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Order of operations

In mathematics and computer programming, the order of operations (sometimes called operator precedence) is a rule used to clarify unambiguously which procedures should be performed first in a given mathematical expression.

For example, in mathematics and most computer languages multiplication is done before addition; in the expression 2 + 3 4, the answer is 14. Brackets, "( and ), { and }, or [ and ]", which have their own rules, may be used to avoid confusion, thus the preceding expression may also be rendered 2 + (3 4), but the brackets are unnecessary as multiplication still has precedence without them.

Since the introduction of modern algebraic notation, multiplication has taken precedence over addition.[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 + 52 = 28 and 3 × 52 = 75. To change the order of operations, originally a vinculum (an overline or underline) was used. Today, parentheses or brackets are used to explicitly denote precedence by grouping parts of an expression that should be evaluated first. Thus, to force addition to precede multiplication, we write (2 + 3) × 4 = 20, and to force addition to precede exponentiation, we write (3 + 5)2 = 64.

The standard order of operations

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

terms inside parentheses or brackets
exponents and roots
multiplication and division

This means that if a mathematical expression is preceded by one 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.

It is helpful to treat division as multiplication by the reciprocal (multiplicative inverse) and subtraction as addition of the opposite (additive inverse). Thus 3/4 = 3   4 = 3    ; in other words the quotient of 3 and 4 equals the product of 3 and   . Also 3   4 = 3 + ( 4); in other words the difference of 3 and 4 equals the sum of positive three and negative four. With this understanding, we can think of 1 3 + 7 as the sum of 1, negative 3, and 7, and add in any order: (1 3) + 7 = 2 + 7 = 5 and in reverse order (7 3) + 1 = 4 + 1 = 5. The important thing is to keep the negative sign with the 3.

The root symbol, , requires a symbol of grouping around the radicand. The usual symbol of grouping is a bar (called vinculum) over the radicand. Other functions use parentheses around the input to avoid ambiguity. The parentheses are sometimes omitted if the input is a monomial. Thus, sin x = sin(x), but sin x + y = sin(x) + y, because x + y is not a monomial. Calculators usually require parentheses around all function inputs.

Stacked exponents are applied from the top down.

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.

Examples

\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.

For ease in reading, other grouping symbols such as braces, sometimes called curly braces { }, or brackets, sometimes called square brackets [ ], are often used along with parentheses ( ). For example,

[(1+2)-3]-(4-5) = [3-3]-(-1) = 1. \,

Exceptions to the standard

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 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] In cases where there is the possibility that the notation might be misinterpreted, parentheses are usually used to clarify the intended meaning.

Similarly, there can be ambiguity in the use of the slash ('/') symbol. The string of characters "1/2x" is interpreted by the above conventions as (1/2)x. The contrary interpretation should be written explicitly as 1/(2x). Again, the use of parentheses clarifies the meaning and avoids the possibility of misinterpretation.

The precedence of an implied multiplication e.g. 2x being 2 × x also varies by source. For example, Wolfram Alpha considers that implied multiplication precedes division, e.g. 2x 2x gives 1 instead of x ,[5] except where parentheses are adjacent, e.g. 48 2(9+3) gives 288 instead of 2.[6]

Mnemonics

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 and the UK uses BIDMAS or BODMAS. In Canada and other English speaking countries, Parentheses may be called Brackets, or symbols of inclusion and Exponentiation 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 BEDMAS, BIDMAS, BODMAS, BERDMAS, PERDMAS, and BPODMAS.

These mnemonics may be misleading when written this way, especially 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 also give the wrong answer.

10 - 3 + 2 \,

The correct answer is 9 (and not 5, which we get when we add 3 and 2 first to get 5,and then subtract it from 10 to get the final answer of 5), which is best understood by thinking of the problem as the sum of positive ten, negative three, and positive two.

10 + (-3) + 2 \,

An alternative way to write the mnemonic is:

P

E

MD

AS

Or, simply as PEMA, where it is taught that multiplication and division inherently share the same precedence; and that addition and subtraction inherently share the same precedence. PEMA is one of the mnemonics taught in New Zealand.

This makes the equivalence of multiplication and division and of addition and subtraction clear.

Another potentially misleading aspect of this mnemonic is the inclusion of P for parentheses. First of all, parentheses are grouping symbols, not operation symbols. Also, within the grouping symbols, there may be expressions involving several operations, which would need to be evaluated according to the correct order or operations which appear AFTER the P in the mnemonic. It is probably better to teach that grouping symbols are not operations themselves, but rather are used to change the precedence of operations from the default.

Once logs are introduced they should be given the same precedence as exponents.

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 23! = 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^{b^c} = a^{(b^c)} \ne (a^b)^c. \,

Sometimes a heavy dot is used as a multiplication sign which indicates that the entire expression before the heavy dot is multiplied by the entire expression after the heavy dot, but this notation may be misunderstood. Thus x + y a + b may be used for (x + y)(a + b), but the latter notation is more common.

Calculators

Different calculators follow different orders of operations. Most non-scientific 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 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 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 ^ b ^ 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 ^ b ^ c = a ^ (b ^ c) = a^{(b^c)} = a^{b^c}

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

a ^ b ^ c = (a ^ b) ^ 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.[7] 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, so 1/2x is read one divided by two and the answer multiplied by x.

When the user is unsure how a calculator will interpret an expression, it is a good idea to use parentheses so there is no ambiguity.

Programming languages

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 considers to be unsatisfactory.[8] However, many programmers have become accustomed to this order. The relative precedence levels of operators found in many C-style languages are 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
15 , Comma operator

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) )

The accuracy of software developer knowledge about binary operator precedence has been found to closely follow their frequency of occurrence in source code.[9]

References

1. [Allen R. Angel, Elementary Algebra for College Students 8/E; Chapter 1, Section 9, Objective 3]
2. Dennis M. Ritchie: The Development of the C Language. In History of Programming Languages, 2nd ed., ACM Press 1996.
3. "Developer beliefs about binary operator precedence" Derek M. Jones, CVu 18(4):14–21

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