Order of operations

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The standard order of operations[edit]

The order of operations used throughout mathematics, science, technology and many computer programming languages is expressed here:[2]

exponents and roots
multiplication and division
addition and subtraction

This means that if a mathematical expression is preceded by one binary operator and followed by another, the operator higher on the list should be applied first. The commutative and associative laws of addition and multiplication allow terms to be added in any order and factors to be multiplied in any order, but mixed operations must obey the standard order of operations.



Stacked exponents are applied from the top down, i.e., from right to left.

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.



A horizontal fractional line also acts as a symbol of grouping:




Exceptions to the standard[edit]

There exist differing conventions concerning the unary operator − (usually read "minus"). In written or printed mathematics, the expression −32 is interpreted to mean −(32) = −9,[3] but in some applications and programming languages, notably the BASIC programming language, the application Microsoft Office Excel and the programming language bc, unary operators have a higher priority than binary operators, that is, the unary minus (negation) has higher precedence than exponentiation, so in those languages −32 will be interpreted as (−3)2 = 9.[4] Note this does not apply to the binary operator −; for example while the formulas =-2^2 and =0+-2^2 return 4 in Microsoft Excel, the formula =0-2^2 returns −4. In cases where there is the possibility that the notation might be misinterpreted, parentheses are usually used to clarify the intended meaning, however due to the syntax of most major programming languages, it is usually hard or impossible to be ambiguous.

Similarly, there can be ambiguity in the use of the slash ('/') symbol in expressions such as 1/2x. If one rewrites this expression as 1 ÷ 2 × x and then interprets the division symbol as indicating multiplication by the reciprocal, this becomes

1 \div 2 \times x = 1 \times \tfrac{1}{2} \times x = \tfrac{1}{2}x.

Hence, with this interpretation we have that 1/2x is equal to (1/2)x, and not 1/(2x). However, there are examples, including in published literature, where implied multiplication is interpreted as having higher precedence than division, so that 1/2x equals 1/(2x), not (1/2)x. For example, the manuscript submission instructions for the Physical Review journals state that multiplication is of higher precedence than division with a slash,[5] 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.[6] Wolfram Alpha changed in early 2013 to treat implied multiplication the same as explicit multiplication (formerly, implied multiplication without parentheses was assumed to bind stronger than explicit multiplication). 2x/2x, 2*x/2*x, and 2(x)/2(x) now all yield x2.[7] The TI 89 and TI 86 calculators also yield x2 in all three cases.


Mnemonics are often used to help students remember the rules, but the rules taught by the use of acronyms can be misleading. In the United States the acronym PEMDAS is common. It stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. PEMDAS is often expanded to "Please Excuse My Dear Aunt Sally" with the first letter of each word creating the acronym PEMDAS. Canada uses BEDMAS. It stands for Brackets, Exponents, Division, Multiplication, Addition, Subtraction. The UK and Australia[8] use BODMAS or BIDMAS.

In Canada and other English speaking countries, Parentheses may be called Brackets, or symbols of inclusion and Exponents may be called either Indices, Powers or Orders, which have the same precedence as Roots or Radicals. Since multiplication and division are of equal precedence, M and D are often interchanged, leading to such acronyms as BOMDAS. The original order of operations in most countries was BODMAS which stood for Brackets, Orders, Division, Multiplication, Addition, Subtraction. This mnemonic was used until exponentials were added into the mnemonic.


10 - 3 + 2 \,.

The correct answer is 9 (and not 5, which we get when we do the addition first and then the subtraction). The best way to understand a combination of addition and subtraction is to think of the subtraction as addition of a negative number. In this case, we see the problem as the sum of positive ten, negative three, and positive two.

10 + (-3) + 2 \,

To emphasize that addition and subtraction have the same precedence (and multiplication and division have the same precedence) the mnemonic is sometimes written P E MD AS; or, simply as PEMA.

All of these acronyms conflate two different ideas, operations on the one hand and symbols of grouping on the other, which can lead to confusion.

Special cases[edit]

If exponentiation is indicated by stacked symbols, the usual rule is to work from the top down, thus


which typically is not equal to (a^b)^c. However, some computer systems may resolve the ambiguous expression differently. For example, Microsoft Office Excel evaluates a^b^c as (a^b)^c which is opposite of normally accepted convention of top-down order of execution for exponentiation. If a=4, p=3, and q=2, a^{p^q} is evaluated to be 4096 in Microsoft Excel 2013, the same as (a^p)^q. The expression  a^{(p^q)} , on the other hand, results in 262144 using the same program.



1 + 2 \times 3 = 9, \;


1 + 2 \times 3 = 7. \;

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

The non-scientific calculator 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. For example, the expression a ^ b ^ c on the TI-92, the TI-30XII and the TI-30XS MultiView (all Texas Instruments calculators) associate two different ways:

The TI-92 and the TI-30XS MultiView in "MathPrint Mode" associate to the right, that is


whereas, the TI-30XII and the TI-30XS MultiView in "Classic Mode" associate to the left, that is


An expression like 1/2x is interpreted as 1/(2x) by TI-82, but as (1/2)x by TI-83.[9] [10] While the first interpretation may be expected by some users, only the latter is in agreement with the standard rule that multiplication and division are of equal precedence,[citation needed] so 1/2x is read one divided by two and the answer multiplied by x.


Calculators that utilize reverse Polish notation, also known as postfix notation, use stack to enter formulas without the need for parentheses.

Programming languages[edit]

Many programming languages use precedence levels that conform to the order commonly used in mathematics, though some, such as APL and Smalltalk, have no operator precedence rules (in APL, evaluation is strictly right to left; in Smalltalk, it's strictly left to right).

The logical bitwise operators in C (and all programming languages that borrowed precedence rules from C, for example, C++, Perl and PHP) have a precedence level that the creator of the C language considered to be unsatisfactory.[11] However, many programmers have become accustomed to this order. The relative precedence levels of operators found in many C-style languages are as follows:

3 *   /   % Multiplication, division, modulo
4 +   - Addition and subtraction
5 <<   >> Bitwise shift left and right
7 ==   != Comparisons: equal and not equal
8 & Bitwise AND
9 ^ Bitwise exclusive OR
10 | Bitwise inclusive (normal) OR
11 && Logical AND
12 || Logical OR
13  ?:   =   +=   -=   *=   /=   %=   &=   |=   ^=   <<=   >>= Conditional expression (ternary) and assignment operators
14 , Comma operator


  • A + B * CA + (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.[12]


See also[edit]


  1. ^ "Ask Dr. Math". Math Forum. 22 November 2000. Retrieved 5 March 2012. 
  2. ^ "Order of Operations Lessons". Algebra.Help. Retrieved 5 March 2012. 
  3. ^ Allen R. Angel, Elementary Algebra for College Students 8/E; Chapter 1, Section 9, Objective 3
  4. ^ "Formula Returns Unexpected Positive Value". Support.microsoft.com. 15 August 2005. Retrieved 5 March 2012. 
  5. ^ "Physical Review Style and Notation Guide". American Physical Society. Section IV–E–2–e. Retrieved 5 August 2012. 
  6. ^ For example, the third edition of Mechanics by Landau and Lifshitz contains expressions such as hPz/2π (p. 22), and the first volume of the Feynman Lectures contains expressions such as 1/2N (p. 6–8). In both books these expressions are written with the convention that the solidus is evaluated last.
  7. ^ "2x/2x, 2*x/2*x, 2(x)/2(x) - Wolfram|Alpha". Wolframalpha.com. Retrieved 11 February 2013. 
  8. ^ http://syllabus.bos.nsw.edu.au/assets/global/files/maths_s3_sampleu1.doc
  9. ^ "Implied Multiplication Versus Explicit Multiplication on TI Graphing Calculators". Texas Instruments Incorporated. 16 January 2011. Retrieved 29 April 2011. [citation not found]
  10. ^ "Google cache for: Implied Multiplication Versus Explicit Multiplication on TI Graphing Calculators". Texas Instruments Incorporated. 23 Apr 2013. Retrieved 10 May 2013. 
  12. ^ 6÷2(1+2)=? Andy Li's Blog. 2 May 2011. Retrieved 31 December 2012.

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