Order of operations

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. Thus, the expression is interpreted to have the value , and not . When exponents were introduced 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. Thus and .
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, forces addition to precede multiplication, while 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 .
Definition.
The order of operations, which is used throughout mathematics, science, technology and many computer programming languages, is expressed here:
This means that if, in a mathematical expression, a subexpression appears between two operators, the operator that is higher in the above list should be applied first.
The commutative and associative laws of addition and multiplication allow adding terms in any order, and multiplying factors in any order—but mixed operations must obey the standard order of operations.
In some contexts, it is helpful to replace a division by multiplication by the reciprocal (multiplicative inverse) and a subtraction by addition of the opposite (additive inverse). For example, in computer algebra, this allows one to handle fewer binary operations, and makes it easier to use commutativity and associativity when simplifying large expressions (for more, see ). Thus ; in other words, the quotient of 3 and 4 equals the product of 3 and . Also ; in other words the difference of 3 and 4 equals the sum of 3 and −4. Thus, can be thought of as the sum of , and the three summands may be added in any order, in all cases giving 5 as the result.
The root symbol √ is traditionally prolongated by a bar (called vinculum) over the radicand (this avoids the need for parentheses around the radicand). Other functions use parentheses around the input to avoid ambiguity. The parentheses can be omitted if the input is a single numerical variable or constant (as in the case of and . Another shortcut convention that is sometimes used is when the input is monomial; thus, rather than , but , because is not a monomial. This, however, is ambiguous and not universally understood outside of specific contexts.
Some calculators and programming languages require parentheses around function inputs, some do not.
Symbols of grouping can be used to override the usual order of operations. Grouped symbols can be treated as a single expression. Symbols of grouping can be removed using the associative and distributive laws, also they can be removed if the expression inside the symbol of grouping is sufficiently simplified so no ambiguity results from their removal.
Examples.
A horizontal fractional line also acts as a symbol of grouping:
For ease in reading, other grouping symbols, such as curly braces } or square brackets , are often used along with parentheses . For example:
Mnemonics.
Mnemonics are often used to help students remember the rules, involving the first letters of words representing various operations. Different mnemonics are in use in different countries.
These mnemonics may be misleading when written this way. For example, misinterpreting any of the above rules to mean "addition first, subtraction afterward" would incorrectly evaluate the expression
When evaluating expressions that only include addition and subtraction, such as the expression above, addition and subtraction may be performed sequentially, working left to right, but of the six major arithmetic operations, only addition and multiplication are commutative and associative, and so professionals usually view subtraction, especially in algebra and in higher mathematics, as addition of a negative number. Either working left to right or treating the subtraction as adding a signed number will produce the correct answer; performing subtraction in the wrong order will result in the incorrect answer. Mnemonics do not reflect the grouping of addition/subtraction or multiplication/division so their use can result in this misunderstanding.
A similar ambiguity exists in the case of serial division, for example, the expression might be read multiple ways but they may not always come to the same answer. For example, evaluated from left to right would be three divided by one equals three, divided by two equals three halves, but using division as multiplication by the reciprocal gives three divided by one half equals three multiplied by two equals six. There are three possibilities, none universally agreed upon. Either each of the several different division symbols must be given complicated rules of precedence, or imitating the rule for subtraction as addition of the opposite, division can be defined as multiplication by the reciprocal, or parentheses can be used to avoid ambiguities.
Since this is controversial, we summarize the above:
The first rule is that if there are multiple divisions in a row, the order of calculation may go from left to right:
But this is not universally accepted.
The second rule, treating division as multiplication by a reciprocal, greatly reduces the frequency of ambiguous division. 
But the only sure way to avoid ambiguity is to use parentheses.
Special cases.
Serial exponentiation.
If exponentiation is indicated by stacked symbols using superscript notation, the usual rule is to work from the top down:
which typically is not equal to ("a""b")"c".
However, when using operator notation with a caret (^) or arrow (↑), there is no common standard. For example, Microsoft Excel and computation programming language MATLAB evaluate codice_1 as ("a""b")"c", but Google Search and Wolfram Alpha as "a"("b""c"). Thus codice_2 is evaluated to 4,096 in the first case and to 262,144 in the second case.
Unary minus sign.
There are differing conventions concerning the unary operator − (usually read "minus"). In written or printed mathematics, the expression −32 is interpreted to mean .
In some applications and programming languages, notably Microsoft Excel, PlanMaker (and other spreadsheet applications) and the programming language bc, unary operators have a higher priority than binary operators, that is, the unary minus has higher precedence than exponentiation, so in those languages −32 will be interpreted as . This does not apply to the binary minus operator −; for example in Microsoft Excel while the formulas codice_3, codice_4 and codice_5 return 4, the formula codice_6 and codice_7 return −4.
Mixed division and multiplication.
Similarly, there can be ambiguity in the use of the slash symbol / in expressions such as 1/2"x". If one rewrites this expression as and then interprets the division symbol as indicating multiplication by the reciprocal, this becomes:
With this interpretation is equal to . However, in some of the academic literature, multiplication denoted by juxtaposition (also known as implied multiplication) is interpreted as having higher precedence than division, so that equals , not .
For example, the manuscript submission instructions for the "Physical Review" journals state that multiplication is of higher precedence than division with a slash, and this is also the convention observed in prominent physics textbooks such as the "Course of Theoretical Physics" by Landau and Lifshitz and the "Feynman Lectures on Physics".
Calculators.
Different calculators follow different orders of operations. Many simple calculators without a stack implement chain input working left to right without any priority given to different operators, for example typing
while more sophisticated calculators will use a more standard priority, for example typing
The "Microsoft Calculator" program uses the former in its standard view and the latter in its scientific and programmer views.
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 to the right. For example, the expression codice_1 is interpreted as "a"("b""c") 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 codice_11 is interpreted as 1/(2"x") by TI-82, as well as many modern Casio calculators, but as (1/2)"x" by TI-83 and every other TI calculator released since 1996, 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, 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.
Programming languages.
Some programming languages use precedence levels that conform to the order commonly used in mathematics, 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 codice_12 is interpreted as rather than ; 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, codice_13.
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. 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:
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.
The accuracy of software developer knowledge about binary operator precedence has been found to closely follow their frequency of occurrence in source code.