameter. (π = 3.141...). The number e (e = 2.718...) is Euler's number, the base of natural logarithms, which occurs widely in mathematical analysis. The number i is the imaginary unit of the complex numbers, which by definition satisfies i2 = −1. Euler idetity equation is simple to look at and yet incredibly profound, it comprises the five most important mathematical constants - zero (additive identity), one (multiplicative identity), e and pi (the two most common transcendental numbers) and i (fundamental imaginary number). It also comprises the three most basic arithmetic operations - addition, multiplication and exponentiation."

"We study mathematics for play, for beauty, for truth, for justice and for love." - Francis Su

Leonhard Euler was a Swiss mathematician, physicist, astronomer, logician and engineer who made important and influential discoveries in many branches of mathematics like infinitesimal calculus and graph theory while also making pioneering contributions to several branches such as topology and analytic number theory. (15 April 1707 – 18 September 1783)

Euler Characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. 101 (different meanings).

Imaginary Unit or unit imaginary number (i) is a solution to the quadratic equation x2 + 1 = 0. Although there is no real number with this property, i can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of i in a complex number is 2 + 3i. Imaginary numbers are an important mathematical concept, which extend the real number system R to the complex number system C, which in turn provides at least one root for every nonconstant polynomial P(x). (See Algebraic closure and Fundamental theorem of algebra.) The term "imaginary" is used because there is no real number having a negative square. There are two complex square roots of −1, namely i and −i, just as there are two complex square roots of every real number other than zero, which has one double square root. In contexts where i is ambiguous or problematic, j or the Greek ι is sometimes used (see § Alternative notations). In the disciplines of electrical engineering and control systems engineering, the imaginary unit is normally denoted by j instead of i, because i is commonly used to denote electric current. For the history of the imaginary unit, see Complex number § History.

The Institute of Mathematics and its Applications.

Mathematical Sciences Research Institute

Jim Simons: A rare interview with the Mathematician who cracked Wall Street (video)

Chern-Simons Theory (wiki)

Math is Beautiful to the Mind

Eugene Wigner was a Hungarian-American theoretical physicist, engineer and mathematician. He received half of the Nobel Prize in Physics in 1963 "for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles. (November 17, 1902 – January 1, 1995).

Hermann Minkowski was a Jewish German mathematician, professor at Königsberg, Zürich and Göttingen. He created and developed the geometry of numbers and used geometrical methods to solve problems in number theory, mathematical physics, and the theory of relativity. (22 June 1864 – 12 January 1909).

Pontryagin's Minimum Principle is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls.

Pontryagin Duality in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact groups, such as R, the circle, or finite cyclic groups. The Pontryagin duality theorem itself states that locally compact abelian groups identify naturally with their bidual.

Pontryagin Class lies in cohomology groups with degree a multiple of four. It applies to real vector bundles.

Symmetry is an agreement in dimensions and arrangement. A sense of harmonious and beautiful proportion and balance, even though there is still diversity and differences in self-assembly or self organizing systems.

Proportionate is exhibiting equivalence or correspondence among constituents of an entity or between different entities. The correct, attractive, or ideal relationship in size or shape between one thing and another or between the parts of a whole. The relationship of one thing to another in terms of quantity, size, or number; the ratio.

Facial Symmetry is one specific measure of bodily asymmetry or the absence of symmetry.

Vitruvian Man - Perfect Proportions - Golden Ratio - Pi - Flower

Symmetry Number of an object is the number of different but indistinguishable or equivalent arrangements or views of the object.

Platonic Solids - Patterns - Time Symmetry

Order is the arrangement or disposition of people or things in relation to each other according to a particular sequence, pattern, or method. A state in which everything is in its correct or appropriate place. The opposite of chaos.

Symmetry in biology is the balanced distribution of duplicate body parts or shapes within the body of an organism. Right Brain and Left Brain.

Molecular Symmetry in chemistry describes the symmetry present in molecules and the classification of molecules according to their symmetry. Molecular symmetry is a fundamental concept in chemistry, as it can be used to predict or explain many of a molecule's chemical properties, such as its dipole moment and its allowed spectroscopic transitions. Many university level textbooks on physical chemistry, quantum chemistry, and inorganic chemistry devote a chapter to symmetry. DNA (CTAG)

Bilateria are animals with bilateral symmetry, i.e., they have a front ("anterior") and a back ("posterior") as well as an upside ("dorsal") and downside ("ventral"); therefore they also have a left side and a right side. In contrast, radially symmetrical animals like jellyfish have a topside and a downside, but no identifiable front or back.

Symmetry in physics of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation.

Radiolaria are protozoa of diameter 0.1–0.2 mm that produce intricate mineral skeletons, typically with a central capsule dividing the cell into the inner and outer portions of endoplasm and ectoplasm. The elaborate mineral skeleton is usually made of silica. They are found as zooplankton throughout the ocean, and their skeletal remains make up a large part of the cover of the ocean floor as siliceous ooze. Due to their rapid change as species, they represent an important diagnostic fossil found from the Cambrian onwards. Some common radiolarian fossils include Actinomma, Heliosphaera and Hexadoridium. Sound Shapes.

Discrete Symmetry is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square's original appearance. Discrete symmetries sometimes involve some type of 'swapping', these swaps usually being called reflections or interchanges. In mathematics and theoretical physics, a discrete symmetry is a symmetry under the transformations of a discrete group—e.g. a topological group with a discrete topology whose elements form a finite or a countable set. One of the most prominent discrete symmetries in physics is parity symmetry. It manifests itself in various elementary physical quantum systems, such as quantum harmonic oscillator, electron orbitals of Hydrogen-like atoms by forcing wavefunctions to be even or odd. This in turn gives rise to selection rules that determine which transition lines are visible in atomic absorption spectra.

Parity in physics is the flip in the sign of one spatial coordinate. In three dimensions, it is also often described by the simultaneous flip in the sign of all three spatial coordinates (a point reflection).

Symmetry in mathematics occurs not only in Geometry, but also in other branches of mathematics. Symmetry is a type of invariance: the property that something does not change under a set of transformations.

A Physical Model for Forming Patterns in Pollen. Physicists have developed a model that describes how patterns form on pollen spores, the first physically rigorous framework that details the thermodynamic processes that lead to complex biological architectures.

Octahedral Symmetry. A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the dual of an octahedron. The group of orientation-preserving symmetries is S4, the symmetric group or the group of permutations of four objects, since there is exactly one such symmetry for each permutation of the four pairs of opposite faces of the octahedron.

Symmetry in geometry is a circle rotated about its center that will have the same shape and size as the original circle—all points before and after the transform would be indistinguishable. A circle is said to be symmetric under rotation or to have rotational symmetry. If the isometry is the reflection of a plane figure, the figure is said to have reflectional symmetry or line symmetry; moreover, it is possible for a figure/object to have more than one line of symmetry.

Reflection Symmetry is symmetry with respect to reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. Mirroring.

Orthographic Projection is a means of representing three-dimensional objects in two dimensions. It is a form of parallel projection, in which all the projection lines are orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface. The obverse of an orthographic projection is an oblique projection, which is a parallel projection in which the projection lines are not orthogonal to the projection plane.

Plane Curve is a plane curve which is mirror-symmetrical and is approximately U-shaped.

Universal Parabolic Constant (wiki) - Parabola (wiki).

E8 Polytope can be visualized as symmetric orthographic projections in Coxeter planes of the E8 Coxeter group, and other subgroups. Uniform 8-Polytope (wiki).

Rotational Symmetry is a center point around which the object is turned (rotated) a certain number of degrees and the object looks the same. The number of positions in which the object looks exactly the same is called the order of the symmetry.

Gauge Symmetry in mathematics any Lagrangian system generally admits gauge symmetries, though it may happen that they are trivial. In theoretical physics, the notion of gauge symmetries depending on parameter functions is a cornerstone of contemporary field theory.

Continuous Symmetry is an intuitive idea corresponding to the concept of viewing some symmetries as motions, as opposed to discrete symmetry, e.g. reflection symmetry, which is invariant under a kind of flip from one state to another. However, a discrete symmetry can always be reinterpreted as a subset of some higher-dimensional continuous symmetry, e.g. reflection of a 2 dimensional object in 3 dimensional space can be achieved by continuously rotating that object 180 degrees across a non-parallel plane.

Spontaneous Symmetry Breaking is a mode of realization of symmetry breaking in a physical system, where the underlying laws are invariant under a symmetry transformation, but the system as a whole changes under such transformations, in contrast to explicit symmetry breaking. It is a spontaneous process by which a system in a symmetrical state ends up in an asymmetrical state. It thus describes systems where the equations of motion or the Lagrangian obey certain symmetries, but the lowest-energy solutions do not exhibit that symmetry. Chaos.

Translational Symmetry In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation. Discrete translational symmetry is invariant under discrete translation

Supersymmetry is a proposed type of spacetime symmetry that relates two basic classes of elementary particles: bosons, which have an integer-valued spin, and fermions, which have a half-integer spin. Each particle from one group is associated with a particle from the other, known as its superpartner, the spin of which differs by a half-integer. In a theory with perfectly "unbroken" supersymmetry, each pair of superpartners would share the same mass and internal quantum numbers besides spin.

Space Group is the symmetry group of a configuration in space, usually in three dimensions. In three dimensions, there are 219 distinct types, or 230 if chiral copies are considered distinct. Space groups are also studied in dimensions other than 3 where they are sometimes called Bieberbach groups, and are discrete cocompact groups of isometries of an oriented Euclidean space.

Golden Ratio is a common mathematical ratio found in nature. It is a special number found by dividing a line into two parts so that the longer part divided by the smaller part is also equal to the whole length divided by the longer part. The Golden Ration is also known as the divine proportion, golden mean, or golden section. Phi is the symbol for the golden ratio. Irrational Number. 1.618 - 1.100111100011011101 - PDF.

Golden Angle is the smaller of the two angles created by sectioning the circumference of a circle according to the golden ratio; that is, into two arcs such that the ratio of the length of the larger arc to the length of the smaller arc is the same as the ratio of the full circumference to the length of the larger arc. 137.5 Degrees. Algebraically, let a+b be the circumference of a circle, divided into a longer arc of length a and a smaller arc of length b such that.

Creating The Never-Ending Bloom (youtube) - John Edmark's sculptures are both mesmerizing and mathematical. Using meticulously crafted platforms, patterns, and layers, Edmark's art explores the seemingly magical properties that are present in spiral geometries. In his most recent body of work, Edmark creates a series of animating “blooms” that endlessly unfold and animate as they spin beneath a strobe light. 137.5 degrees.

Fine-Structure Constant commonly denoted by α (the Greek letter alpha), is a dimensionless physical constant characterizing the strength of the electromagnetic interaction between elementary charged particles. It is related to the elementary charge e, which characterizes the strength of the coupling of an elementary charged particle with the electromagnetic field, by the formula 4πε0ħcα = e2. Being a dimensionless quantity, it has the same numerical value in all systems of units, which is approximately (1 over 137 or 1/137) or α = e2/(hc)=1/137.03599976.

137 (one hundred [and] thirty-seven) is the natural number following 136.

Planck Constant is a physical constant that is the quantum of electromagnetic action, which relates the energy carried by a photon to its frequency. A photon's energy is equal to its frequency multiplied by the Planck constant. The Planck constant is of fundamental importance in quantum mechanics, and in metrology it is the basis for the definition of the kilogram.

Golden Section is the division of a line so that the whole is to the greater part as that part is to the smaller part (i.e. in a ratio of 1 to 1/2 (√5 + 1)), a proportion which is considered to be particularly pleasing to the eye.

The Golden Ratio (why it is so irrational) - Numberphile (youtube)

Patterns in Nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically. Natural patterns include symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks and stripes. Phyllotaxis is the arrangement of leaves on a plant stem. Circles.

Logarithmic Spiral is a self-similar spiral curve which often appears in nature. Infinity.

Golden Spiral is a logarithmic spiral whose growth factor is φ, the golden ratio. That is, a golden spiral gets wider (or further from its origin) by a factor of φ for every quarter turn it makes.

Spirograph is a geometric drawing device that produces mathematical roulette curves of the variety technically known as hypotrochoids and epitrochoids. The well-known toy version was developed by British engineer Denys Fisher and first sold in 1965.

Logarithm is the inverse operation to exponentiation, just as division is the inverse of multiplication. That means the logarithm of a number is the exponent to which another fixed number, the base, must be raised to produce that number.

Spiral is a curve which emanates from a point, moving farther away as it revolves around the point. Torus - LI Patterns.

Fibonacci Number are the numbers in the following integer sequence, called the Fibonacci Sequence, and characterized by the fact that every number after the first two is the sum of the two preceding ones: 1+1=2, 1+2=3, 2+3=5, 3+5=8, 5+8=13, 8+13=21....34, 55, 89, 144, …Fibonacci Sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89.... Fibonacci Zoetrope Sculptures (youtube).

Why do Prime Numbers make these Spirals? (youtube)

Mathematics of Plant Leaves. Unusual Japanese plant inspires recalculation of equation used to model leaf arrangement patterns. Common patterns are symmetrical and have leaves arranged at regular intervals of 90 degrees (basil or mint), 180 degrees (stem grasses, like bamboo), or in Fibonacci golden angle spirals (like the needles on some spherical cacti, or the succulent spiral aloe). The angles between O. Japonica leaves are 180 degrees, 90 degrees, 180 degrees, 270 degrees, and then the next leaf resets the pattern to 180 degrees. No matter what values are put into the DC2 equation, certain uncommon leaf arrangement patterns are never calculated. The Fibonacci spiral leaf arrangement pattern is by far the most common spiral pattern observed in nature, but is only modestly more common than other spiral patterns calculated by the DC2 equation.

Rule of Thirds guideline which applies to the process of composing visual images such as designs, films, paintings, and photographs. The guideline proposes that an image should be imagined as divided into nine equal parts by two equally spaced horizontal lines and two equally spaced vertical lines, and that important compositional elements should be placed along these lines or their intersections. Proponents of the technique claim that aligning a subject with these points creates more tension, energy and interest in the composition than simply centering the subject.

Everything in our reality possesses a star tetrahedral energy field, and planets are no exception. The points of the bases of the two tetrahedrons in the star tetrahedron touch an enclosing sphere at 19.47 degrees. At each planet’s 19.47 degree latitudes we have the intersection between the light body of the planet and its surface, and since light-bodies have the ability to connect us to other dimensions, at these latitudes we have an energetic predisposition for inter-dimensional experience. Polygon.

Tetrahedron Grid Points on Planet Earth (image)

Pythagoras was an Ionian Greek philosopher, mathematician, and putative founder of the Pythagoreanism movement. He is often revered as a great mathematician and scientist and is best known for the Pythagorean theorem, which is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the long side opposite the right angle) is equal to the sum of the squares of the other two sides. (570 – c. 495 BC).

"All is Number"

Pythagoreanism originated in the 6th century BC, based on the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans, who were considerably influenced by mathematics and mysticism. Later revivals of Pythagorean doctrines led to what is now called Neopythagoreanism or Neoplatonism. Pythagorean ideas exercised a marked influence on Aristotle, and Plato, and through them, all of Western philosophy. Tam.

Intelligent Design - Everything is Connected

Vitruvian Man is a drawing based on the correlations of ideal human proportions with geometry described by the ancient Roman architect Vitruvius in Book III of his treatise De architectura. The drawing was by Leonardo da Vinci around 1490. The drawing depicts a man in two superimposed positions with his arms and legs apart and inscribed in a circle and square. The drawing and text are sometimes called the Canon of Proportions or, less often, Proportions of Man. Translated to "The proportions of the human body according to Vitruvius"), or simply L'Uomo Vitruviano. Perfect Proportions - four fingers equal one palm, four palms equal one foot, six palms make one cubit, four cubits equal a man's height, four cubits equal one pace, 24 palms equal one man. Symmetry.

Divina Proportione is a book on mathematics written by Luca Pacioli and illustrated by Leonardo da Vinci, composed around 1498 in Milan and first printed in 1509. Its subject was mathematical proportions (the title refers to the golden ratio) and their applications to geometry, visual art through perspective, and architecture. The clarity of the written material and Leonardo's excellent diagrams helped the book to achieve an impact beyond mathematical circles, popularizing contemporary geometric concepts and images. Flower of Life.

Leonardo da Vinci was an Italian polymath of the Renaissance whose areas of interest included invention, drawing, painting, sculpting, architecture, science, music, mathematics, engineering, literature, anatomy, geology, astronomy, botany, writing, history, and cartography. He has been variously called the father of palaeontology, ichnology, and architecture, and he is widely considered one of the greatest painters of all time. Sometimes credited with the inventions of the parachute, helicopter, and tank, he epitomised the Renaissance humanist ideal. (15 April 1452 – 2 May 1519).

Mandelbrot Set is a particular set of complex numbers that has a highly convoluted fractal boundary when plotted. The set of values of c in the complex plane for which the orbit of 0 under iteration of the quadratic map remains bounded. That is, a complex number c is part of the Mandelbrot set if, when starting with z0 = 0 and applying the iteration repeatedly, the absolute value of zn remains bounded however large n gets. The Mandelbrot set is generated by what is called iteration, which means to repeat a process over and over again. In mathematics this process is most often the application of a mathematical function. For the Mandelbrot set, the functions involved are some of the simplest imaginable: they all are what is called quadratic polynomials and have the form f(x) = x2 + c, where c is a constant number. As we go along, we will specify exactly what value c takes. . To iterate x2 + c, we begin with a seed for the iteration. This is a number which we write as x0. Applying the function x2 + c to x0 yields the new number. x1 = x02 + c. Now, we iterate using the result of the previous computation as the input for the next. That is x2 = x12 + c and then x3 = x22 + c and then x4 = x32 + c and then x5 = x42 + c and so forth. The list of numbers x0, x1, x2,... generated by this iteration has a name: it is called the orbit of x0 under iteration of x2 + c. Plot the Mandelbrot Set By Hand.

Mandelbulb is a three-dimensional fractal. A canonical 3-dimensional Mandelbrot set does not exist, since there is no 3-dimensional analogue of the 2-dimensional space of complex numbers. It is possible to construct Mandelbrot sets in 4 dimensions using quaternions and bicomplex numbers.

Fractal is a geometric pattern that is repeated at every scale and so cannot be represented by classical geometry.

Fractal is a mathematical set that exhibits a repeating pattern that displays at every scale. It is also known as expanding symmetry or evolving symmetry. If the replication is exactly the same at every scale, it is called a self-similar pattern. An example of this is the Menger Sponge. A fractal is a curve or geometric figure, each part of which has the same statistical character as the whole. Fractals are useful in modeling structures (such as eroded coastlines or snowflakes) in which similar patterns recur at progressively smaller scales, and in describing partly random or chaotic phenomena such as crystal growth, fluid turbulence, and galaxy formation. One way that fractals are different from finite geometric figures is the way in which they scale. Doubling the edge lengths of a polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the dimension of the space the polygon resides in). Likewise, if the radius of a sphere is doubled, its volume scales by eight, which is two (the ratio of the new to the old radius) to the power of three (the dimension that the sphere resides in). However, if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer This power is called the fractal dimension of the fractal, and it usually exceeds the fractal's topological dimension. Analytically, fractals are usually nowhere differentiable. An infinite fractal curve can be conceived of as winding through space differently from an ordinary line – although it is still 1-dimensional, its fractal dimension indicates that it also resembles a surface.

Iterated Function is a function X ? X (that is, a function from some set X to itself) which is obtained by composing another function f : X ? X with itself a certain number of times. The process of repeatedly applying the same function is called iteration. In this process, starting from some initial number, the result of applying a given function is fed again in the function as input, and this process is repeated. Iterated functions are objects of study in computer science, fractals, dynamical systems, mathematics and renormalization group physics.

Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. The sequence will approach some end point or end value. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. In mathematics and computer science, iteration (along with the related technique of recursion) is a standard element of algorithms.

Barnsley Fern is a fractal named after the British mathematician Michael Barnsley who first described it in his book Fractals Everywhere. He made it to resemble the Black Spleenwort, Asplenium adiantum-nigrum.

Affine Transformation is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. Examples of affine transformations include translation, scaling, homothety, similarity transformation, reflection, rotation, shear mapping, and compositions of them in any combination and sequence.

Earth Fractals - Fractal Jigsaw (youtube)

Sierpinski Triangle is a fractal and attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. Originally constructed as a curve, this is one of the basic examples of self-similar sets, i.e., it is a mathematically generated pattern that can be reproducible at any magnification or reduction. It is named after the Polish mathematician Wacław Sierpiński, but appeared as a decorative pattern many centuries prior to the work of Sierpiński.

Crystals - Snowflakes - Intelligent Design - Frame Rate - Turing Complete

Conway's Game of Life is a cellular automaton devised by the British mathematician John Horton Conway in 1970. The game is a zero-player game, meaning that its evolution is determined by its initial state, requiring no further input. One interacts with the Game of Life by creating an initial configuration and observing how it evolves, or, for advanced players, by creating patterns with particular properties.

Zero-player Game is a game that has no sentient players.

Ulam-Warburton Automaton is a 2-dimensional fractal pattern that grows on a regular grid of cells consisting of squares. Starting with one square initially ON and all others OFF, successive iterations are generated by turning ON all squares that share precisely one edge with an ON square. This is the von Neumann neighborhood. The automaton is named after the Polish-American mathematician and scientist Stanislaw Ulam and the Scottish engineer, inventor and amateur mathematician Mike Warburton.

Visualizations - Mind Maps - Wolfram - Digital Physics - Universal Computing

Cellular Automaton consists of a regular grid of cells, each in one of a finite number of states, such as on and off (in contrast to a coupled map lattice). The grid can be in any finite number of dimensions. For each cell, a set of cells called its neighborhood is defined relative to the specified cell. An initial state (time t = 0) is selected by assigning a state for each cell. A new generation is created (advancing t by 1), according to some fixed rule (generally, a mathematical function) that determines the new state of each cell in terms of the current state of the cell and the states of the cells in its neighborhood. Typically, the rule for updating the state of cells is the same for each cell and does not change over time, and is applied to the whole grid simultaneously, though exceptions are known, such as the stochastic cellular automaton and asynchronous cellular automaton. Cell Division.

Replicator is a Life-like cellular automaton where a cell survives or is born if there are an odd number of neighbors. It is one of two Life-like Fredkin replicator rules. Under this ruleset, every pattern self-replicates; furthermore, every pattern will eventually produce an arbitrary number of copies of itself, all arbitrarily far away from each other.

Life-like Cellular Automaton. A cellular automaton (CA) is Life-like (in the sense of being similar to Conway's Game of Life) if it meets the following criteria: The array of cells of the automaton has two dimensions. Each cell of the automaton has two states (conventionally referred to as "alive" and "dead", or alternatively "on" and "off"). The neighborhood of each cell is the Moore neighborhood; it consists of the eight adjacent cells to the one under consideration and (possibly) the cell itself. In each time step of the automaton, the new state of a cell can be expressed as a function of the number of adjacent cells that are in the alive state and of the cell's own state; that is, the rule is outer totalistic (sometimes called semitotalistic).

Still Life is a pattern that does not change from one generation to the next. The term comes from the art world where a still life painting or photograph depicts an inanimate scene. In cellular automata, a still life can be thought of as an oscillator with unit period.

Continuous Spatial Automaton have a continuum of locations, while the state of a location still is any of a finite number of real numbers. Time can also be continuous, and in this case the state evolves according to differential equations.

Model of Computation is a model which describes how an output of a mathematical function is computed given an input. A model describes how units of computations, memories, and communications are organized. The computational complexity of an algorithm can be measured given a model of computation. Using a model allows studying the performance of algorithms independently of the variations that are specific to particular implementations and specific technology.

Coupled Map Lattice is a dynamical system that models the behavior of non-linear systems (especially partial differential equations). They are predominantly used to qualitatively study the chaotic dynamics of spatially extended systems. This includes the dynamics of spatiotemporal chaos where the number of effective degrees of freedom diverges as the size of the system increases.

Power of Two is a number of the form 2n where n is an integer, i.e. the result of exponentiation with number two as the base and integer n as the exponent. In a context where only integers are considered, n is restricted to non-negative values, so we have 1, 2, and 2 multiplied by itself a certain number of times. Because two is the base of the binary numeral system, powers of two are common in computer science. Written in binary, a power of two always has the form 100…000 or 0.00…001, just like a power of ten in the decimal system.

Attractor is a set of numerical values toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain close even if slightly disturbed. In finite-dimensional systems, the evolving variable may be represented algebraically as an n-dimensional vector. The attractor is a region in n-dimensional space. In physical systems, the n dimensions may be, for example, two or three positional coordinates for each of one or more physical entities; in economic systems, they may be separate variables such as the inflation rate and the unemployment rate. If the evolving variable is two- or three-dimensional, the attractor of the dynamic process can be represented geometrically in two or three dimensions, (as for example in the three-dimensional case depicted to the right). An attractor can be a point, a finite set of points, a curve, a manifold, or even a complicated set with a fractal structure known as a strange attractor (see strange attractor below). If the variable is a scalar, the attractor is a subset of the real number line. Describing the attractors of chaotic dynamical systems has been one of the achievements of chaos theory.

Scale Invariance s a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, thus represent a universality. The technical term for this transformation is a dilatation (also known as Dilation), and the dilatations can also form part of a larger conformal symmetry.

Dilatation is the state of being stretched beyond normal dimensions. The act of expanding an aperture.

Mysterium Cosmographicum - Johannes Kepler claimed to have had an epiphany on July 19, 1595. He realized that regular polygons bound one inscribed and one circumscribed circle at definite ratios, which, he reasoned, might be the geometrical basis of the universe. After failing to find a unique arrangement of polygons that fit known astronomical observations (even with extra planets added to the system), Kepler began experimenting with 3-dimensional polyhedra. He found that each of the five Platonic solids could be uniquely inscribed and circumscribed by spherical orbs; nesting these solids, each encased in a sphere, within one another would produce six layers, corresponding to the six known planets—Mercury, Venus, Earth, Mars, Jupiter, and Saturn. By ordering the solids correctly—octahedron, icosahedron, dodecahedron, tetrahedron, cube—Kepler found that the spheres could be placed at intervals corresponding (within the accuracy limits of available astronomical observations) to the relative sizes of each planet’s path, assuming the planets circle the Sun. Kepler also found a formula relating the size of each planet’s orb to the length of its orbital period: from inner to outer planets, the ratio of increase in orbital period is twice the difference in orb radius. However, Kepler later rejected this formula, because it was not precise enough.

Law of Squares (order out of chaos) - Searl Effect

Chaos Game - Numberphile (youtube)

Number 9

The sum of all digits 1 through 8=36 3+6=9

9 plus any digit returns the same digit 9+5=14 1+4=5

360 degrees in a circle 3+6+0=9

180 degrees in a circle 1+8+0=9

90 degrees in a circle 9+0=9

45 degrees in a circle 4+5=9

22.5 degrees in a circle 2+2+5=9

The resulting angle always reduces to 9

Sum of angles on polygons vectors communicate outward divergence. Nine reveals a linear duality, it's both singularity and the vacuum. Nine models everything and nothing simultaneously. 432 HZ.

0.999 is also written 0.9, among other ways, denotes the repeating decimal consisting of infinitely many 9s after the decimal point (and one 0 before it). This repeating decimal represents the smallest number no less than all decimal numbers 0.9, 0.99, 0.999, etc. This number can be shown to equal 1. In other words, "0.999..." and "1" "0.999... = 1" represent the same number. There are many ways of showing this equality, from intuitive arguments to mathematically rigorous proofs. The technique used depends on target audience, background assumptions, historical context, and preferred development of the real numbers, the system within which 0.999... is commonly defined. (In other systems, 0.999... can have the same meaning, a different definition, or be undefined.)

Earth has 92 different atoms and is 92.96 million miles from the Sun.

Nikola Tesla 3 6 9 (youtube)

Torus - Vortex Based Math (wiki) - Peter cullinane passing on {360* vortex maths divine symmetry} (youtube)

Flower of Life is one of the basic sacred geometry shapes. The symbol is believed to represent the cycle of creation, and reminds us of the unity of everything. Flowers.

Sacred Geometry ascribes symbolic and sacred meanings to certain geometric shapes and certain geometric proportions.

Overlapping Circles Grid is a geometric pattern of repeating, overlapping circles of an equal radius in two-dimensional space. Commonly, designs are based on circles centered on triangles (with the simple, two circle form named vesica piscis) or on the square lattice pattern of points. Lattice.

Arc is any smooth curve joining two points. The length of an arc is known as its arc length. In a graph, a graph arc is an ordered pair of adjacent vertices. An arc whose endpoints lie on a diameter of a circle is called a semicircle. Symmetry.

Six-Petal Rosette is a design with six-fold dihedral symmetry composed from six vesica piscis lenses arranged radially around a central point, often shown enclosed in a circumference of another six lenses. The design is found as a rosette ornament in artwork dating back to at least the Late Bronze Age. Golden Ratio.

Number Circles or Magic Circle is the arrangement of natural numbers on circles where the sum of the numbers on each circle and the sum of numbers on diameter are identical. One of his magic circles was constructed from 33 natural numbers from 1 to 33 arranged on four concentric circles, with 9 at the center.

Modular Arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus (plural moduli). A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Usual addition would suggest that the later time should be 7 + 8 = 15, but this is not the answer because clock time "wraps around" every 12 hours. Because the hour number starts over after it reaches 12, this is arithmetic modulo 12. According to the definition below, 12 is congruent not only to 12 itself, but also to 0, so the time called "12:00" could also be called "0:00", since 12 is congruent to 0 modulo 12.

Cymatics is when patterns emerge in the excitatory medium depending on the geometry of the plate and the driving frequency. - Frequencies (HZ) - Sound Shapes.

Gematria assigns numerical value to a word/name/phrase in the belief that words or phrases with identical numerical values bear some relation to each other or bear some relation to the number itself as it may apply to Nature, a person's age, the calendar year, or the like.

Feynman Diagram are pictorial representations of the mathematical expressions describing the behavior of subatomic particles. Virtual Particle.

Gauge Theory is a type of field theory in which the Lagrangian is invariant under a continuous group of local transformations.

Lorentz Covariance is "the feature of nature that says experimental results are independent of the orientation or the boost velocity of the laboratory through space". In everyday language, it means that the laws of physics stay the same for all observers that are moving with respect to one another with a uniform velocity.

Geometric Progression is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2.

Crystallization - Ice.

Eightfold way in physics is a theory organizing subatomic hadrons. Group Theory - Intelligent Design.

Constant is something continuing forever or for an indefinitely long time. A number having an unchanging value. A number representing a quantity assumed to have a fixed value in a specified mathematical context. A quantity that does not vary. Consistent - Resilient.

Mathematical Constant is a special number that is "significantly interesting in some way". Constants arise in many areas of mathematics, with constants such as e and π occurring in such diverse contexts as geometry, number theory, and calculus. What it means for a constant to arise "naturally", and what makes a constant "interesting", is ultimately a matter of taste, and some mathematical constants are notable more for historical reasons than for their intrinsic mathematical interest. The more popular constants have been studied throughout the ages and computed to many decimal places. All mathematical constants are definable numbers and usually are also computable numbers (Chaitin's constant being a significant exception).

Mathematical Proof - Theory

"In the 1930's, mathematician, Kurt Godel, established that there are statements which cannot be proved true or untrue within the axioms of a mathematical system. For a mathematical 'proof' only has meaning within the limited definitions, rules and conventions of the language of mathematics. So meaning cannot be found in numbers themselves, although patterns of order amongst them obviously can and may imply meanings."

The Primal Code (PDF)

Godels Incompleteness Theorem are two theorems of mathematical logic that demonstrate the inherent limitations of every formal axiomatic system containing basic arithmetic. Axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. Prime Numbers.

Gödel Numbering is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number, called its Gödel number. The concept was used by Kurt Gödel for the proof of his incompleteness theorems. (Gödel 1931). A Gödel numbering can be interpreted as an encoding in which a number is assigned to each symbol of a mathematical notation, after which a sequence of natural numbers can then represent a sequence of symbols. These sequences of natural numbers can again be represented by single natural numbers, facilitating their manipulation in formal theories of arithmetic. Since the publishing of Gödel's paper in 1931, the term "Gödel numbering" or "Gödel code" has been used to refer to more general assignments of natural numbers to mathematical objects.

Archimedes was a Greek mathematician, physicist, engineer, inventor, and astronomer. (287 – c. 212 BC)

Numerology is any belief in the divine, mystical relationship between a number and one or more coinciding events. Astrology.

19.47 Latitude

Finite Subdivision Rule is a recursive way of dividing a polygon or other two-dimensional shape into smaller and smaller pieces. Subdivision rules in a sense are generalizations of fractals. Instead of repeating exactly the same design over and over, they have slight variations in each stage, allowing a richer structure while maintaining the elegant style of fractals. Subdivision rules have been used in architecture, biology, and computer science, as well as in the study of hyperbolic manifolds. Substitution tilings are a well-studied type of subdivision rule.

E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8.

Lie Group is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.

LI Patterns Depicted in Ancient Cultures - Patterns in Nature

Lichtenberg Figure are branching electric discharges that sometimes appear on the surface or in the interior of insulating materials. Lichtenberg figures are often associated with the progressive deterioration of high voltage components and equipment. The study of planar Lichtenberg figures along insulating surfaces and 3D electrical trees within insulating materials often provides engineers with valuable insights for improving the long-term reliability of high voltage equipment. Lichtenberg figures are now known to occur on or within solids, liquids, and gases during electrical breakdown. Lichtenberg.

Brownian Tree are mathematical models of dendritic structures associated with the physical process known as diffusion-limited aggregation. A Brownian tree is built with these steps: first, a "seed" is placed somewhere on the screen. Then, a particle is placed in a random position of the screen, and moved randomly until it bumps against the seed. The particle is left there, and another particle is placed in a random position and moved until it bumps against the seed or any previous particle, and so on.

Triskelion is a motif consisting of a triple spiral exhibiting rotational symmetry. The spiral design can be based on interlocking Archimedean spirals, or represent three bent human legs.

Julia Set are two complementary sets (Julia 'laces' and Fatou 'dusts') defined from a function. Informally, the Fatou set of the function consists of values with the property that all nearby values behave similarly under repeated iteration of the function, and the Julia set consists of values such that an arbitrarily small perturbation can cause drastic changes in the sequence of iterated function values. Thus the behavior of the function on the Fatou set is 'regular', while on the Julia set its behavior is 'chaotic'.

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Math is at the base of all creation. The universe started with zero, then a one, and then a 2, and so on and so on. Everything in the universe, including humans, is part of a mathematical structure. All matter is made up of particles, which have properties such as charge and spin, but these properties are purely mathematical.

Plato believes that the truths of mathematics are absolute, necessary truths. He believes that, in studying them, we shall be in a better position to know the absolute, necessary truths about what is good and right, and thus be in a better position to become good ourselves. Count the things that matter.

Mathematics is based on logical reasoning and the assumption of premises called axioms. So, if we had to justify mathematics, we would need to explain the use of axioms as well as the use of logic itself.

"Mathematics no more describes the universe than atoms describe the objects they compose; rather math is the universe."

"Our reality isn't just described by mathematics, it is mathematics."

Where did math come from? When did math come into being? We did not create math, we realized math when someone realized thousands of years ago that there where patterns in life that can be measured and predicted using numbers as symbols that represent increments. History of Mathematics (wiki)

Math is the hidden secret to understanding the world: Roger Antonsen (video and interactive text)

Most of us never get to see the real mathematics because our current math curriculum is more than 1,000 years old. For example, the formula for solutions of Quadratic Equations was in Al-Khwarizmi's book published in 830, and Euclid laid the foundations of Euclidean Geometry around 300 BC. If the same time warp were true in physics or biology, we wouldn't know about the solar system, the atom or DNA.

"Everything that can be counted does not necessarily count; everything that counts cannot necessarily be counted." (Albert Einstein 1879 - 1955, German-born Theoretical Physicist)

Math is every where in nature and every where in human life. Mathematics is more then a language of measurement, math is the ability to encode and decode information. 1-1=0, If you keep subtracting from what you have you will eventually end up with nothing, which is the path that most of us are on. We are blindly ignoring one of the most constant things in the universe, which is math and our ability to calculate cause and effects.

Though understanding basic math is essential, understanding advanced math helps us to expand our minds and shows us that there are seemingly no real limits to human thinking. The possibilities and the potential of the human mind seems to be endless. It seems like our universe is expanding almost at the same speed as our understanding. So don't underestimate the power of math. Artificial Intelligence or almost any intelligence, can be best explained using math, but math alone can not explain everything about intelligence, math mostly helps us to measure particular elements and behaviors that can help us to understand what being intelligent is.

Though math explains many aspects of reality, math does not explain everything about reality. But math does help us start a conversation using the language of math, which all humans can equally understand, and ultimately help us to define certain elements in our reality. That is to a certain degree of course, because good communication depends on each persons level of knowledge and education. We can say that math can not explain consciousness, but we can use math to explain many of the elements that makes consciousness possible.

Math does more than just give us definitions to elements in our reality, math also gives us meaning and helps us to explain things and describe things on many different levels, which helps humans to communicate at a higher degree of understanding. Don't underestimate the power of math.

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