# Math Operators

Math operators define the basic operations that act on numbers and other math constructs. Typically, operators take between one and two numbers as input and return a number as output. In order for two mathematicians to produce the same result when given an expression, an order of operations is defined so that the result is unambigous.

## Arithmetic

The basic arithmetic operators are addition, subtraction, multiplication, and division. Introduced in elementary mathematics, they describe ways to manipulate numbers. As the notion of a number becomes more complex, their definitions are expanded beyond discrete numbers and properties are introduced to make the concept of a number and operations more cohesive. For example, in algebra the operators are expanded to work with fractions and partial numbers.

Addition is a basic operation in mathematics for combining two numbers together. It is a binary operation denoted with the plus symbol with an expression on the left and an expression on the right.

Subraction is a basic arithmetic operation of taking away one number from another number.

Multiplication is a basic arithmetic operation performed on two numbers. Multiplying a number by another number is the same as taking n groups of the other number.

The division operator returns the result of dividing one number by another number.

## Algebra

These advanced operators cover some more complicated patterns that appear in mathematics. For example, the factorial operator represents the patterns found in combinations and permutations. Another example is the exponent and logarithm operators which describe expoenential growth and decay.

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

Returns the square root of the provided expression.

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

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.

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 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.

The modulus operator returns the remainder of dividing the first expression by the second expression.

The summation operator is represented by the symbol Σ (capital sigma) and represents the operation of summing a sequence of expressions together. The operator is used in math to represent the sequence and series.

The product operator is represented by the Π (capital pi) symbol and is used to represent the operation of multiplying a sequence of expressions together.

## Linear Algebra

The cross product operates on two vectors and produces another vector as a result.

The magnitude of a vector operator, denoted by two vertical lines on either side of the expression, returns the length of the vector.

The dot product takes in two vectors as input and produces a number as output.

Matrix multiplication composes two matrices together to create a third matrix function.

The transpose of a matrix is an operator which reflects a matrix accross its diagonal.

The deterimnant operator calculates a scalar value from a square matrix

## Calculus

The syntax for a limit is a the abbreviation "lim" followed an expression. Underneath the letters "lim" is the value the variable approaches within the expression denoted as the variable with an arrow to the value it is approaching.

An integral can be geometrically interpretted as the area under the curve of a function between the two points a and b. The concept is used throughout physics and higher level mathematics.

The derivative of a functin with respect to a variable returns a function that represents the change in the function with repsect to time.

The gradient operator returns a vector representing the change in a function at a point. The operator is similar to the derivative operator of calculus, the difference being that it operates on functions of higher dimension.

## Boolean Logic

The boolean logic operators operate on boolean expressions - values that are either true or false. Typically, the binary boolean operators take in two boolean values and return a boolean value as a result. In computing, numbers and more complex data can be compared with data of the same type in order to test for lexigraphical order (less than, greater than) and equality to produce a boolean value.

The logical and operator returns true if both the left side expression and the left side expression evaluate to true, otherwise the operator returns false.

The logical "or" operator returns true if either the left side expression evaluates to true or the right side expression evaluates to true, otherwise returns false.

The logical exclusive or (abreviated as xor) operator returns true if the left side evaluates to true and the right side evaluates to false. The operator also returns true if the left side evaluates to false and the right side evaluates to true. Otherwise, returns false.

The logical implication operators returns true if the left and right hand side expressions evaluate to true, or if the left-hand expression is false.

The negation symbol is used to reperesent the unary operator for negation, which inverts the value of the expression it is applied to.

The symbol < represents the logic expression that the left side is less than the right side.

The symbol > represents the logical expression that the left side is greater than the right side.

Two stacked horizontal lines respresents the equals symbol in mathematics. The two expressions on either side are equal, or the same, when the equal sign is placed in between them.

## Set

The set operators are binary operators used in set theory to operate on sets.

The set union operation is denoted using the cup symbol. The union of two sets returns the combined elements of both sets. Duplicates are ignored.

The set intersection operator returns the shared elements between two sets. The operator is denoted using the cap symbol. - set theory

The minus symbol is used in set theory to represent the difference operator for two sets. The operation removes all elements found in one set from another and returns the resulting set.

The subset operator is denoted using a U shapes symbol rotated ninety degrees to the righ with a horizontal line underneath.

The superset operator in set theory is denoted using the superset symbol which looks like a U turned ninety degrees counter clockwise with a horizontal line underneath.

A proper subset is denoted by the subset symbol which looks like a U rotated ninety degrees to the right.

A proper superset is denoted by the superset symbol which looks like a U rotated ninety degrees to the left.

The element of symbol describes membership to a set. When reading an equation the symbol can be read as "in" or "belongs to".