Order of operations

In mathematics, logic and computer science, operator order, operator valence, operator priority or operator precedence refers to a defined half-order in which the operators of an expression in infix notation are to be evaluated.

The operator rank order is not a total order, but a half order, because there does not have to be a strict order between all operators. There can also be more than one operator on the same rank. For example, in arithmetic, the rank of multiplication and division is the same but higher than the rank of addition and subtraction ("dot arithmetic before dash arithmetic").

Bracketing offers the possibility of giving priority to a part of a formula: The bracketed range, i.e., the range enclosed by a pair of brackets "( ... )", is to be computationally executed first and replaced by the corresponding partial result. The parentheses must contain the operators together with their necessary operands. Thus, if notated, the bracket expression is to be calculated first, i.e. the sum is to be formed before is multiplied by this sum.

Ranking can save explicit bracketing. Thus, in arithmetic is equivalent to , because the multiplication operator has a higher rank. However, other rankings may be defined for other applications of these operator symbols.

In the case of non-commutative operators, additional conventions are required as to whether equal-ranking subexpressions are to be evaluated from left to right or right to left, in order to unambiguously determine the order of calculation.

Ranking of different operators

For the standard arithmetic operations of mathematics, the following order of precedence (in descending order of priority) is common:

1. Exponentiation
2. Multiplication and division ("point calculation")
3. Addition and subtraction ("stroke calculation")

In programming languages and computer programs for formula evaluation (e.g., the Unix utility bc), there are additional categories. One of them is the sign, which usually enjoys an even higher priority over exponentiation. Thus, while in mathematical formulas the expression read as , in the expressions of such evaluation programs it is often read as .

In logic, it is not always common to define an operator precedence. Where this is done, the following is usually chosen (in descending priority):

1. Negation
2. Conjunction
3. Disjunction
4. Conditional
5. Biconditional

Also in many programming languages this order of precedence is chosen for Boolean operators.

For example, after applying the above operator rank sequences, the arithmetic expression evaluated as , the logical expression as .

Sequence of equivalent operators

In addition, an associativity can be specified for operations, which determines the order in which adjacent, equivalent operators are to be evaluated. An operator is called left-associative if A op B op C op D is evaluated as ((A op B) op C) op D; an operator is called right-associative if A op B op C op D is evaluated as A op (B op (C op D)). Of the above arithmetic operators, exponentiation is defined to be right-associative, that is:

.

Likewise the arrow operator:

The remaining two-digit operators are defined as left-associative, i.e., for example, .

In logic, junctors are usually defined in a left-associative way, but there are also authors who use at least the conditional in a right-associative way.

There are also programming languages, such as Occam, that set all operators to the same rank and evaluate them from left to right.