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Mathematics is the science of Measuring things and Calculating things in our world.  A science (or group of related Sciences) dealing with the logic of quantity (numbers), structure, patterns, space, change, shape and arrangement. A tool that is used to help make sense of the world. Used in Engineering, Reasoning, Decision Making, Planning and Problem Solving, to name a few.  Teaching Math

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

Mathematical intelligence is being number Smart and being good at Reasoning using Math. The ability to determine the number or amount of. The ability to correctly apply Mathematics when needed. Consists of the Capacity to Analyze Problems Logically, carry out Mathematical Operations, 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.

What Math Skills Are Needed to Become an Engineer?


Mathematics Education is the practice of teaching and learning mathematics, along with the associated scholarly research.
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.
Outline of Mathematics (PDF)
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.
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.
Discrete Mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous.
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.
Mathematical Visualization is an aspect of geometry which allows one to understand and explore mathematical phenomena via visualization.
Physics Math
Combinatorial Game Theory  is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Math Games
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.
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).
Mathematical Analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.
Formula is a concise way of expressing information symbolically as in a mathematical or chemical 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.
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, 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.
Math Mnemonics (PDF)

Evaluation (testing)

Films about Math - Math Videos


The branch of Pure Mathematics dealing with the theory of numerical Calculations.
Calculation is to judge to be probable. Predict in advance. 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. Measurement
Mental Calculation comprises arithmetical calculations using only the human brain, with no help from calculators, computers, or pen and paper.
Counting is the action of finding the number of elements of a finite set of objects.
Equation is 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 Solving finding an answer to a set of variables using a mathematical function like adding or subtraction.
Combination is a way of selecting items from a collection.
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.
Computation is the procedure of calculating; determining something by mathematical or logical methods. Problem solving that involves numbers or quantities.

Calculator is a small, portable electronic device used to perform operations ranging from basic arithmetic to complex mathematics.
Download Calculators for PC
Mathematical Integrals Calculator
Wolfram Alpha


Numbers Written 1-100 Number is a mathematical object used to count, measure, and label.
Prime Number is a Natural Number greater than 1 that has no positive divisors other than 1 and itself. (5 is a Prime)
Composite Number is a positive integer that can be formed by multiplying together two smaller Positive Integers.
Complex Number (wiki)
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.
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 42 = (−4)2 = 16.
Number Theory is a branch of pure mathematics devoted primarily to the study of the integers.
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 (wiki)

Large Numbers
More Numbers

"Crunching the Numbers"

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.

Plus Sign for Addition

Adding - Addition

Addition Worksheet 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. Make an addition (to); join or combine or unite with others; increase the quality, quantity, size or scope of. 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 judgements of numbers performed for small numbers of items.
List of Numbers
Enumeration is a complete, ordered listing of all the items in a collection.
Summation is the addition of a sequence of numbers; the result is their sum or total.
Sigma Summation Symbol

Minus Sign for Subtracting

Subtracting - Minus

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.

Divide Symbol


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 dividing or partitioning; separation by the creation of a boundary that divides or keeps apart.
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, 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.


Equivalent Fractions
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

Multiplication Sign (Times)


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.
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.

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:


Algebra is when several of the factors of a problem are known and one or more are unknown. Algebra uses alphabetic charactersAlgebra Worksheet representing a number which is either arbitrary or not fully specified or unknown.
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 {\displaystyle x+2=5} x+2=5 the letter x {\displaystyle x} x is unknown, but the law of inverses can be used to discover its value: x = 3 {\displaystyle x=3} x=3. In E = mc2, the letters E {\displaystyle E} E and m {\displaystyle m} m are variables, and the letter c {\displaystyle c} 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.
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 (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
Symbols (letters)
Logic Symbols (wiki)
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.
Mathematical Symbols (wiki)

Deductive Reasoning
Converse (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."


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.Natural Number
Euclidean geometry consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these.
Congruence (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.
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".
Grade-school students teach a robot to help themselves learn geometry

Shapes  -  Dimensions  -  Patterns

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.
Dimensions (space)
Mathematica Object is an abstract object arising in mathematics.
File:Schlegel wireframe 8-cell.png 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.
Tetrahedron is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces.
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.
Edge (geometry) is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope.
Vertex (geometry) is a point where two or more curves, lines, or edges meet.
Rectangle is a quadrilateral with four right angles.
Triangle is a polygon with three edges and three vertices.
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 is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.
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

Circumference Diameter Radius Pi 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 centre is called the radius. Circle
Radius 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.
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)

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.
Spiral is a curve which emanates from a point, moving farther away as it revolves around the point.
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.
Water Spheres in Space

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.

Polygon Polygon is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit. These segments are called its edges or sides, and the points where two edges meet are the polygon's vertices (singular: vertex) or corners. The interior of the polygon is sometimes called its body. An n-gon is a polygon with n sides; for example, a triangle is a 3-gon. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions.
Octagon is an eight-sided polygon or 8-gon.
Pythagorean Theorem
Truncated Octahedron
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.
Weaire-Phelan Structure is a complex 3-dimensional structure representing an idealised foam of equal-sized bubbles.
Pentagram is the shape of a five-pointed star drawn with five straight strokes.
Three-Dimensional Space
Pentagon Tiling is a tiling of the plane where each individual piece is in the shape of a pentagon.
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.
The 15th kind of Pentagon that can Tile a Plane 
List of Geometric Shapes
Geometric Shapes
Goldberg Polyhedron is a convex polyhedron made from hexagons and pentagons.
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 (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.
Tetrahedron 64Mathematics of Paper Folding - Box Pleat
Programmable Matter

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 neighbourhoods.

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.

Factorization is to reduce something to “basic building blocks”, such as numbers to prime numbers, or polynomials to irreducible polynomials.
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.

Abacuses 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.

Problem Solving

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

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.

Mathematical Model is a description of a system using mathematical concepts and language.
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.
Probabilistic Graphical Models (coursera)

Axiom is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek axíōma (ἀξίωμα) 'that which is thought worthy or fit' or 'that which commends itself as evident.

Quadratic Equation (wiki)

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.

Graph a means of representing data.
Graph Theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices, nodes, or points which are connected by edges, arcs, or lines. A graph may be undirected, meaning that there is no distinction between the two vertices associated with each edge, or its edges may be directed from one vertex to another.
Charts is a graphical representation of data, in which "the data is represented by symbols, such as bars in a bar chart, lines in a line chart, or slices in a pie chart". A chart can represent tabular numeric data, functions or some kinds of qualitative structure and provides different info.
Scatter Plot is a type of plot or mathematical diagram using Cartesian coordinates to display values for typically two variables for a set of data. If the points are color-coded, one additional variable can be displayed. The data is displayed as a collection of points, each having the value of one variable determining the position on the horizontal axis and the value of the other variable determining the position on the vertical axis
Funnel Plot is a graph designed to check for the existence of publication bias; funnel plots are commonly used in systematic reviews and meta-analyses. In the absence of publication bias, it assumes that studies with high precision will be plotted near the average, and studies with low precision will be spread evenly on both sides of the average, creating a roughly funnel-shaped distribution. Deviation from this shape can indicate publication bias.
Image Plot Maker
Wave Metrics
Mind Maps


Trigonometry is the science of measuring triangles, which 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.
A 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.
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
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). Trigonometric Functions
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.
Deductive Reasoning


Calculating changes and calculating the actions needed to correctly adjust to these changes.Calculus Problem Derivative
Calculus is the mathematical study of change, in 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.
Differential Calculus
Integral Calculus
Multivariable Calculus
MIT 2006 Integration Bee Competitive Calculus (youtube)
Deductive Reasoning

Operation (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 isDifferentials Time the instantaneous change of one quantity relative to another; df(x)/dx A quality that differentiates between similar things. The result of mathematical differentiation; 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. 
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.
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.


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.
Statistics WorksheetErrors
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 Significance is attained whenever the observed p-value of a test statistic is less than the significance level defined for the study.
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.
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.
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
Mind Maps
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 (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 is the property of a mathematical relationship or function which means that it can be graphically represented as a straight line. Examples are the relationship of voltage and current across a resistor (Ohm's law), or the mass and weight of an object. Proportionality implies linearity, but linearity does not imply proportionality.
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 (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.
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
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.

Sample (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.
Sampling (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.
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

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.

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

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.
Odds is calculating the likelihood that the event will happen or not happen, using a numerical expression usually expressed as a pair of numbers.
Ratio is a relationship between two numbers indicating how many times the first number contains the second. Scale
Variables is an alphabetic character representing a number, called the value of the variable, which is either arbitrary or not fully specified or unknown.
Probability Distribution is a mathematical description of a random phenomenon in terms of the probabilities of events.
Propensity Probability a physical propensity, or disposition, or 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.
Reliability (statistics) is consistency that produces similar results under consistent conditions.

Average is 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.
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.
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).
Luck - Comparisons

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.
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.
Approximation is anything that is similar but not exactly equal to something else.
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
Approximate Number System an adult could distinguish 100 items versus 115 items without counting.
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

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.
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.

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
Errors and Residuals
Observation Errors

Google Analytics
Watson Analytics

Piwik Analytics Software
Web Analytics
Mind Maps 

Calibration is the process of finding a relationship between two quantities that are unknown (when the measurable quantities are not given a particular value for the amount considered or found a standard for the quantity). When one of quantity is known, which is made or set with one device, another measurement is made as similar way as possible with the first device using a second device. The measurable quantities may differ in two devices which are equivalent. The device with the known or assigned correctness is called the standard. The second device is the unit under test, test instrument, or any of several other names for the device being calibrated.

"It's good to have data, but remember to always know how the data was collected because the numbers could be misleading. There may also be bias in the research or even mistakes made, so always check for accuracy before making a decision on what action to take or when determining how to use data."

Videos that Teach Math  -  Math Films

Khan Academy: Learning 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
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
Math Symbols
Math Trick: Multiply Numbers Close To Each Other In Your Head (youtube)

Teaching Math

Not Just on Paper

If you are teaching Math then you should use real life examples that relate 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 you use real life situations when learning math is the associations. When you associate knowledge with other knowledge that is used more often, you remember it more often, so the knowledge stays with you. 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.

Real life Preparation has to be the goal in all educational courses.
Example Choice when students see a connection between physics and the real world, they learn easier because the subject is more interesting and relevant to their daily life.
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.
Demonstration Teaching involves showing by reason or proof, explaining or making clear by use of examples or experiments.
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.
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.
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
Learning Methods
Math for America Classroom Lessons
Count the things that Matter
Read to Learn Real Life Examples

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.

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.

Mathematical Proof

"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 

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

"All is Number"
Pythagoreanism (wiki)
Archimedes (wiki) 

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.

We did not create math, we discovered math. 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.

Math Contests - Math Competitions

Mathematical Olympiad
The International Mathematical Olympiad
MOSP (wiki)
USA Math Camp Advanced Mathematics ("cool math")

How to Multiply Large Numbers in your Head

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

Systems  -  Principles  -  Standards

Order of Operations  PEMDAS  Please Excuse My Dear Aunt Sally 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
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

Some of the things that Math can Do

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. (psychology) the performance of some composite cognitive activity; an operation that affects mental contents. (mathematics) calculation by mathematical methods.

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 (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.

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.
Level of Measurement is a classification that describes the nature of information within the numbers assigned to variables.
Metrology is the science of measurement and includes all theoretical and practical aspects of measurement.
Calibration - Statistics
Metric System by Country

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
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.
Quantification (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 (linguistics) is a type of determiner, such as all, some, many, few, a lot, and no, (but not numerals) that indicates quantity.
Quantifier (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.

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.

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.

Weight is the vertical force exerted by a mass as a result of gravity. 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.

Dimension is one of three Cartesian coordinates that determine a position in space. A construct whereby objects or individuals can be distinguished. (physics) 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 (how big 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)

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)

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 Height

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.

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

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. 
Cycle (wiki)
Seasons (earth)
Life-Cycle Assessment

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).

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.
Measuring Value

Count the things that Matter

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 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.  

"If you don't count the things that matter, then knowing how to count won't matter." 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.

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
Hidden Costs

Capstone Project # 1

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 then 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   ..."If you don't measure the things that count, then knowing how to measure will not benefit you."

Eventually we'll have a human action calculator on your computer, or an app on your smart phone, that when you put in your action, and your numbers, it calculates time, people, resources, impacts, skills, tools, assistance, and any other parameters that may apply to the current condition, then displays this information so that it can be quickly understood. This way people can easily see the reality of a particular action along with the positive and negative effects of this action. People can then make better decisions and plan more effectively, and also see what other choices and options there may be. 
Rating System

Critical Thinking and Technology
Cause and Effect
Problem Solving

Investigative Dashboard

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

Students must still learn to Verify and Confirm these Calculations manually, and also, fully understand the process.
This way we can minimize errors, while at the same time, not be vulnerable or totally dependent on our electronic technology tools.

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.
Pattie Maes demos the Sixth Sense (youtube)
Rating System
Responsibly Produced Rating

"Criminals know how to use a calculator better then the general public does. That's one of the reasons why education fails to prepare students effectively. If you educate students to be smarter then 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 then 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.

Cause and Effect
Stop Teaching Calculating, Start Learning Maths! - Conrad Wolfram

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.

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.

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.

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

Life by the Numbers

PiPi Symbol  3.14159 is the ratio of a circle's circumference to its diameter.
Tau or Pi (youtube)
Approximations of Pi
Phi also used as a symbol for the Golden Ratio and on other occasions in math and science.

Fractals - Mandelbrot Set

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 CurvesInfinity 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.
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.

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.

Pi Plate "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.
Bytes - Size
Names of Large Numbers
Law of Large Numbers

Natural Number are those used for counting.  Numbers
Ordinal Number is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another.
Cardinal Number is the number of elements in a mathematical set; denotes a quantity but not the order. Cardinal Number are a generalization of the natural numbers used to measure the cardinality (size) of sets.
Trinity is the cardinal number that is the sum of one and one and one. Three people considered as a unit.
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 (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 s 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.
Prime Number is a natural number greater than 1 that 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.
Largest known Prime Number a number with 17,425,170 digits
Great Internet Mersenne Prime Search
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, ornatural 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
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 an + bn = cn for any integer value of n greater than two. The cases n = 1 and n = 2 have been known to have infinitely many solutions since antiquity.
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.
PolynomialPhi Symbol 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

Euler Identity
Euler Identity - "It 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."
Leonhard Euler (15 April 1707 – 18 September 1783) 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.
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)

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
Math is Beautiful to the Mind

Eugene Wigner (November 17, 1902 – January 1, 1995), 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

Hermann Minkowski (22 June 1864 – 12 January 1909) 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.

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.

Beautiful Numbers

Face Symmetry Symmetry agreement in dimensions, due proportion, arrangement. A sense of harmonious and beautiful proportion and balance
Symmetry Number of an object is the number of different but indistinguishable (or equivalent) arrangements (or views) of the object.
Symmetry in Biology is the balanced distribution of duplicate body parts or shapes within the body of an organism.
Facial Symmetry is one specific measure of bodily asymmetry. 1.618

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.

Gauge Symmetry (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.

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.

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.

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.

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 1.618
Flower of Life 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. Torus
Nikola Tesla 3 6 9 (youtube)
Vortex Based Math  
Peter cullinane passing on {360* vortex maths divine symmetry} (youtube)

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

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.

Leonardo da Vinci's Vitruvian Man (ca. 1487)

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.
Mandelbrot Set Fracta Plot the Mandelbrot Set By Hand

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.
Earth Fractals
Fractal Jigsaw (youtube)
Scale Invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor.
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.

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.

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. Early Greek philosophers studied pattern, with Plato, Pythagoras and Empedocles attempting to explain order in nature. The modern understanding of visible patterns developed gradually over time.

E8 Lie Group Crystallization is the (natural or artificial) process where a solid forms where the atoms or molecules are highly organized in a structure known as a crystal. Some of the ways which crystals form are through precipitating from a solution, melting or more rarely deposition directly from a gas. Crystallization is also a chemical solid–liquid separation technique, in which mass transfer of a solute from the liquid solution to a pure solid crystalline phase occurs. In chemical engineering crystallization occurs in a crystallizer. Crystallization is therefore related to precipitation, although the result is not amorphous or disordered, but a crystal.

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.

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.

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

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.

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
Cymatics is when patterns emerge in the excitatory medium depending on the geometry of the plate and the driving frequency

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 (for example English) using technical terms and grammatical conventions that are peculiar to mathematical discourse (Mathematical jargon), supplemented by a highly specialized symbolic notation for mathematical formulas. Language of Math
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.

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

Virtual Particle is a transient fluctuation that exhibits many of the characteristics of an ordinary particle, but that exists for a limited time. The concept of virtual particles arises in perturbation theory of quantum field theory where interactions between ordinary particles are described in terms of exchanges of virtual particles. Any process involving virtual particles admits a schematic representation known as a Feynman diagram, in which virtual particles are represented by internal lines

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 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , 89 , 144 , …
Fibonacci Zoetrope Sculptures (youtube)

Frequencies (HZ)
Sound Shapes

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

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.

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'.

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.

The Thinker Man