# Concepts

This page contains a collection of concepts relating to mathematics. Each concept is linked together with relating concepts and examples.

## Algebra

Chebyshev are the roots to a special series of polynomial equations.

In math equality is an assertion that the left expression and the right expression are equivalent.

The standard form of a circle is given by the radius and center point of the circle.

The equation of a line can be expressed in the form of y is equal to ax plus b.

The shape of a parabola is formed by the quadratic equation which is a polynomial equation of degree two.

The standard form of the equation of a circle is given in terms of its radius and center point.

There are six properties of the exponent operator: the zero property, identity property, negative property, product property, quotient property, and the power property.

This page contains the properties of logarithms.

The general form of the quadratic equation describes all possible equations of the second degree. The shape formed by the quadratic equation is called a parabola.

The reciprocal of a number is given by one divided by the number. When the number and its reciprocal are multiplied together the result should be one.

## Arithmetic

Counting is the process of calculating the quantity of a group of objects.

The fundamental theory of arithmetic states that every number greater than 1 is either a prime number or composed by a unique product of prime numbers.

In math an interval represents the valid values between an upper limit and a lower limit.

There are three propreties of addition: the commutative property, associative property, and the identity property.

There are four properties of the multiplication operation in mathematics: the commutative property, associative property, identity property, and distributive property.

## Calculus

The fundamental theorem of calculus relates integration to differentiation by defining the integral of a continuous function on a closed bounded interval.

The intermediate value theorem states that if f is a continuous function an contains the interval [a,b], then f takes on every value at least once between f(a) and f(b) at some point in the interval.

The mean value theorem states that for a function f that is continuous on the closed interval [a,b] and differentiable on the interval (a,b) there exists some point c where a < c < b whose slope is equal to the slope formed by the points a and b.

Riemann Sum is a method for approximating the area underneath a continuous function. To find the Riemann's sum, divide the area under the curve into n equal width rectangles. Then calculate the area of each rectangle and sum the results together.

A series is the operation of adding an infinite number expressions together. A series is denoted using the summation operator and the index variable k.

## Geometry

An acute angle is an angle that is smaller than 90 degrees or PI fourths.

An angle is defined as the amount of rotation between two rays. Angles are measured using degrees and radians. A full rotation in degrees is 360°. A full rotation in radians is approximately 6.283 radians or τ (tau) radians.

Area is the physical amount of two dimensional space that a shape takes up. Area is measured in square units.

The Cartesian Coordinate System describes space of one, two, and three dimensions. Each point in space is represented by its distance relative to the origin of the system.

Chebyshev are the roots to a special series of polynomial equations.

This page describes the basic elements and properties of a circle.

Geometric construction is a classic form of math that studies building forms and shapes using a compass and straight edge.

Degrees are a unit that measures angle of rotation as a fraction of the number 360.

The golden ratio is a number represented by the greek letter ϕ (phi). The value of ϕ is approximately 1.618 and is a naturally occuring number in nature. The golden ratio is often associated with the golden rectangle whose sides form a ratio equal to ϕ.

The golden rectangle is a rectangle whose width divided by height is equal to the golden number (approximately 1.618).

In math an interval represents the valid values between an upper limit and a lower limit.

An obtuse angle is an angle that is larger than 90 degrees or PI fourths.

A Perpendicular angle, sometimes also referred to as a square angle, is exactly 90 degrees or PI fourths.

A Perpendicular angle, sometimes also referred to as a square angle, is exactly 90 degrees or PI fourths.

The greek letter π (pi) is a naturally occuring number that is defined by any circle's circumference divided by its diameter.

A point represents a position in space. In modern mathematics, space is represented using the cartesian coordinate system where there is an origin and the position of a point in space is measured by its distance from the origin.

The Polar Coordinate System describes points in space using an angle and radius relative to the origin.

The pythagorean theorem is an equation that equates the square of the sides of a right triangle together.

Radians are a unit that measure angles using the radius of a circle. One radian is equal to the amount of rotation required to travel the length of one radius along the circumference of the circle.

Slope is the concept of how much a function is changing with respect to x.

This page describes the basic elements and properties of a sphere.

τ (tau) is a geometric constant approximately equal to 6.283. The number is naturally occuring as any circle's circumference divided by its radius.

A triangle is a three sided geometric shape. The shape forms a basis for the subject of trigonometry and is used throughout mathematics and programming.

There are six total trigonometric functions that relate to the geometry of the right-triangle sine, cosine, tangent, cosecant, secant, and cotangent. The functions take the angle of a right triangle as input and return a ratio of two of its sides.

Volume measures the amount of three dimensional space an object occupies.

## Linear Algebra

The determinant is a function that maps the values in a matrix to a number. Using this number, certain properties can be detirmined. For example, like whether or not the matrix has an inverse.

The identity matrix is a function that when given a vector as input outputs an identitical vector as ouput.

The image of a mapping between two vector spaces is the set of all vectors that map to a non-zero vector.

The kernel represents the set of vectors that map to the zero vector when a matrix is applied to a vector.

In math, a matrix is a function that maps between two vector spaces. A matrix can also be thought of as a shorthand way to write a system of linear equations.

Polynomial interpolation is the process of approximating an unknown function by fitting a polynomial curve to data.

A vector has direction and magnitude.

A vector space is ...

## Logic

Induction is used in a math as a technique for proving differenct concepts and theories. The process is divided into three steps: base case(s), induction hypothesis, and the induction step.

## Numbers

Counting is the process of calculating the quantity of a group of objects.

Mathematical constants are symbols that represent useful numbers.

A prime number is a number that is only divisible by itself and one. The set of prime numbers is infinitely big. The first prime numbers are 2, 3, 5, 7, 11, ... and continues on forever.

The set of complex numbers contains all possible complex numbers. Each complex number has a real part and an complex part.

The set of integers is made up of the set of counting integers and each of their negative counter parts.

The set of natural numbers, also called the counting numbers, contains the numbers 0, 1, 2, 3, ...

The set of real numbers contains the set of rational numbers as well as irrational numbers like pi, e, and the square root of two.

## Probability

A random variable represents an event whose outcome is unknown.

Bayes theorem describes a way of expressing conditional probability.

## Set Theory

Sets of numbers are often discussed in mathematics in relation to the domain of certain problems and applications.

The set of complex numbers contains all possible complex numbers. Each complex number has a real part and an complex part.

The set of integers is made up of the set of counting integers and each of their negative counter parts.

The set of natural numbers, also called the counting numbers, contains the numbers 0, 1, 2, 3, ...

The set of real numbers contains the set of rational numbers as well as irrational numbers like pi, e, and the square root of two.

## Trigonometry

The unit circle is a unifying idea in mathematics that connects many useful concepts together. This article goes over the basic properties of the circle using interactive examples and explains how they connect to the trigonometric functions and pythagorean theorem.

The triangle defined by the three angles: 30 degrees, 60 degrees, and 90 degrees is a special triangle that has meaningful properties in mathematics.

The triangle defined by the three angles: 45 degrees, 45 degrees, and 90 degrees is a special triangle that has meaningful properties in mathematics.

An angle is defined as the amount of rotation between two rays. Angles are measured using degrees and radians. A full rotation in degrees is 360°. A full rotation in radians is approximately 6.283 radians or τ (tau) radians.

Degrees are a unit that measures angle of rotation as a fraction of the number 360.

The law of cosines is a more general form of pythagoreans theorem that relates the squares of the sides together using the cosine function.

The law of sines is an equation that relates the three sides of a triangle with the three angles of a triangle using the sine function.

The pythagorean identity relates the sides of the right triangle together using only the angle of the right triangle. The identity is derived using pythagorean's theorem and the properties of the unit circle.

The pythagorean theorem is an equation that equates the square of the sides of a right triangle together.

Radians are a unit that measure angles using the radius of a circle. One radian is equal to the amount of rotation required to travel the length of one radius along the circumference of the circle.

A right triangle is a triangle where one of the three angles is a perpendicular angle. There are three sides of the right triangle: the adjacent, opposite, and hypotenuse sides.

Similar Triangles are two triangles that share the same three angles making them proportional to each other.

There are two special right triangles in geometry defined by their three angles the 45 45 90 degrees and the 30 60 90 degrees.

The double angle identity...

A triangle is a three sided geometric shape. The shape forms a basis for the subject of trigonometry and is used throughout mathematics and programming.

There are six total trigonometric functions that relate to the geometry of the right-triangle sine, cosine, tangent, cosecant, secant, and cotangent. The functions take the angle of a right triangle as input and return a ratio of two of its sides.

The trigonometric identites are a set of equations derived from the properties of the right triangle.

The unit circle chart shows the position of the points along the circle that are formed by dividing the circle into eight and twelve parts.