Editing Order of operations (section)

{{short description|Performing order of mathematical operations}}
{{distinguish|Operations order}}
[[File:Order of operations.svg|thumb|'''Order of operations''']]
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In [[mathematics]] and [[computer programming]], the '''order of operations'''  is a collection of rules that reflect conventions about which operations to perform first in order to evaluate a given [[mathematical expression]].

These rules are formalized with a ranking of the operations. The rank of an operation is called its '''precedence''', and an operation with a ''higher'' precedence is performed before operations with ''lower'' precedence. [[Calculator]]s generally perform operations with the same precedence from left to right,{{r|Microsoft}} but some [[programming language]]s and calculators adopt different conventions.

For example, multiplication is granted a higher precedence than addition, and it has been this way since the introduction of modern [[mathematical notation|algebraic notation]].{{r|BS}}{{r|Peterson}} Thus, in the expression {{nowrap|1 + 2 × 3}}, the multiplication is performed before addition, and the expression has the value {{nowrap|1 + (2 × 3) {{=}} 7}}, and not {{nowrap|(1 + 2) × 3 {{=}} 9}}. When exponents were introduced in the 16th and 17th centuries, they were given precedence over both addition and multiplication and placed as a superscript to the right of their base.{{r|BS}} Thus {{nowrap|3 + 5<sup>2</sup> {{=}} 28}} and {{nowrap|3 × 5<sup>2</sup> {{=}} 75}}.

These conventions exist to avoid notational [[ambiguity]] while allowing notation to remain brief.{{r|Swokowski}} Where it is desired to override the precedence conventions, or even simply to emphasize them, [[Bracket#Parentheses|parentheses]] ( ) can be used. For example, {{nowrap|(2 + 3) × 4 {{=}} 20}} forces addition to precede multiplication, while {{nowrap|(3 + 5)<sup>2</sup> {{=}} 64}} forces addition to precede [[exponentiation]]. If multiple pairs of parentheses are required in a mathematical expression (such as in the case of nested parentheses), the parentheses may be replaced by other types of [[bracket]]s to avoid confusion, as in {{nowrap|[2 × (3 + 4)] − 5 {{=}} 9}}.

These rules are meaningful only when the usual notation (called [[infix notation]]) is used. When [[functional notation|functional]] or [[Polish notation]] are used for all operations, the order of operations results from the notation itself.