What is (-4) × (-3)?
The Question
Find the value of (-4) × (-3).
Many students find this confusing — “How can a negative times a negative be positive?” Let’s understand WHY, not just remember the rule.
First — What We Already Know
We’re comfortable with:
4 × 3 = 12 (positive × positive = positive)
What about 4 × (-3)? Think of it as repeated addition:
4 × (-3) = (-3) + (-3) + (-3) + (-3) = -12
So positive × negative = negative.
Now the real question: what is negative × negative?
Understanding Through Patterns
The best way to see why negative × negative = positive is through a pattern.
Let’s look at what happens when we multiply by 4, then 3, then 2, then 1, then 0, then -1:
| Calculation | Result | Pattern |
|---|---|---|
| 4 × 3 | 12 | |
| 3 × 3 | 9 | Decreasing by 3 each time |
| 2 × 3 | 6 | |
| 1 × 3 | 3 | |
| 0 × 3 | 0 | |
| (-1) × 3 | -3 | Continuing: subtract 3 more |
| (-2) × 3 | -6 | |
| (-3) × 3 | -9 | |
| (-4) × 3 | -12 |
The pattern decreases by 3 at each step. Continuing below zero gives negative values. This confirms negative × positive = negative.
Now, let’s do the same with multiplying by (-3):
| Calculation | Result | Pattern |
|---|---|---|
| 4 × (-3) | -12 | |
| 3 × (-3) | -9 | Increasing by 3 each step |
| 2 × (-3) | -6 | |
| 1 × (-3) | -3 | |
| 0 × (-3) | 0 | |
| (-1) × (-3) | +3 | Continuing: add 3 more |
| (-2) × (-3) | +6 | |
| (-3) × (-3) | +9 | |
| (-4) × (-3) | +12 |
The pattern increases by 3 each step as we go below zero. So (-4) × (-3) = +12!
The pattern forces this result. It’s not a made-up rule — it follows logically.
The “Opposite of Opposite” Explanation
Here’s an intuitive way to think about it:
- Multiplying by -1 means “take the opposite.”
- (-1) × 5 = -5 (opposite of 5 is -5). ✓
- (-1) × (-5) = +5 (opposite of -5 is +5). ✓
Now, (-4) = (-1) × 4.
So: (-4) × (-3) = (-1) × 4 × (-3) = (-1) × (-12) = +12.
Taking the opposite of -12 gives us +12.
Real Life Story: Video of a Man Walking Backwards
Imagine a video of a man walking backwards (in reverse direction).
- Man walks forward (+) at normal speed (+): He moves forward. (+) × (+) = (+)
- Man walks backward (-) at normal speed (+): He moves backward. (-) × (+) = (-)
- Man walks forward (+) but video is played in reverse (-): He appears to move backward. (+) × (-) = (-)
- Man walks backward (-) and video is played in reverse (-): He appears to move forward! (-) × (-) = (+)
The double reversal brings you back to the forward direction!
The Sign Rule
- (+) × (+) = (+)
- (+) × (−) = (−)
- (−) × (+) = (−)
- (−) × (−) = (+)
Summary: Same signs → Positive. Different signs → Negative.
Solving (-4) × (-3)
Now that we understand why, let’s solve it formally.
Step 1: Both numbers are negative — same signs.
Step 2: Multiply the values (ignoring signs):
4 × 3 = 12
Step 3: Same signs → result is positive.
(-4) × (-3) = +12
What About More Than Two Negatives?
Interesting question! When we multiply several negative numbers:
- Two negatives: positive
- Three negatives: negative
- Four negatives: positive
- Five negatives: negative
Rule: Count how many negative numbers are being multiplied.
- Even count → positive result
- Odd count → negative result
Examples:
(-2) × (-3) × (-4) = +6 × (-4) = -24 (3 negatives = odd = negative) (-1) × (-2) × (-3) × (-4) = 2 × 12 = +24 (4 negatives = even = positive)
Common mistake: Writing (-4) × (-3) = -12.
Many students automatically put a negative sign because they see two negative numbers. But the rule is: same signs give a positive result.
Always ask: “Are the signs the same or different?” Same → positive. Different → negative.
Try These Similar Problems
Problem 1: (-6) × (-7) = ?
Same signs (both negative). 6 × 7 = 42. Answer: +42
Problem 2: (-5) × 3 = ?
Different signs. 5 × 3 = 15. Answer: -15
Problem 3: (-2) × (-3) × (-5) = ?
Step 1: (-2) × (-3) = +6 (same signs → positive) Step 2: (+6) × (-5) = -30 (different signs → negative) Answer: -30
Shortcut: 3 negative signs = odd number = negative result. 2 × 3 × 5 = 30, so answer is -30.
Exam tip: The most common trap in multiplication of integers is forgetting that negative × negative = positive. Before writing your answer, always check: same signs → positive, different signs → negative. This one rule covers all cases.