A student claims that (−18) ÷ (−3) = −6, reasoning that 'two negatives in division must give a negative, like subtracting a negative.' What is wrong with this reasoning?
ANothing — two negatives in division do produce a negative result
BDivision sign rules are the inverse of multiplication sign rules, so two negatives give a positive
CThe reasoning is wrong because division sign rules are unrelated to multiplication sign rules
DThe sign rules for division are the same as multiplication, so (−18) ÷ (−3) = 6, because division is the inverse of multiplication and must be consistent with it
Division is the inverse of multiplication and must use identical sign rules for consistency. To check: what number times (−3) gives (−18)? That number is 6, because 6 × (−3) = −18. So (−18) ÷ (−3) = 6. The student's confusion likely comes from mixing up different contexts where 'two negatives' have different effects — subtracting a negative does add, but dividing two negatives gives a positive, just like multiplying two negatives.
Question 2 Multiple Choice
Which of the following expressions is UNDEFINED?
A0 ÷ (−7)
B(−5) ÷ 5
C(−12) ÷ (−4)
D(−9) ÷ 0
(−9) ÷ 0 is undefined because you are asking: what number times 0 gives −9? Nothing times zero ever equals a nonzero number, so no answer exists. The other expressions all have valid answers: 0 ÷ (−7) = 0 (zero divided by anything nonzero is 0), (−5) ÷ 5 = −1, and (−12) ÷ (−4) = 3. The confusion between 0 ÷ n (= 0) and n ÷ 0 (undefined) is one of the most common errors.
Question 3 True / False
The sign rules for dividing integers are the same as the sign rules for multiplying integers: same signs give a positive result, different signs give a negative result.
TTrue
FFalse
Answer: True
This is precisely the key insight of the topic. Division is the inverse of multiplication, so the sign rules must be identical for the two operations to be consistent. If same-sign multiplication gives positive, then same-sign division must also give positive — otherwise the operations would contradict each other.
Question 4 True / False
Dividing zero by any number is undefined, just like dividing any number by zero.
TTrue
FFalse
Answer: False
These two cases are completely different. Dividing zero BY a nonzero number gives zero: 0 ÷ 5 = 0, because 0 × 5 = 0. Dividing a nonzero number BY zero is what is undefined, because no number times 0 can give a nonzero result. The inverse-of-multiplication perspective makes this clear: 0 ÷ n asks 'what times n gives 0?' — the answer is always 0. But n ÷ 0 asks 'what times 0 gives n?' — impossible.
Question 5 Short Answer
A classmate says: 'Since (−3) × (−4) = 12, I must memorize a separate sign rule for division to figure out (−12) ÷ (−3).' Why is this reasoning wrong, and what is the correct answer?
Think about your answer, then reveal below.
Model answer: (−12) ÷ (−3) = 4. No separate rule is needed — division is the inverse of multiplication, so the sign rules are forced to be the same. To compute (−12) ÷ (−3), ask: what number times (−3) gives (−12)? The answer is 4, because 4 × (−3) = −12. Since same-sign multiplication gives positive, same-sign division must also give positive. The sign rules aren't separate facts to memorize; they follow automatically from the relationship between the two operations.
The power of understanding division as the inverse of multiplication is that you don't need to memorize sign rules separately. Every division problem can be checked by converting it to a multiplication question. This is the conceptual core of the topic and the reason sign rules for division and multiplication are identical.