Order of Operations
(Priority Rules for Arithmetic)
To ensure that even the most complicated mathematical
expressions have the same meaning forevery reader, the following
rule is always observed in evaluating an expression:
For all numerical or algebraic expressions, the order of
evaluation is:
1. parentheses or brackets first (starting with the innermost)
2. exponentials or powers next3. multiplications and divisions
next
4. additions and subtractions last
5. if an expression involves three or more operations at the
same level of priority, those operations are done from left to
right.
These rules are important. If you don’t obey them when
evaluating numerical expressions, orwhen manipulating algebraic
expressions, you will get results that the rest of the world
considersto be incorrect.
Common Error #1:
Be careful with sequences of additions and subtractions. For
example, if we proceeded asfollows:
| 5 - 3 + 6 |
|
5 - 9 |
Add the 3 to the 6 to get 9 – this
violates thefifth part of the rule. Addition and
subtractionare at the same priority level, so in this
case weshould do the ‘-‘ between the 5 and the
3before doing the ‘+’ found further to the
right. |
| |
|
- 4 |
The wrong answer! |
| Instead, the
rules require |
|
| 5 - 3 + 6 |
|
2 + 6 |
The subtract and add are at the same
level ofpriority, so the subtract, being the leftmost
ofthe two operations, is done first. Thus, 5 –
3gives 2 |
| |
|
8 |
Only one operation is left, so add 2 to 6
to get8, the correct answer! |
Common Error #2:
Another common error is to overlook higher priority operations
that may not be obviously present.This often happens when
multiplications are overlooked, because no specific multiply
operator ispresent. For example
| 3 + 5 (10 - 6) |
|
8 (10 - 6) |
Add the 3 and the five to the left. This
is anerror, because the 5 itself is to be multipliedonto
the result of evaluating the bracketedexpression, and
both the evaluation of thebracketed expression and the
multiplication arehigher priority than the addition. The
‘+’between the 3 and 5 is the lowest
priorityaccording to the rule, and yet was done firsthere
in error |
| |
|
8 (4) |
Evaluate the bracketed expression |
| |
|
32 |
which is an incorrect final result. |
| Instead, the evaluation of
this expression should have proceeded as follows: |
| |
|
|
Evaluate the bracketed expression – this is
thehighest priority operation present |
| |
|
|
Do the multiplication – it has higher
prioritythan the addition |
| |
|
|
Finally, do the remaining addition, to get thecorrect
final result of 23. |
|
