Chapter 3: Order of Operations and Integers 37When you see a minus sign in front of parentheses, as in 14 – (8 + 3), it says to do what’s in the
parentheses and subtract the result from the number before the minus sign. The distributive
property gives you the option to distribute the minus and rewrite the problem as 14 – 8 – 3.
Taking away the sum of 8 and 3 is the same as taking away 8 and taking away 3.
CHECK POINT
Tell whether it’s easier to perform the operation in parentheses first or to distribute
first. Then solve the problem.- 2(35 + 14)
- 3(20 + 8)
- 7(100 – 2)
9. 15(40 – 14)
10. 250(1,000 – 400)
By now you know how to do all your arithmetic, and you’ve memorized your addition facts and
multiplication tables. You know that the commutative, associative, and distributive properties are
handy helpers, and you’ve got PEMDAS totally under control. What else is there to say about
arithmetic? Well, a lot actually. Some of it can wait for a later chapter, but this is a good time to
tackle the half-truth you probably learned about subtraction.
As soon as you begin to subtract, you run into a problem. You can subtract 5 – 3 and get 2, but
what happens if you subtract 3 – 5? The easy answer is, “you can’t take a bigger number away
from a smaller one.” In some sense that’s true. If you only have 3 cookies, you can’t give me 5
cookies, which is sad, because I love cookies.
But what if you promised me 5 cookies? You owe me 5 cookies. If you give me 3 cookies, you
still owe me 2 more cookies. You don’t have cookies, but you owe 2 cookies. In a sense, you have
less than no cookies, because even if you get more cookies, you have to give me 2. How do we
write, in symbolic form, that you owe me 2 cookies? That opposite-of-having idea is written by
putting a negative sign in front of the 2. You have -2 cookies. The number -2, or negative 2, is the
opposite of 2.
This notion of needing a way to express opposites leads you to a bigger set of numbers. Every
number has an opposite, and the whole numbers didn’t take that into account. So you need a
larger set of numbers that will include all the whole numbers and their opposites. Those numbers
are called the integers.