Let’s take a close look at the processes, also called operations, known as addition and subtrac-
tion. Much of this material will seem like a review of arithmetic to you, but you’ll need to
know it “forward and backward” to work with the algebra to come later.Moving Up and Down
Adding a number to another, or subtracting a number from another, are sophisticated ways of
counting. When you do these operations with integers, it’s like moving up or down, point-by-
point, on a vertical number line of the sort you saw in the last chapter.Absolute value
Imagine the “number reflector” from Chap. 3 as a flat plane perpendicular to the number
line and passing through 0, as shown in Fig. 4-1. Every number is a certain distance above or
below the “number reflector.” The distance of an integer from the “number reflector” is called
theabsolute value of the integer.
To denote absolute value, you enclose a numeral or expression between vertical lines. The
absolute value of 2 is written |2|, and the absolute value of −3 is written |−3|. If you have any
quantity, no matter how complicated, you can always indicate its absolute value by putting
vertical lines on either side of the set of symbols that represents it.
There’s no such thing as negative absolute value, because it is an expression of distance
without taking direction into account. To say that a number has an absolute value of −3 is like
saying that your house is −3 miles from your cousin’s house on the other side of town. That’s
nonsense! You can talk about direction as well as distance. Then negative values are possible.
When you move or travel over a certain distance in a certain direction, it is called displace-
ment, and it can be positive or negative. It can even go off in other directions, such as west, or
straight up, or toward the sun, or toward your cousin’s house.
The absolute value of any natural number (or nonnegative integer) is equal to that natural
number. The absolute value of any negative integer is its “image” in the “number reflector.” If51CHAPTER4 Addition and Subtraction
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