Order of operations

In mathematics and computer programming, an expression or string of symbols intended to represent a numerical value must follow commonly accepted and unambiguous rules. For example, the rule for evaluating 2 + 3 * 4 in mathematics and in most computer languages is to do the multiplication first, so the correct answer is 14. Sometimes parentheses, which have their own rules, may be used to avoid confusion, thus: 2 + (3 * 4). When a term in an expression is both preceded and followed by an operator such as minus or times, the convention needed to clarify which operator should be applied first is known as a precedence rule or, more informally, order of operation. From the earliest use of mathematics notation multiplication took precedence over addition, whichever side of a number it appeared on.^[1] Thus 3 + 4 × 5 = 4 × 5 + 3 = 23. When exponents were first introduced, in the 16th and 17th centuries, exponents took precedence over both addition and multiplication, and could be placed only as a superscript to the right of their base. Thus 3 + 5 ² = 28 and 3 × 5 ² = 75. To change the order of operations, a vinculum (an overline or underline) was originally used. Today parentheses are used. Thus, to force addition to precede multiplication, write (3 + 4) × 5 = 35.

Contents 1 The standard order of operations 2 Examples 2.1 Specific cases 2.2 An example worked out in detail 3 Mnemonics 4 Proper use of parentheses and other grouping symbols 5 Special cases 6 Calculators 7 Programming languages 8 References and notes 9 See also 10 External

The standard order of operations

The standard order of operations, or precedence, is expressed in the following chart.

exponents and roots

multiplication and division

addition and subtraction

This means that (in the absence of parentheses, horizontal fraction lines, a bar over a radicand, or other symbols of grouping) one should perform all exponentiation and root-taking first; then perform all multiplication and division (including multiplications implied by juxtaposition, e.g. ab for (a × b); finally, perform all addition and subtraction. In each of these stages, when two operators have the same precedence, they are normally applied from left to right. The exception is exponentiation: if it is indicated by symbols places at different heights in a display, stacked exponents are evaluated from the top down, and if indicated by a caret, the operators are evaluated from right to left. Thus a typewritten string "4^3^2" and a display 4³² are evaluated as equal to 4^(3^2), i.e. 4^9 or 262144.)

It is helpful to treat division as multiplication by the reciprocal (multiplicative inverse) and subtraction as addition of the negative (additive inverse). Thus 3/4 = 3 ÷ 4 = 3 • ¼ and 3 − 4 = 3 + (−4), that is, the sum of positive three and negative four.

If an expression involves multiple parentheses, these indicate that the arithmetic inside the innermost pair of parentheses should be performed first and then the entire set of symbols is replaced by the result of that computation. Root symbols have a bar (called vinculum) over the radicand which acts as a symbol of grouping:

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 the standard round parentheses, e.g.

Unfortunately, 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, but in some applications and programming languages, notably 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. [3]. In any case where there is a possibility that the notation might be misinterpreted, it is advisable to use parentheses to clarify which interpretation is intended.

Similarly, one occasionally sees expressions which omit parentheses for brevity even though the resulting expression would not be properly evaluated by the conventions above; for example, the string of symbols "1/2x" should by these conventions mean (1/2) × x = (x/2) but in context might be understood to mean 1/(2x). Again, the use of parentheses would clarify what is meant and should be used if there is any chance of confusion.

Examples

Specific cases

Brackets [ ] are used here to indicate what will be evaluated next.

1. Evaluate subexpressions contained within parentheses, starting with the innermost expressions.

2. Evaluate exponential powers; for iterated powers, start from the right:

3. Evaluate multiplications, divisions and "of" (referring to multiplication by a fraction), starting from the left:

4. Evaluate additions and subtractions, starting from the left:

5. Evaluate negation on the same level as subtraction, starting from the left:^[2]

An example worked out in detail

• Given:

• Evaluate the innermost subexpression (7 + 1):

• Evaluate the subexpression within the remaining parentheses (5 − 8):

• Evaluate the power of (−3)²:

• Evaluate the multiplication 9 × (−5):

• Evaluate the subtraction 3-(-45):

• Evaluate the addition :(48)+3

Mnemonics

Mnemonics are often used to help students remember the rules. One common strategy is to build an acronym from the names of the operations. For instance, in the United States, the acronym PEMDAS (for Parentheses, Exponentiation, Multiplication/Division, Addition/Subtraction) is used, sometimes expressed as the sentence "Please Excuse My Dear Aunt Sally" or one of many other variations. In other English speaking countries, Parentheses may be called Brackets, and Exponentiation may be called Indices, Powers or Orders. Also, as Multiplication and Division are of equal precedence, M and D may be interchanged in the mnemonic. (This does not mean that multiplication and division can be performed in any arbitrary order, just that it is wrong to say "M always comes before D", or "D comes before M". See below.) The same comments apply to Addition and Subtraction. Finally, "Of" can be used to refer to multiplication as in "3/4 of 8 is 6". Thus, we also have BEDMAS, BIDMAS, BIMDAS, BIODMAS, BODMAS, BOMDAS and BPODMAS. one other way to remember the precedence rule is by memorizing simple sentences.

However, all these mnemonics are misleading if the user is not aware that multiplication and division are of equal precedence, as are addition and subtraction. Using any of the above rules in the order "addition first, subtraction afterward" would give the wrong answer to

The correct answer is 9, which is best understood by thinking of the problem as the sum of positive ten, negative three, and positive two.

It is usual, wherever one needs to calculate operations of equal precedence to work from left to right. The following rules are useful:

First: perform any calculations inside parentheses (brackets)

Second: Next perform all multiplication and division, working from left to right

Third: Lastly perform all addition and subtraction, working from left to right

Or, you could extend the mnemonic to: "Please Excuse My Dear Aunt Sally, she limps from left to right."

There is a new mnemonic featured in Danica McKellar's books Math Doesn't Suck^[3] and Kiss My Math^[4] that does address this very issue: "Pandas Eat: Mustard on Dumplings, and Apples with Spice." The intention being that Mustard and Dumplings is a "dinner course" and that Apples and Spice is a "dessert course." Then it becomes not a linear string of operations to do one after the other, but rather the "dinner course" operations are considered together and performed left to right, and then addition and subtraction are considered together, again performed again left to right.

With experience, the commutative law, associative law, and distributive law allow shortcuts. For example,

is much easier when worked from right to left, where here the answer is 34.

Proper use of parentheses and other grouping symbols

This section may contain original research or unverified claims. Please improve the article by adding references. See the talk page for details. (February 2008)

When restricted to using a straight text editor, parentheses (or more generally "grouping symbols") must be used generously to make up for the lack of graphics, like square root symbols. Here are some suggestions for doing so:

1) Whenever there is a fraction formed with a slash, put the numerator (the number on top of the fraction) in one set of parentheses, and the denominator (the number on the bottom of the fraction) in another set of parentheses. This is not required for fractions formed with underlines:

y = (x+1)/(x+2)

2) Whenever there is an exponent using the caret (^) symbol, put the base in one set of parentheses, and the exponent in another set of parentheses:

y = (x+1)^(x+2)

3) Whenever there is a trigonometric function, put the argument of the function, typically shown in bold and/or italics, in parentheses:

y = sin(x+1)

4) The rule for trigonometric functions also applies to any other function written in "functional notation", such as "sqrt" for a square root. That is, the argument of the function should be contained in parentheses:

y = sqrt(x+1)

5) An exception to the rules requiring parentheses applies when only one character is present. While correct either way, it is more readable if parentheses around a single character are omitted:

y = (3)/(x) or y = 3/x

y = (3)/(2x) or y = 3/(2x)

y = (x)^(5) or y = x^5

y = (2x)^(5) or y = (2x)^5

y = (x)^(5z) or y = x^(5z)

Calculators generally require parentheses around the argument of any function. Printed or handwritten expressions sometimes omit the parentheses, provided the argument is a single character. Thus, a calculator or computer program requires:

y = sqrt(2)

y = tan(x)

while a printed text may have:

y = sqrt 2

y = tan x.

6) Whenever anything can be interpreted multiple ways, put the part to be computed first inside parentheses, to make it clear.

7) One may alternate use of the different grouping symbols (parentheses, brackets, and braces) to make expressions more readable. For example:

y = { 2 / [ 3 / ( 4 / 5 ) ] }

is more readable than:

y = ( 2 / ( 3 / ( 4 / 5 ) ) )

Note that certain applications, like computer programming, will restrict one to certain grouping symbols.

Special cases

An exclamation mark indicates that one should compute the factorial of the term immediately to its left, before computing any of the lower-precedence operations, unless grouping symbols dictate otherwise. But 2³! means (2³)! = 8! = 40320 while 2^3! = 2⁶ = 64; a factorial in an exponent applies to the exponent, while a factorial not in the exponent applies to the entire power.

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

A function name usually applies to the monomial following the name, thus "sin xy" means sin (xy) but sin x + y means (sin x) + y. Calculators usually require parentheses, and parentheses should be used in complicated expressions to prevent misunderstanding.

Sometimes a dash or a heavy dot is used as a multiplication sign which has higher precedence than division. For example, J/kg K, J/kg-K, and J/kg · K are all equivalent.

Calculators

Different calculators follow different orders of operations. Cheaper calculators without a stack work left to right without any priority given to different operators, for example giving

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

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

The "cheap" 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 only evaluates 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 on the TI-92 and TI-30XII (both Texas Instruments calculators) associates two different ways:

The TI-92 associates to the right, that is

whereas, the TI-30XII associates 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. While the first interpretation may be expected by some users, only the latter is in agreement with the standard rules stated above.

Programming languages

Many programming languages use precedence levels that conform to the order commonly used in mathematics, though some, such as APL or 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 author of the language considers to be inferior, despite that many programmers have become accustomed to such precedence.^[5] The relative precedence levels of operators found in many C-style languages is as follows:

1	()   []   ->   .   ::	Grouping, scope, array/member access
2	!   ~   -   +   *   &   sizeof   type cast ++x   --x	(most) unary operations, sizeof and type casts
3	*   /   %	Multiplication, division, modulo
4	+   -	Addition and subtraction
5	<<   >>	Bitwise shift left and right
6	<   <=   >   >=	Comparisons: less-than, ...
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 operator)
14	=   +=   -=   *=   /=   %=   &=   |=   ^=   <<=   >>=	Assignment operators

Examples:

•  !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 ) )

References and notes

1. ^ http://mathforum.org/library/drmath/view/52582.html
2. ^ Some programs, notably Microsoft Excel and Unix bc, give a higher priority to negation than to exponentiation, which results in −3^2 = (−3)^2 = 9. [1]
3. ^ p.105, Math Doesn't Suck, ISBN 978-0-452-28949-9
4. ^ p.21, Kiss My Math, ISBN 978-0-452-29540-7
5. ^ Dennis M. Ritchie: The Development of the C Language. In History of Programming Languages, 2nd ed., ACM Press 1996. [2]

See also

• Common operator notation (for a more formal description)
• Associativity
• Commutativity
• Distributivity
• Operator overloading
• Operator precedence in C and C++

External

• Order of operations on PlanetMath
• Experiment investigating developer beliefs about operator precedence
• Experiment investigating effect of variable names on operator precedence selection