not done so well. The group leader asks, “How much did you cut back? Go on, don’t be afraid to tell
us.” You blush, clear your throat, hold your head up, and declare, “I reduced my driving last month
by negative 80 miles.”
Here’s a challenge!
Start with the integer 5, add −3 to it, then subtract −6 from that, then subtract 10 from that, then add 14
to that, and finally subtract −21 from that. What’s the result?
Solution
We can break this down step-by-step, paying careful attention to signs and using parentheses when we
need them. Here we go:
5 + (−3)= 5 − 3 = 2
2 − (−6)= 2 + 6 = 8
8 − 10 =− 2
− 2 + 14 = 12
12 − (−21)= 12 + 21 = 33
The Commutative Law for Addition
In basic arithmetic, you learned that you can add a long string of numbers backward or for-
ward, and it doesn’t matter. Good accountants take advantage of this when checking their
work. They’ll add up a column of numbers from top to bottom, then again from bottom to
top, just to be sure they have done the arithmetic correctly.
It works when you add
The fact that you can add two integers in either order and get the same result is called the
commutative law for addition. It means you can commute (interchange) the two numbers you’re
adding, called the addends, and get the same sum either way. In formal terms, a mathematician
would say that for any two integers a and b,
a+b=b+a
This works whether the numbers are positive, negative, or 0. It also works if there are three,
four, five, or more numbers in a sum, as long as the number of addends is not infinite.
It fails when you subtract
In subtraction, the order does matter. It’s easy to find an example that shows why. Consider
this:
3 − 5 =− 2
The Commutative Law for Addition 57
