Algebra Know-It-ALL

Suppose p is an integer and n is a positive integer. From the above facts, you can see

that whenever you multiply p by n, you add p to itself (n− 1) times. On the number line,

you move away from the “number reflector” (n− 1) times by a distance equal to the abso-

lute value of p. If your starting integer p is above the “number reflector,” you move up; if

your starting integer p is below the “number reflector,” you move down. Figure 5-1 illustrates

how this works for 2 × 3 and − 2 × 5. The “number reflector” is shown as a horizontal,

dashed line.

What if n is negative instead of positive? To multiply p by n in this situation, first take the

additive inverse (negative) of your starting integer p, and then move away from the “number

reflector” |n|− 1 times by a distance equal to the absolute value of p. Figure 5-2 shows how

this works for 2 × (−3) and − 2 × (−5).

When you multiply two quantities, you get a product. In a multiplication problem, the

first quantity (the one to be multiplied) is sometimes called the multiplicand, and the second

quantity (the one you are multiplying by) is sometimes called the multiplier. More often, they

are both called factors.

To divide, move in

Now imagine the multiplication process in reverse. For examples, look at Figs. 5-1 and 5-2

and think backward!

In Fig. 5-1, suppose you start with 6 and you want to divide it by 3. You reduce your

distance from the “number reflector” by a factor of 3 but stay on the same side, so you end up

66 Multiplication and Division

Start here

Start here

Finish here

Finish here

12

8

4

• 4


• 8


• 12

Multiply by 3

Add to itself 2 times

Get 3 times as far from 0

Multiply by 5

Add to itself 4 times

Get 5 times as far from 0

Figure 5-1 On top, 2 is multiplied by 3. On the bottom,

−2 is multiplied by 5. To avoid clutter, only

the even-integer points are shown on the

number line.