Math Operators

Math operators define the basic operations that act on numbers and other math constructs. Typically, operators take one or two expressions as input and calculate a result as output. For two mathematicians to produce the result when given the same expression, the order of operations is defined so that the result is unambiguous.

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.

Subtraction 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 operators in the algebra section build upon the previous operators in the arithmetic section and add a variety of operators used to express 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 exponential growth and decay.

The absolute value operator returns the distance from zero on the number line of a number.

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 a 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 summation symbol ∑ and represents the operation of summing a sequence of expressions together.

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

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, linear transformations, and systems of linear equations. These operators act on vectors and matrices to produce results like transformations.

Vector addition is the operation of adding two vectors together. Visually, the operation can be thought of as placing the two vectors "tip to tail" and drawing the resulting vector.

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

The magnitude (or length) of a vector gives a scalar representation of its size, irrespective of its direction.

The dot product, also known as the scalar product, takes two vectors and returns a scalar. It measures the extent to which one vector goes in the direction of another.

Matrix multiplication is a way to compose linear transformations. In order to multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix.

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

The determinant operator calculates a scalar value from a square matrix.

Calculus

These calculus operators define the basic operations such as finding the rate of change of a function or the area underneath a curve.

The limit operator describes the result of an expression as a variable approaches a value. The operator is used in calculus to formalize what mathematicians mean by approach.

An integral can be geometrically interpreted as the area under the curve of a function between the two points a and b. Integrals are a core operator in calculus and are used throughout physics and higher-level mathematics.

The derivative is one of the main operators in calculus.

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 to test for lexicographical 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 (abbreviated 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 operator returns true if the left and right-hand side expressions evaluate to true, or if the left-hand expression is false.

The logical if and only if operator, or iff for short, returns true if both A and B are false or if both A and B are true.

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.