In mathematics and computer programming, the **order of operations** (or **operator precedence**) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression.

For example, in mathematics and most computer languages, multiplication is granted a higher precedence than addition, and it has been this way since the introduction of modern algebraic notation.^{[1]}^{[2]} Thus, the expression 2 + 3 × 4 is interpreted to have the value 2 + (3 × 4) = 14, and not (2 + 3) × 4 = 20. With the introduction of exponents in the 16th and 17th centuries, they were given precedence over both addition and multiplication, and could be placed only as a superscript to the right of their base.^{[1]} Thus 3 + 5^{2} = 28 and 3 × 5^{2} = 75.

These conventions exist to eliminate notational ambiguity, while allowing notation to be as brief as possible. Where it is desired to override the precedence conventions, or even simply to emphasize them, parentheses ( ) can be used to indicate an alternative order of operations (or to simply reinforce the default order of operations). For example, (2 + 3) × 4 = 20 forces addition to precede multiplication, while (3 + 5)^{2} = 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 brackets or braces to avoid confusion, as in [2 × (3 + 4)] − 5 = 9.^{[3]}

Different calculators follow different orders of operations.^{[1]} Many simple calculators without a stack implement chain input working left t

Different calculators follow different orders of operations.^{[1]} Many simple calculators without a stack implement chain input working left to right without any priority given to different operators, for example typing

`1 + 2 × 3`

yields 9,

while more sophisticated calculators will use a more standard priority, for example typing

`1 + 2 × 3`

yields 7.

The *Microsoft Calculator* program uses the former in its standard view and the latter in its scientific and programmer views.

Chain input expects two o

while more sophisticated calculators will use a more standard priority, for example typing

`1 + 2 × 3`

yields 7.

The *Microsoft Calculator* program uses the former in its standard view and the latter in its scientific and programmer views.

Chain input expects two Chain input expects two operands and an operator. When the next operator is pressed, the expression is immediately evaluated and the answer becomes the left hand of the next operator. Advanced calculators allow entry of the whole expression, grouped as necessary, and evaluates only when the user uses the equals sign.

Calculators may associate exponents to the left or to the right depending on the model or the evaluation mode. For example, the expression

is interpreted as *a*^*b*^*c**a*^{(bc)} on the TI-92 and the TI-30XS MultiView in "Mathprint mode", whereas it is interpreted as (*a*^{b})^{c} on the TI-30XII and the TI-30XS MultiView in "Classic mode".

An expression like `1/2`

is interpreted as 1/(2*x**x*) by TI-82, as well as many modern Casio calculators,^{[23]} but as (1/2)*x* by TI-83 and every other TI calculator released since 1996,^{[24]} as well as by all Hewlett-Packard calculators with algebraic notation. While the first interpretation may be expected by some users due to the nature of implied multiplication, the latter is more in line with the standard rule that multiplication and division are of equal precedence,^{[25]}^{[26]} where 1/2*x* is read one divided by two and the answer multiplied by *x*.

When the user is unsure how a calculator will interpret an expression, it is a good idea to use parentheses so there is no ambiguity.

Calculators that use reverse Polish notation (RPN), also known as postfix notation, use a stack to enter expressions in the correct order of precedence without a need for parentheses or any possibly model-specific order of execution.^{[12]}^{[11]}

Some programming languages use precedence levels that conform to the order commonly used in mathematics,^{[19]} though others, such as APL, Smalltalk, Occam and Mary, have no operator precedence rules (in APL, evaluation is strictly right to left; in Smalltalk etc. it is strictly left to right).

In addition, because many operators are not associative, the order within any single level is usually defined by grouping left to right so that `16/4/4`

is interpreted as (16/4)/4 = 1 rather than 16/(4/4) = 16; such operators are perhaps misleadingly referred to as "left associa

In addition, because many operators are not associative, the order within any single level is usually defined by grouping left to right so that `16/4/4`

is interpreted as (16/4)/4 = 1 rather than 16/(4/4) = 16; such operators are perhaps misleadingly referred to as "left associative". Exceptions exist; for example, languages with operators corresponding to the cons operation on lists usually make them group right to left ("right associative"), e.g. in Haskell, `1:2:3:4:[] == 1:(2:(3:(4:[]))) == [1,2,3,4]`

.

The creator of the C language has said of the precedence in C (shared by programming languages that borrow those rules from C, for example, C++, Perl and PHP) that it would have been preferable to move the bitwise operators above the comparison operators.^{[27]} However, many programmers have become accustomed to this order. The relative precedence levels of operators found in many C-style languages are as follows:

Examples: (Note: in the examples below, '≡' is used to mean "is equivalent to", and not to be interpreted as an actual assignment operator used as part of the example expression.)

`!A + !B`

≡`(!A) + (!B)`

`++A + !B`

≡`(++A) + (!B)`

`A + B * C`

≡`A + (B * C)`

`A || B && C`

≡`A || (B && C)`

`A && B == C`

≡`A && (B == C)`

`A & B == C`

≡`A & (B == C)`

Source-to-source compilers that compile to multiple languages need to explicitly deal with the issue of different order of operations across languages. Haxe for example standardizes the order and enforces it by inserting brackets where it is appropriate.^{[28]}

The accuracy of software developer knowledge about binary operator precedence has been found to closely follow their frequency of occurrence in source code.^{[29]}

- Common operator notation (for a more formal description)
- Hyperoperation
- Operator associativity
- Source-to-source compilers that compile to multiple languages need to explicitly deal with the issue of different order of operations across languages. Haxe for example standardizes the order and enforces it by inserting brackets where it is appropriate.
^{[28]}The accuracy of software developer knowledge about binary operator precedence has been found to closely follow their frequency of occurrence in source code.

^{[29]}## See also