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2.6 Dividing Signed Numbers

The rules for dividing signed numbers are exactly analogous to the rules for multiplying them.

When two signed numbers are divided (and the divisor is nonzero)

• if the dividend and divisor have the same sign, the quotient
is positive

• if the dividend and the divisor have opposite signs, the quo-
tient is negative.

In both cases, the absolute value of the quotient is the quotient of
the individual absolute values.

Notice that we have made no mention of remainders here. To do so would require a new definition
of the quotient; this is not worth the trouble. When a division of signed numbers is not exact (i.e, when
the remainder is not zero) it is much better to treat the division as a fraction.We’ll do that in the
next chapter. For now, we consider only exact divisions. In this case, as with positive numbers, we can
restate division in terms of multiplication, and that explains the rules above. For example, because

6 × 4= 24,

we say that 24 ÷ 6= 4 (or 24 ÷ 4 = 6.) Similarly, we say that

24 ÷ −6= −4,

because
24 = (−6) × (−4).

Example 66. Express the statement −72 = −8 · 9 as an exact division of signed numbers.

Solution. We can write
−72 ÷ (−8) = 9
or we can write
−72 ÷ 9= −8.

Example 67. The division 63 ÷ (−7) is exact. Find the quotient.

Solution. Since −63 = 9(−7),
63 ÷ (−7) = −9,
that is, the quotient is −9.

We remind you that 0 cannot be the divisor in any division problem: repeated subtraction of 0 has
no effect on any number (recall the discussion in Section 1.6). This remains true for signed numbers.
Example 68. (−23) ÷ 0is undefined.

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