order of operations

order of operations

The

order of operations

is a convention that tells us how to evaluate mathematical

expressions

(these could be purely numerical). The problem arises because expressions consist of

operators

applied to

variables

or values (or other expressions) that each demand

individual

evaluation

, yet the order in which these individual evaluations are done leads to different outcomes.

A conventional order of operations solves this. One could technically do without memorizing this convention, but the only alternative is to use parentheses to group every single term of an expression and evaluate the innermost

operations

first.

For example, in the expression

a

⋅

b

+

c

, how do we know whether to apply

multiplication

or

addition

first? We could interpret even this simple expression two drastically different ways:

1.

Add

b

and

c

,

2.

Multiply the sum from (1) with

a

.

or

1.

Multiply

a

and

b

,

2.

Add to the

product

in (1) the value of

c

.

One can see the different outcomes for the two cases by selecting some different values for

a

,

b

, and

c

. The issue is resolved by convention in order of operations: the correct evaluation would be the second one.

The nearly

universal

mathematical convention dictates the following order of operations (in order of which operators should be evaluated first):

1.

Factorial

.

2.

Exponentiation

.

3.

Multiplication.

4.

Division

.

5.

Addition.

Any parenthesized expressions are automatically higher “priority” than anything on the above list.

There is also the problem of what order to evaluate repeated operators of the same type, as in:

a

/

b

/

c

/

d

The

solution

in this problem is typically to assume the left-to-right

interpretation

. For the above, this would lead to the following evaluation:

(

(

(

a

/

b

)

/

c

)

/

d

)

In other words,

1.

Evaluate

a

/

b

.

2.

Evaluate (1)/

c

.

3.

Evaluate (2)/

d

.

Note that this isn’t a problem for

associative

operators such as multiplication or addition in the reals. One must still proceed with caution, however, as associativity is a notion bound up with the

concept

of groups rather than just operators. Hence, context is extremely important.

Exponentiation is an exception to the left-to-right

assumption

, as it is evaluated right-to-left. That is,

a

^

b

^

c

is computed as

1.

Evaluate

b

^

c

.

2.

Evaluate

a

^(1).

Of course, this could also have been written as

a

b

c

, and in this form can be thought of as evaluated “highest to lowest”.

For more obscure operations than the ones listed above, parentheses should be used to remove ambiguity. Completely new operations are typically assumed to have the highest priority, but the

definition

of the operation should be accompanied by some sort of explanation of how it is evaluated in

relation

to itself. For example, Conway’s

chained arrow notation

explicitly defines what order repeated applications of itself should be evaluated in (it is right-to-left rather than left-to-right)!

Title

order of operations

Canonical

name

OrderOfOperations

Date of creation

2013-03-22 13:24:28

Last modified on

2013-03-22 13:24:28

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

7

Author

akrowne (2)

Entry type

Definition

Classification

msc 00A99

Synonym

operator precedence

Synonym

precedence of operations

Generated on Tue Feb 6 22:21:28 2018 by

LaTeXML