This article is about the concept of operator precedence. For operator precedence parsing, see operator-precedence parser.
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In programming languages, scientific calculators and similar common operator notation or operator grammar is a way to define and analyse mathematical and other formal expressions. In this model a linear sequence of tokens are divided into two classes: operators and operands.
Operands are objects upon which the operators operate. These include literal numbers and other constants as well as identifiers (names) which may represent anything from simple scalar variables to complex aggregated structures and objects, depending on the complexity and capability of the language at hand as well as usage context. One special type of operand is the parenthesis group. An expression enclosed in parentheses is typically recursively evaluated to be treated as a single operand on the next evaluation level.
Each operator is given a position, precedence, and an associativity. The operator precedence is a number (from high to low or vice versa) that defines which operator takes an operand that is surrounded by two operators of different precedence (or priority). Multiplication normally has higher precedence than addition,[1] for example, so 3+4×5 = 3+(4×5) ≠ (3+4)×5.
In terms of operator position, an operator may be prefix, postfix, or infix. A prefix operator immediately precedes its operand, as in −x. A postfix operator immediately succeeds its operand, as in x! for instance. An infix operator is positioned in between a left and a right operand, as in x+y. Some languages, most notably the C-syntax family, stretches this conventional terminology and speaks also of ternary infix operators (a?b:c). Theoretically it would even be possible (but not necessarily practical) to define parenthesization as a unary bifix operation.
^Bronstein, Ilja Nikolaevič; Semendjajew, Konstantin Adolfovič (1987) [1945]. "2.4.1.1.". In Grosche, Günter; Ziegler, Viktor; Ziegler, Dorothea (eds.). Taschenbuch der Mathematik (in German). Vol. 1. Translated by Ziegler, Viktor. Weiß, Jürgen (23 ed.). Thun and Frankfurt am Main: Verlag Harri Deutsch (and B. G. Teubner Verlagsgesellschaft, Leipzig). pp. 115–120. ISBN 3-87144-492-8.
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