Division is simply the “inverse” (the opposite) of multiplication. If we know that $4 \times 3 = 12$, then we know that $12 \div 4 = 3$. We can use this same logic to figure out how negative signs behave.
Look at these four scenarios:
Positive ÷ Positive: $12 \div 4 = \mathbf{3}$
(Because $4 \times 3 = 12$)
Negative ÷ Positive: $-12 \div 4 = \mathbf{-3}$
(Because $4 \times -3 = -12$)
Positive ÷ Negative: $12 \div -4 = \mathbf{-3}$
(Because $-4 \times -3 = 12$)
Negative ÷ Negative: $-12 \div -4 = \mathbf{3}$
(Because $-4 \times 3 = -12$)
If you don’t want to think about the multiplication inverse every time, you can just memorize these four simple sign rules. They work exactly like the rules for multiplication!
| Signs in Problem | Resulting Sign | Example |
| $(+) \div (+)$ | $(+)$ | Same signs = Positive |
| $(-) \div (-)$ | $(+)$ | Same signs = Positive |
| $(+) \div (-)$ | $(-)$ | Different signs = Negative |
| $(-) \div (+)$ | $(-)$ | Different signs = Negative |
Pro Tip: If the signs are the same, the answer is positive. If the signs are different, the answer is negative.
Let’s look at a larger number to see these rules in action. Let’s take the number 101,010 and divide it by 2.
Both Positive: $\frac{101010}{2} = 50505$
Top Negative: $\frac{-101010}{2} = -50505$
Both Negative: $\frac{-101010}{-2} = 50505$
Bottom Negative: $\frac{101010}{-2} = -50505$