# Algebra Index

Algebra is the branch of mathematics that uses symbols to represent unknown quantities and numbers.

## Notation

In mathematics functions are denoted using the name of the function followed by parentheses which contain the input to the function. If the function has multiple inputs they are separated by a comma.

The notation for taking the absolute value is two vertical lines on either side of the expression being evaluated.

The exclamation mark is used to represent the factorial of a number in mathematics. A factorial is a unary operator.

In mathematics superscript text is used to indicate the exponent of some number.

Logarithms are often abreviated as "log" followed by a subscript number representing the base and the number that the logarithm is being applied to.

The radical symbol by itself is used to denote taking the square root of a number.

The capital Greek letter Pi is used to represent the product operator in mathematics. The operator has three parts: an initial value, an end value, and the expression being evaluated.

A radical is used to represent fractional exponents. By itself it is used to represent the square root of an expression, but it also is used to represent higher roots as well.

Slope is denoted as the change in y over the change in x. The capital greek letter delta (Δ) is used to represent the change in a variable.

The capital Greek letter Sigma is used to represent the summation operator in mathematics.

A variable is a letter or symbol that represents a value that either changes or is unknown.

## Formulas

The Quadratic formula solves for the x-intercepts of a quadratic equation.

The slope formula gives the slope of a line. Slope is defined as the change in y over the change in x. Another way of thinking about slope is the rise over run.

## Operators

The capital letter Sigma Σ represents the summation operation in mathematics, often called sum for short. A sum has three parts: an initial value, an end value, and the expression being summed. The summation starts at the initial value and iterates, adding one to the value for each iteration, stopping when the value reaches the end value.

The notation for taking the absolute value is two vertical lines on either side of the expression being evaluated.

The exponentiation operator is a binary operator. The base is an expression or number that is being raised to some exponent. The exponent expression is denoted using superscript text.

The factorial operator is represented using the exclamation mark. The operator is unary, meaning that it only operates on one expression. The operator is useful when calculating combinations and permutations.

Taking the logarithm of an number is the inverse operation of exponentiation. The subscript number is the base of the logarithm and the expression is the operand.

The product operator multiplies the expression starting at an initial value, incrementing by one for each sub-expression, and ending at the end value.

The radical operator returns the n-th root of the provided expression. The radical operator is an alternative way of writing a fractional exponent.

Returns the square root of the provided expression.

## Concepts

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

A coeffecient is a constant value in a math expression that is given a symbol to represent it instead of a value.

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.

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

A mathematical operator defines operators for numbers and other constructs in math.

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.

A variable is a core concept in algebra where a symbol, usualy a lower case latin letter, is used as a placeholder in a math expression.

## Examples

This example calculates the geometric interpretation of a plus b squared.

To find the absolute value of a number you can calculate the number's distance from zero on the number line.

To approximate the square root of two you can use algebra to derive a recursive formula to come up with an arbitrarily precise number.

To derive the difference of two angles identities, two right triangles are placed next to eachother so their angles sum together to be one angle and one triangle's angle is the difference of the sum and the other.

To derive the formula for the distance between two points in two dimensions you can use the properties of the right triangle and Pythagorean's theorem to solve for the length of the hypotenuse.

The half-angle identities can be derived from the double angle identities by transforming the angles using algebra and then solving for the half-angle expression.

To derive the equation for the law of sines, observe the shared perpendicular line by two angles. The third angle can be included by repeating the process.

The law of sines can be derived using a triangle inscribed on the perimeter of a circle. The proof uses the inscribed angle theorem.

To find the point where two lines intersect set the equations equal together and solve for the x-coordinate. Then substitute the solved for coordinate back into one of the equations to get the y-coordinate.

To derive the quadratic formula, start with the general form of the quadratic equation set equal to zero and solve for the variable x.

The sum of two angles addition formula can be derived using a quadrilateral inscribed on a circle of diameter 1.

To derive the sum of two angles identities, two right triangles are placed next to eachother so their angles sum together, then their proportions are related together.

To find the distance between two points in one dimension, substitue the x values of the points into the formula for the distance between two points.

To find the distance between two points in one dimension, substitue the x values of the points into the formula for the distance between two points.

To calculate the factorial of a number multiply all of the numbers in the range from one to the number together.

To find the equation of a line given two points first calculate the slope of the line and then the y-intercept.

To find the midpoint between two points, average the two points x coordinate together to get the midpoint's x coordinate, then average the two points y coordinate together to get the midpoints y coordinate.

Examples of using the quadratic formula. The quadratic formula is a general way to find the solution to quadratic equation set to zero. This can be visualized in a graph as the points where the parabola crosses the x-axis.

To find where this quadratic equation crosses the x-axis, we can use the quadratic formula and substitute the values of the coefficients of the quadratic equation.

To find where this quadratic equation crosses the x-axis, we can use the quadratic formula and substitute the values of the coefficients of the quadratic equation.

To calculate the summation of the counting numbers 1 + 2 + 3 + ... + n the sequence can be visualized geometrically and solved for by finding the area of the shape formed.