Order of operations

In mathematics and computer programming, the order of operations is a collection of conventions about which arithmetic operations to perform first in order to evaluate a given mathematical expression.
These conventions 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. Calculators generally perform operations with the same precedence from left to right, but some programming languages 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 algebraic notation. Thus, in the expression, the multiplication is performed before addition, and the expression has the value, and not. 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 and.
These conventions exist to avoid notational ambiguity while allowing notation to remain brief. Where it is desired to override the precedence conventions, or even simply to emphasize them, parentheses can be used. For example, forces addition to precede multiplication, while forces addition to precede exponentiation. If multiple pairs of parentheses are required in a mathematical expression, the parentheses may be replaced by other types of brackets to avoid confusion, as in.
These conventions are meaningful only when the usual notation is used. When functional or Polish notation are used for all operations, the order of operations results from the notation itself.

Conventional order

The order of operations, that is, the order in which the operations in an expression are usually performed, results from a convention adopted throughout mathematics, science, technology and many computer programming languages. It is summarized as:

This means that to evaluate an expression, one first evaluates any sub-expression inside parentheses, working inside to outside if there is more than one set. Whether inside parentheses or not, the operation that is higher in the above list should be applied first. Operations of the same precedence are conventionally evaluated from left to right.
If each division is replaced with multiplication by the reciprocal then the associative and commutative laws of multiplication allow the factors in each term to be multiplied together in any order. Sometimes multiplication and division are given equal precedence, or sometimes multiplication is given higher precedence than division; see below. If each subtraction is replaced with addition of the opposite, then the associative and commutative laws of addition allow terms to be added in any order.
The radical symbol which signifies a square root is traditionally extended by a bar 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. Traditionally this convention extends to monomials; thus, and even, but, because is not a monomial. However, this convention is not universally understood, and some authors prefer explicit parentheses. Some calculators and programming languages require parentheses around function inputs, while others do not.
Parentheses and alternate symbols of grouping can be used to override the usual order of operations or to make the intended order explicit. Grouped symbols can be treated as a single expression.

Examples

Multiplication before addition:
Parenthetical subexpressions are evaluated first:
Exponentiation before multiplication, multiplication before subtraction:
When an expression is written as a superscript, the superscript is considered to be grouped by its position above its base:
The operand of a root symbol is determined by the overbar:
A horizontal fractional line forms two grouped subexpressions, one above divided by another below:
Parentheses can be nested, and should be evaluated from the inside outward. For legibility, outer parentheses can be made larger than inner parentheses. Alternately, other grouping symbols, such as curly braces or square brackets, are sometimes used along with parentheses. For example:

Special cases

Unary minus sign

There are differing conventions concerning the unary operation . In written or printed mathematics, the expression −3² is interpreted to mean.
In some applications and programming languages, notably Microsoft Excel, PlanMaker and the programming language bc, unary operations have a higher priority than binary operations, 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 for example in Microsoft Excel the formulas =-2^2, =^2 and =0+-2^2 return 4, but the formulas =0-2^2 and =- return −4.

Mixed division and multiplication

There is no universal convention for interpreting an expression containing both division denoted by '÷' and multiplication denoted by '×'. Proposed conventions include assigning the operations equal precedence and evaluating them from left to right, or equivalently treating division as multiplication by the reciprocal and then evaluating in any order; evaluating all multiplications first followed by divisions from left to right; or eschewing such expressions and instead always disambiguating them by explicit parentheses.
Beyond primary education, the symbol '÷' for division is seldom used, but is replaced by the use of algebraic fractions. These are most explicitly and unambiguously written "vertically" with the numerator stacked above the denominator separated by a fraction bar. But they can also be written "horizontally" with the numerator and denominator separated by the slash symbol '/'. That is, expressions such as are avoided in favor of or.
Multiplication denoted by juxtaposition creates a visual unit and is often given higher precedence than most other operations. In academic literature, when inline fractions are combined with implied multiplication without explicit parentheses, the multiplication is conventionally interpreted as having higher precedence than division, so that e.g. is interpreted to mean rather than. For instance, the manuscript submission instructions for the Physical Review journals directly state that multiplication has precedence over division, and this is also the convention observed in physics textbooks such as the Course of Theoretical Physics by Landau and Lifshitz and mathematics textbooks such as Concrete Mathematics by Graham, Knuth, and Patashnik. However, some authors recommend against expressions such as, preferring the explicit use of parenthesis.
More complicated cases are more ambiguous. For instance, the notation could plausibly mean either
or. Sometimes interpretation depends on context. The Physical Review submission instructions recommend against expressions of the form ; more explicit expressions or are unambiguous.
File:Precedence62xplus.jpg|thumb|6÷2 is interpreted as 6÷ by a fx-82MS, and × by a TI-83 Plus calculator, respectively.
This ambiguity has been the subject of Internet memes such as "", for which there are two conflicting interpretations: 8 ÷ = 1 and · = 16. Mathematics education researcher Hung-Hsi Wu points out that "one never gets a computation of this type in real life", and calls such contrived examples "a kind of Gotcha! parlor game designed to trap an unsuspecting person by phrasing it in terms of a set of unreasonably convoluted rules".

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 ^c. This convention is useful because there is a property of exponentiation that ^c = a^bc, so it's unnecessary to use serial exponentiation for this.
However, when exponentiation is represented by an explicit symbol such as a caret or arrow, there is no common standard. For example, Microsoft Excel and computation programming language MATLAB evaluate a^''b^c'' as ^c, but Google Search and Wolfram Alpha as a. Thus 4^3^2 is evaluated to 4,096 in the first case and to 262,144 in the second case.

Mnemonics

are often taught in primary schools to help students remember the order of operations. The acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication/Division, Addition/Subtraction, is common in the United States and France. Sometimes the letters are expanded into words of a mnemonic sentence such as "Please Excuse My Dear Aunt Sally". The United Kingdom and other Commonwealth countries may use BODMAS, standing for Brackets, Of, Division/Multiplication, Addition/Subtraction, with "of" meaning fraction multiplication. Sometimes the O is instead expanded as Order, meaning exponent or root, or replaced by I for Indices in the alternative mnemonic BIDMAS. In Canada and New Zealand BEDMAS is common.
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 as, while the correct evaluation is. These values are different when.
In Germany, the convention is simply taught as Punktrechnung vor Strichrechnung, "dot operations before line operations" referring to the graphical shapes of the taught operator signs,, and,. This avoids the potential for the above misunderstanding.
Mnemonic acronyms have been criticized for not developing a conceptual understanding of the order of operations, and not addressing student questions about its purpose or flexibility. Students learning the order of operations via mnemonic acronyms routinely make mistakes, as do some pre-service teachers. Even when students correctly learn the acronym, a disproportionate focus on memorization of trivia crowds out substantive mathematical content. The acronym's procedural application does not match experts' intuitive understanding of mathematical notation: mathematical notation indicates groupings in ways other than parentheses or brackets and a mathematical expression is a tree-like hierarchy rather than a linearly "ordered" structure; furthermore, there is no single order by which mathematical expressions must be simplified or evaluated and no universal canonical simplification for any particular expression, and experts fluently apply valid transformations and substitutions in whatever order is convenient, so learning a rigid procedure can lead students to a misleading and limiting understanding of mathematical notation.