Order of Operations

In mathematics and computer programming, the order of operations (sometimes called operator precedence) is a rule used to clarify which procedures should be performed first in a given mathematical expression.

For example, in mathematics and most computer languages multiplication is done before addition; in the expression 2 + 3 × 4, the answer is 14. Brackets, "( and ), { and }, or ", which have their own rules, may be used to avoid confusion, thus the preceding expression may also be rendered 2 + (3 × 4), but the brackets are unnecessary as multiplication still has precedence without them.

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 + 52 = 28 and 3 × 52 = 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, to force addition to precede multiplication, we write (2 + 3) × 4 = 20, and to force addition to precede exponentiation, we write (3 + 5)2 = 64.