32 Part 1: The World of Numbers
Order of Operations ...................................................................................................................
What does 2 + 3 v 7 equal? Some people might say 2 + 3 is 5 and 5 times 7 is 35. Those people
started at the left and did what they saw as they saw it, moving across the line, a logical enough
approach. Other people would do the multiplying first and say that’s 21, then add on the 2 to get- Those people might have thought that multiplication is repeated addition, so do that first,
then the simple addition.
And what about the order of the numbers? You learned that addition is associative and commuta-
tive, so you can rearrange an addition problem. And you can rearrange a multiplication problem,
too, so can you rearrange one that has both operations? Can you change 2 + 3 v 7 to 2 + 7 v 3?
If you do, the multiplication-first people will get the same answer, but the left to right people will
get a different one than they got the first time.
Those are great questions, and you certainly don’t want 2 + 3 v 7 to have two (or more) different
values, depending on who does the arithmetic. That wouldn’t be practical. So what’s the solution?
One way to communicate what should be done first is to add parentheses. The expression
(2 + 3) v 7 tells you to add first, then multiply. (2 + 3) v 7 = 5 v 7 = 35. On the other hand,
2 + (3 v 7) says multiply first, so you get 2 + (3 v 7) = 2 + 21 = 23.
That’s helpful, but you still need to know what to do, even if people writing the problem don’t
use parentheses. So there’s an agreement among people who do arithmetic about what to do first,
second, and so on.
That agreement is called the order of operations. What’s in parentheses will always get done first,
if there are parentheses, and after that, the order will be exponents (think of them like super-
multiplication), then multiplication and division, and finally addition and subtraction.
DEFINITION
The order of operations is an agreement among mathematicians that operations
enclosed in parentheses or other grouping symbols should be done first, and then
exponents should be evaluated. After that, do multiplication and division as you meet
them moving left to right, and finally do addition and subtraction as you meet them,
moving left to right.