# Concepts

This page contains a collection of math concepts grouped by subject.

## Algebra

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.

A function takes input and produces output. The idea is a useful way to abstract away complexity and, especially in the age of computers, is a practical tool to solve problems.

The general form of the equation of a circle is given in terms of coefficients.

## Arithmetic

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.

## Calculus

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.

## Geometry

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.

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

Degrees are a unit of measure for angles. A full rotation is equal to 360 degrees. In the XY Cartesian Coordinate System, degrees are measured starting from the rightmost edge of the circle.

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 occurring number in nature.

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

The inscribed angle theorem states that an inscribed angle in a circle is equal to one-half the central angle.

The Pythagorean theorem equates the square of the sides of a right triangle together.

Radians are a unit that measures angle 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.

The circle constant τ (tau) is a geometric constant approximately equal to 6.283. The numeric value is defined as the length of any circle's circumference divided by the length of its radius.

There are six 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.

There are six 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.

## Linear Algebra

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.

## 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 rational numbers can be defined by the quotient of two numbers belonging to the set of integers, where the divisor is non-zero.

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

The normal distribution is a continuous probability distribution that appears naturally in applications of statistics and probability. The shape of the normal distribution forms a "bell curve".

A probability density function models probability over a continuous range. The probability of an event occurring between thresholds is equal to the area under the curve between the thresholds.

The standard normal distribution is the special case of a normal distribution with a mean of 0 and a standard deviation of 1. The distribution has historical significance because it allows standardized values to be referenced in a look-up table rather than calculated by hand.

## 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 rational numbers can be defined by the quotient of two numbers belonging to the set of integers, where the divisor is non-zero.

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 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.

Degrees are a unit of measure for angles. A full rotation is equal to 360 degrees. In the XY Cartesian Coordinate System, degrees are measured starting from the rightmost edge of the circle.

The difference of two angles identities expresses the cosine and sine of the difference of the two angles in terms of their individual components.

The double angle identities give the sine and cosine of a double angle in terms of the sine and cosine of a single angle.

The half angle identities give the sine and cosine of a half angle in terms of the sine and cosine of an angle.

The inscribed angle theorem states that an inscribed angle in a circle is equal to one-half the central angle.

The law of cosines is a more general form of the Pythagorean 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.

Ptolemy's Theorem relates the diagonal length of an inscribed quadrilateral to the lengths of its sides.

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 equates the square of the sides of a right triangle together.

Radians are a unit that measures angle 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.

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 sum of two angles identities are trigonometric identities that express the cosine and sine of the sum of two angles in terms of the trigonometry of the individual angles.

The circle constant τ (tau) is a geometric constant approximately equal to 6.283. The numeric value is defined as the length of any circle's circumference divided by the length of its radius.

There are six 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 are a set of equations derived from the properties of triangles and circles. The identities are used to transform and manipulate math expressions.

The unit circle is a circle of radius one placed at the origin of the coordinate system. A point's geometry on the unit circle corresponds to the input and the output of the trigonometric functions.

The unit circle chart shows the position of the points along the circle that are formed by dividing the circle into equal.

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

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.

The unit circle table contains values for the points along the unit circle. Each point is described by an angle and corresponds with a x and y component.