What is "order of operations"

n. (context mathematics English) The sequence in which the various operations in a mathematical expression are to be evaluated.

In mathematics and computer programming, the order of operations (or operator precedence) is a collection of rules that define which procedures to perform first in order to evaluate a given mathematical expression.

For example, in mathematics and most computer languages, multiplication precedes addition. Thus, in the expression 2 + 3 × 4, the answer is 14 (not 20). Brackets, ( and ), { and }, or [ and ] — which have their own rules — can indicate an alternate order or reinforce the default order to avoid confusion, thus the preceding expression could be changed to (2 + 3) × 4 to produce 20, or 2 + (3 × 4) to produce 14 (the default if there are no brackets).

Since the introduction of modern algebraic notation, multiplication has taken precedence over addition. Thus 3 + 4 × 5 = 4 × 5 + 3 = 23. When exponents were first introduced in the 16th and 17th centuries, exponents took precedence over both addition and multiplication and could be placed only as a superscript to the right of their base. Thus 3 + 5 = 28 and 3 × 5 = 75. To change the order of operations, originally a vinculum (an overline or underline) was used. Today, parentheses or brackets are used to explicitly denote precedence by grouping parts of an expression that should be evaluated first. Thus resulting in (2 + 3) × 4 = 20 to force addition to precede multiplication or (3 + 5) = 64 to force addition to precede exponentiation.