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In mathematics and computer programming, when a number or expression is both preceded and followed by an operator such as minus or times, a rule is needed to specify which operator should be applied first; this rule is known as a precedence rule, or more informally order of operation. From the earliest use of mathematical notation[citation needed], multiplication took precedence over addition, whichever side of a number it appeared on. Thus 3 + 4 × 5 = 5 × 4 + 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 one uses parentheses (). Thus, if one wants to force addition to precede multiplication, one writes (3 + 4) × 5 = 35.

Contents

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

In the absence of parentheses, horizontal fraction lines, a bar over a radicand, or other symbols of grouping, do all exponents and roots first. Stacked exponents must be done from the top down. After taking all exponents and roots, then do all multiplication and division. Next, do all addition and subtraction. Finally, when two operators have the same precedence, then work from the left to right.

It is helpful to treat division as multiplication by the reciprocal and subtraction as addition of the opposite. 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, then do the arithmetic inside the innermost pair of parentheses first and work outward. Root symbols have a bar (called vinculum) over the radicand which acts as a symbol of grouping: \sqrt{1+3}+5=\sqrt4+5=2+5=7. A horizontal fractional line also acts as a symbol of grouping: \frac{1+2}{3+4}+5=\frac37+5.

The order in which the unary operator − (usually read "minus") acts is often problematical. In written or printed mathematics, -3^2=-(3^2)=-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 -3^2=(-3)^2=9, . [3].

Examples

Specific cases

1. Evaluate subexpressions contained within parentheses, starting with the innermost expressions. (Brackets [ ] are used here to indicate what is evaluated next.)
(4+10/2)/9=(4+[10/2])/9=[4+5]/9=[9/9]=1 \,
2. Evaluate exponential powers; for iterated powers, start from the right:
2^{3^2}=2^{[3^2]}=[2^9]=512 \,
3. Evaluate multiplications, divisions and "of" (referring to multiplication by a fraction), starting from the left:
1/2\,\text{ of }\,8/2\times3=[8/4]\times3=[2\times3]=6 \,
4. Evaluate additions and subtractions, starting from the left:
7-2-4+1=[7-2]-4+1=[5-4]+1=[1+1]=2 \,
5. Evaluate negation on the same level as subtraction, starting from the left:[1]
-3^2=-[3^2]=-9 \,

An example worked out in detail

  • Given:
3-[5-(7+1)]^2\times(-5)+3 \,
  • Evaluate the innermost subexpression (7 + 1):
3-(5-8)^2\times(-5)+3 \,
  • Evaluate the subexpression within the remaining parentheses (5 − 8):
3-(-3)^2\times(-5)+3 \,
  • Evaluate the power of (−3)2:
3-9\times(-5)+3 \,
  • Evaluate the multiplication 9 × (−5):
3-(-45)+3 \,
  • Evaluate the subtraction 3 − (−45):
48 + 3 \,
  • Evaluate the addition 48 + 3:
48 + 3 = 51 \,

Mnemonics

Mnemonics are often used to help learners 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.

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

10 - 3 + 2 \,

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.

10 + (-3) + 2 \,

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

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

17 \times 24 / 12 \,

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

Proper use of parentheses and other grouping symbols

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 rules 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 trig function, put the argument of the function, typically shown in bold and/or italics, in parentheses:

y = sin(x+1)

4) The rule for trig functions also applies to any other function, such as 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 done first in 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

In the case of a factorial in an expression, it is evaluated before exponents and roots, unless grouping symbols dictate otherwise. When new operations are defined, they are generally presumed to take precedence over other operations, unless overridden by grouping symbols.

Sometimes a dash or a heavy dot is used as a multiplication sign which has higher precedence than division. For example, \mathrm{J/(kg\ K)}, \mathrm{J/kg\mbox{-}K}, and \mathrm{J/kg\bullet 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

1 + 2 \times 3 = 9, \;

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

1 + 2 \times 3 = 7. \;

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 a \wedge b \wedge c on the TI-92 and TI-30XII (both Texas Instruments calculators) associates two different ways:

The TI-92 associates to the right, that is

a \wedge b \wedge c=a \wedge (b \wedge c) = a^{(b \wedge c)} = a^{(b^{c})} = a^{b^{c}}

whereas, the TI-30XII associates to the left, that is

a \wedge b \wedge c=(a \wedge b) \wedge c=(a^{b})^{c}.

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.

Mathematical precedence

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 less than ideal.[2] 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 \equiv (!A) + (!B)
  • ++A + !B \equiv (++A) + (!B)
  • A * B + C \equiv (A * B) + C
  • A || B && C \equiv A || (B && C)
  • ( A && B == C ) \equiv ( A && ( B == C ) )

References and notes

  1. ^ 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]
  2. ^ Dennis M. Ritchie: The Development of the C Language. In History of Programming Languages, 2nd ed., ACM Press 1996. [2]

See also

External links


Wiktionary

Up to date as of January 15, 2010

Definition from Wiktionary, a free dictionary

English

Noun

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Wikipedia

order of operations

  1. (mathematics) The sequence in which the various operations in a mathematical expression are to be evaluated

See also

  • precedence rule

Simple English

The order of operations is a mathematical and algebraic set of rules. It is used to evaluate (solve) and simplify expressions and equations. The order of operations is the order that different mathematical operations are done. The standard mathematical operations are addition (+), subtraction (), multiplication (* or ×), division (/), brackets (which are grouping symbols, like parentheses () or []) and exponentiation (^n or n, also called orders or indices).

Mathematicians have agreed on a correct order to use operations, and it is very important that they know these rules. When people are solving a problem with more than one operation, they will need to know the correct order to solve the problem correctly. Otherwise the answer will be wrong.

Contents

Rules

Follow all the rules in this order from left to right in the equation.

Brackets and indices

Use operations inside brackets and solve any indices. You should always solve brackets first when solving an equation.

Example:

(2 + 3) * (4 -1) + 23
(2 + 3) * (4 -1) + 23
5 * (4 -1) + 23
5 * (4 -1) + 23
5 * 3 + 23
5 * 3 + 8

Multiplication and division

Solve any multiplication and division in the problem.

Example:

5 * 4 - 9 / 3
5 * 4 - 9 / 3
20 - 9 / 3
20 - 9 / 3
20 - 3

Addition and subtraction

Lastly, solve any addition or subtraction.

Example of all rules

(1 + 8) * (4 - 1) + 16 / 23
(1 + 8) * (4 - 1) + 16 / 23
9 * (4 - 1) + 16 / 23
9 * 3 + 16 / 23
9 * 3 + 16 / 8
9 * 3 + 16 / 8
27 + 16 / 8
27 + 2
29

Ways to remember it

Please Excuse My Dear Aunt Sally

A good way to remember the order of operations is by remembering this phrase. The first letter of each word is also the first letter of a rule in the order.

Please: Parentheses (or brackets)

Excuse: Exponents (or indices)

My: Multiplication

Dear: Division

Aunt: Addition

Sally: Subtraction

BIDMAS/BODMAS

Another good way of remembering the order of operations is the word BIDMAS, or BODMAS. The letters of the word are the same as the first letter for each rule.

B stands for Brackets

I stands for Indices, O stands for Orders

D stands for Division

M stands for Multiplication

A stands for Addition

S stands for Subtraction








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