What Is a Negative Number?
A negative number is any number less than zero. On a number line, negatives are to the left of zero, positives to the right. They came into widespread use partly to represent things that can go below a baseline — temperatures below zero, money owed (debt), altitudes below sea level, or any change going in the "opposite" direction.
The number line makes it easy to visualize: moving right means adding, moving left means subtracting. Negative numbers extend the line indefinitely to the left.
Absolute Value
The absolute value of a number is its distance from zero — always positive. Written with vertical bars: |n|.
|5| = 5
|0| = 0
Absolute value ignores the sign and just tells you the size. Useful when you care about the magnitude but not the direction (e.g., "the temperature changed by 12 degrees" — that's the absolute change regardless of whether it went up or down).
Adding and Subtracting Negative Numbers
This is where most confusion starts. The key insight is that adding a negative is the same as subtracting, and subtracting a negative is the same as adding.
Adding a Negative Number
5 + (−3) = 5 − 3 = 2
−4 + (−6) = −4 − 6 = −10
Think of it as: you have £5, someone takes away £3 — you're left with £2. Adding a negative is like taking something away.
Subtracting a Negative Number
8 − (−3) = 8 + 3 = 11
−5 − (−2) = −5 + 2 = −3
Why does subtracting a negative turn into adding? Think of it as removing a debt. If you owe someone £3 (a "−3" to your finances) and they cancel the debt (subtract that negative), you effectively gained £3. Removing a loss is a gain.
Summary of Sign Rules for Addition/Subtraction
| Operation | Becomes | Example |
|---|---|---|
| + positive | + (add) | 5 + 3 = 8 |
| + negative | − (subtract) | 5 + (−3) = 2 |
| − positive | − (subtract) | 5 − 3 = 2 |
| − negative | + (add) | 5 − (−3) = 8 |
Adding Two Negatives
When you add two negative numbers, you add the absolute values and keep the negative sign.
(−3) + (−3) = −6
Adding a Positive and a Negative
Subtract the smaller absolute value from the larger one, and keep the sign of whichever had the larger absolute value.
(−3) + 9: |9| > |−3|, so answer is positive → +(9−3) = 6
Multiplying and Dividing Negative Numbers
The rules here are simpler: just track the signs separately from the numbers.
The Sign Rules
| Signs | Result Sign | Example |
|---|---|---|
| Positive × Positive | Positive | 4 × 3 = 12 |
| Positive × Negative | Negative | 4 × (−3) = −12 |
| Negative × Positive | Negative | (−4) × 3 = −12 |
| Negative × Negative | Positive | (−4) × (−3) = 12 |
The same rules apply for division. Divide the absolute values, then apply the sign rule.
(−15) ÷ (−3) = 5
20 ÷ (−4) = −5
Why Does Negative × Negative = Positive?
This is the rule that feels most mysterious. Here's an intuitive explanation:
Start with the pattern for multiplying by −3:
2 × 3 = 6
1 × 3 = 3
0 × 3 = 0
−1 × 3 = −3
−2 × 3 = −6
Each step down, the result decreases by 3. Now apply the same pattern but multiply by −3:
1 × (−3) = −3
0 × (−3) = 0
−1 × (−3) = ?
−2 × (−3) = ?
Each step down now increases by 3 (the results are going up: −6, −3, 0, so next must be +3, then +6). So −1 × −3 = 3 and −2 × −3 = 6. The pattern forces the answer to be positive.
Another way to think about it: if "negative" means "the opposite direction," then doing the opposite twice brings you back to the original direction. Two reversals equal no reversal.
Negative Numbers in Real Life
| Context | Negative Means | Example |
|---|---|---|
| Temperature | Below freezing | −10°C (10 degrees below zero) |
| Finance | Debt / loss | Account balance: −£250 |
| Elevation | Below sea level | Dead Sea: −430m |
| Coordinates | Left or below origin | Point (−3, −2) in a graph |
| Time | Before a reference point | −5 seconds (countdown) |
| Velocity | Moving in reverse | −20 km/h (moving backwards) |
Negative Exponents
Negative exponents follow directly from the rules above — a negative power means the reciprocal. So 2⁻³ = 1/8, not −8. The negative sign in the exponent position means something completely different from a negative number.
3⁻² = 1/3² = 1/9
10⁻³ = 1/1000 = 0.001
Ordering Negative Numbers
One thing people occasionally mix up: −10 is less than −2. On the number line, −10 is further left (further from zero in the negative direction). When comparing negatives, the one with the larger absolute value is actually the smaller number.
Using the Calculator
The SolveCalc calculator handles negative numbers — use the ± button (or type a minus before the number) to enter negatives. For expressions like (−3)² vs −3², be careful with parentheses as they give different results: (−3)² = 9 but −3² = −9 by the order of operations.
Conclusion
Negative numbers become intuitive once you have a solid mental model — the number line really does help. The rules for addition and subtraction make sense through the lens of "removing a removal = adding back." The multiplication sign rules (same signs positive, different signs negative) just need to be learned, though the pattern-extension argument shows why they're logically necessary. And negative × negative = positive is a consequence of mathematical consistency, not an arbitrary decision. Master these basics and negatives stop being a source of errors in your calculations.