Algebra Know-It-ALL

at the point representing 2. If you start with −10 and divide by 5, you cut your distance from
the “number reflector” by a factor of 5 and stay on the same side, so you end up at −2.
Dividing by a negative number is trickier. In Fig. 5-2, suppose you start with −6 and
divide it by −3. You jump to the other side of the “number reflector” and then reduce your
distance from it by a factor of 3, finishing at 2. If you start with 10 and divide by −5, you
jump to the other side of the “number reflector” and then cut your distance from it by a factor
of 5, finishing at −2.
In these situations, the integers divide each other “cleanly” without remainders. Remain-
ders occur when one nonzero integer divides another integer and the result is not an integer.
“Messy division” produces fractions, which we’ll study in Chap. 6. An example is the division
of 5 by 3. Another example is the division of −7 by −12.
You can add, subtract, or multiply any two integers and always end up with another inte-
ger. In math jargon, the operations of addition, subtraction, and multiplication are closed over
the set of integers. When you do a division problem with integers, you don’t always get another
integer. Therefore, the operation of division is not closed over the set of integers.
When you divide a quantity by another quantity, you get a quotient. Sometimes it is called
aratio. In a division problem, the first number is sometimes called the dividend, and the sec-
ond number is sometimes called the divisor.

Division by 0

What happens if you try to divide an integer by 0? You don’t get any integer, or any other
known type of number. Look at this problem “inside-out.” What must you multiply 0 by if

Moving Out and In 67

Start here

Start here

Finish here

Finish here

12

8

4

• 4


• 8


• 12

Multiply by - 5

Take additive inverse

Add to itself 4 times

Get 5 times as far from 0

Multiply by - 3

Take additive inverse

Get 3 times as far from 0

Add to itself 2 times

Figure 5-2 At lower left, 2 is multiplied by −3. At upper right,

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

even-integer points are shown on the number line.