Order of operations

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a> and <a href="/facts/Computer_programming/QmSsFnu0">computer programming</a>, the order of operations is a collection of rules that reflect conventions about which operations to perform first in order to evaluate a given <a href="/facts/Mathematical_expression/MPrMlYbE">mathematical expression</a>.
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. <a href="/facts/Calculator/eeLfRS8L">Calculators</a> generally perform operations with the same precedence from left to right, but some <a href="/facts/Programming_language/cWTYbgWa">programming languages</a> 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 <a href="/facts/Mathematical_notation/vHgEE8cE">algebraic notation</a>. Thus, in the expression 1 + 2 × 3, the multiplication is performed before addition, and the expression has the value 1 + (2 × 3) = 7, and not (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. Thus 3 + 52 = 28 and 3 × 52 = 75.
These conventions exist to avoid notational <a href="/facts/Ambiguity/mgn4oEoQ">ambiguity</a> while allowing notation to remain brief. Where it is desired to override the precedence conventions, or even simply to emphasize them, <a href="/facts/Bracket/yzGM9J8A">parentheses</a> ( ) can be used. For example, (2 + 3) × 4 = 20 forces addition to precede multiplication, while (3 + 5)2 = 64 forces addition to precede <a href="/facts/Exponentiation/pac6QY2g">exponentiation</a>. 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 <a href="/facts/Bracket/yzGM9J8A">brackets</a> to avoid confusion, as in [2 × (3 + 4)] − 5 = 9.
These rules are meaningful only when the usual notation (called <a href="/facts/Infix_notation/LY0dMhay">infix notation</a>) is used. When <a href="/facts/Functional_notation/CdReEKCh">functional</a> or <a href="/facts/Polish_notation/dUzpf2Ys">Polish notation</a> are used for all operations, the order of operations results from the notation itself.

Order of operations open-in-new

Order of operations