# Operators

Math operators define the basic operations that act on numbers and other such math constructs. Typically, operators take between one and two numbers as input and return a number as output.

## Basic Operators

The basic arithmetic operators are addition, subtraction, multiplication, and division. They are typically introduced in elementary mathematics and describe the basic manipulation of numbers. As the notion of a number becomes more complex, the details of how they work are expanded to include the ability to work on more complex numbers. For example, in algebra the operators are expanded to work on fractions as well as the regular counting 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.

Division is the process of dividing a number into equal parts.

## Advanced Operators

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

## Boolean Logical Operators

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 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 Operators

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 operation is denoted using the cap symbol.

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".

## Calculus Operators

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 is a concept related to calculus and the slope of a function.

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.