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THE STANDARD ORDER OF OPERATIONS


The order of operations is expressed in the following chart.

:::exponents and '''roots'''
:::
:::multiplication and '''division'''
:::
:::addition and '''subtraction'''

In the absence of parentheses, do all exponents and roots first. Stacked exponents must be done from the top down. Root symbols have a bar (called Vinculum ) over the Radicand which acts as a symbol of grouping. After taking all exponents and roots, then do all multiplication and division. Finally, do all addition and subtraction.

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 -4 + 3 is the sum of negative four and positive three.

If an expression involves parentheses, then do the arithmetic inside the innermost pair of parentheses first and work outward, or use the Distributive Law to remove parentheses.

Acting contrary to the standard order of operations, some programs, notably Microsoft Office Excel and The Programming Language Bc , give unary operators a higher priority than binary operators, e.g. the unary minus (negation) has higher precedence than exponentiation. {Link without Title} .


EXAMPLES FROM ARITHMETIC


: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+ {Link without Title} )/9= {Link without Title} /9=1 \,

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

::2^{3^2}=2^{ {Link without Title} }= {Link without Title} =512 \,

:3. Evaluate multiplications and divisions, starting from the left:

::8/2 imes3=[8/2] imes3=[4 imes3]=12 \,

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

::7-2-4+1= {Link without Title} -4+1= {Link without Title} +1= {Link without Title} =2 \,

:5. Evaluate negation on the same level as subtraction, starting from the left: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. {Link without Title}

::-3^2=- {Link without Title} =-9 \,

B=Brackets
E=Exponents
D=Division
M=Multiplication
A=Addition
S=Subtraction


Acronyms

In the United States , the Acronym ''PEMDAS'' (for ''P''arentheses, ''E''xponentiation, ''M''ultiplication/''D''ivision, ''A''ddition/''S''ubtraction) is used instead, sometimes expressed as the mnemonic "''P''lease ''E''xcuse ''M''y ''D''ear ''A''unt ''S''ally", "''P''lease ''E''xecute ''M''y ''D''umb ''A''ssassin ''S''am", "''P''anda ''E''xpress ''M''akes ''D''elicious ''A''vocado ''S''oup, or "''P''lease ''E''xcuse ''M''y ''D''eadly ''A''ngry ''S''amurai."

In Canada , an acronym ''BEDMAS'' is often used as a Mnemonic for ''B''rackets, ''E''xponents, ''D''ivision, ''M''ultiplication, ''A''ddition, and ''S''ubtraction.

In the UK , Australia and New Zealand , the acronym ''BODMAS'' is commonly used for ''B''rackets, ''O''rders, ''D''ivision, ''M''ultiplication, ''A''ddition, ''S''ubtraction. Since multiplication and division are of the same rank, this is sometimes written as ''BOMDAS'', ''BIDMAS'' or ''BIMDAS'' where ''I'' stands for ''I''ndices.

Warning: Multiplication and division are of equal precedence, and addition and subtraction are of equal precedence. 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.

It is usual, wherever you need to calculate operations of equal precedence to work from left to right. The following rules of thumb 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 imes 24 / 12 \,

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


More examples


  • Given:

    • ::3-(5-(7+1))^2 imes(-5)+3 \,


  • Evaluate the innermost subexpression (7 + 1):

    • ::3-(5-8)^2 imes(-5)+3 \,


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

    • ::3-(-3)^2 imes(-5)+3 \,


  • Evaluate the power of (−3)2:

    • ::3-9 imes(-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 \,



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


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


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

y


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


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


y


y


y


y


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


y


while a printed text may have:

y


y


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

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

y


is more readable than:

y


Note that certain applications, like computer programming, will restrict you 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.

In the case where repeated operators of the same type are used, such as in

::a/b/c

the expression is evaluated from left to right and is said to associate to the left. It is therefore equal to

::(a/b)/c.

With index notation for exponentiation, however, exponents are evaluated from right to left.


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 imes 3 = 9, \;
while more sophisticated calculators will use a more standard priority, for example giving
:1 + 2 imes 3 = 7. \;
The Microsoft '' Calculator '' program uses the former in its standard view and the latter in its scientific view.

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}.


MATHEMATICAL PRECEDENCE


Many programming languages conform to mathematical order of operations, however some, such as APL use different operator precedence rules.

The logical bitwise operators in C (and all programming languages that borrowed precedence rules from C) are known to have incorrect precedence (i.e. they should be placed above the comparison operators) due to a historical quirk and backwards compatibility with that quirk. {Link without Title} The order in C-style languages is as follows:

Examples:














REFERENCES AND NOTES



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