# Concepts

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

## Algebra

Chebyshev Extrema | Concept

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

Equality | Concept

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

Equation of Circle | Concept

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

Equation of Line | Concept

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

Equation of Parabola | Concept

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

General Equation of Circle | Concept

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

Properties of Exponents | Concept

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

Properties of Logarithms | Concept

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.

Reciprocal | Concept

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 | Concept

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

Fundamental Theorem of Arithmetic | Concept

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.

Interval | Concept

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.

Properties of Multiplication | Concept

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

## Calculus

Fundamental Theorem of Calculus | Concept

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

Intermediate Value Theorem | Concept

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.

Mean Value Theorem | Concept

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 | Concept

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.

Series | Concept

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

Acute Angle | Concept

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

Angle | Concept

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 | Concept

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

Cartesian Coordinate System | Concept

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 Extrema | Concept

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

Circle | Concept

Compass and Straight Edge Construction | Concept

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

Degrees | Concept

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

Golden Ratio | Concept

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.

Golden Rectangle | Concept

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

Interval | Concept

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

Obtuse Angle | Concept

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

Perpendicular Angle | Concept

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

Perpendicular Lines | Concept

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

Pi | Concept

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

Point | Concept

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.

Polar Coordinate System | Concept

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

Pythagorean Theorem | Concept

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 | Concept

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

Sphere | Concept

Tau | Concept

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.

Triangle | Concept

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.

Trigonometric Functions | Concept

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 | Concept

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

## Linear Algebra

Determinant | Concept

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.

Identity Matrix | Concept

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

Image | Concept

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

Kernel | Concept

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

Matrix | Concept

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 | Concept

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

Vector | Concept

A vector has direction and magnitude.

Vector Space | Concept

A vector space is ...

## Logic

Induction | Concept

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 | Concept

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

Constants | Concept

Mathematical constants are symbols that represent useful numbers.

Prime Numbers | Concept

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.

Set of Complex Numbers | Concept

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

Set of Integers | Concept

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

Set of Natural Numbers | Concept

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

Set of Real Numbers | Concept

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

Random Variable | Concept

A random variable represents an event whose outcome is unknown.

Bayes Theorem | Concept

Bayes theorem describes a way of expressing conditional probability.

## Set Theory

Number Sets | Concept

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

Set of Complex Numbers | Concept

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

Set of Integers | Concept

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

Set of Natural Numbers | Concept

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

Set of Real Numbers | Concept

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

Unit Circle | Concept

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.

30 60 90 Triangle | Concept

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

45 45 90 Triangle | Concept

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

Angle | Concept

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 | Concept

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

Law of Cosines | Concept

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

Law of Sines | Concept

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.

Pythagorean Identity | Concept

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.

Pythagorean Theorem | Concept

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.

Right Triangle | Concept

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 | Concept

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

Special Triangle | Concept

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

Sum of Two Angles Identities | Concept

The double angle identity...

Triangle | Concept

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.

Trigonometric Functions | Concept

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.

Trigonometric Identities | Concept

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

Unit Circle Chart | Concept

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.