Operations with Negative Numbers

What is a negative number?

A negative number is a number less than zero. You see them in everyday life: temperatures below freezing (−5°C), money owed (a balance of −£20), or floors below ground level in a lift (−2).

On a number line, negatives sit to the left of zero and positives to the right:

... −5 −4 −3 −2 −1 0 +1 +2 +3 +4 +5 ...

The single most useful idea in this whole lesson is this: adding moves you right, subtracting moves you left. If you ever get stuck, picture the line and step along it. For a refresher on the basics, see introduction to integers.

Adding and subtracting

To add or subtract, find your starting number, then move along the line.

Example 1 — Calculate −3 + 5.

1. Start at −3.
2. We are adding, so move 5 steps right: −2, −1, 0, 1, 2.
3. You land on 2.

Example 2 — Calculate 2 − 6.

1. Start at 2.
2. We are subtracting, so move 6 steps left: 1, 0, −1, −2, −3, −4.
3. You land on −4.

The double-negative rule

What happens when a subtraction sign meets a negative sign, like 5 − (−3)? The two negatives sit next to each other and cancel into a plus:

5 − (−3) = 5 + 3 = 8

Why? Subtracting means "take away," and taking away a debt of 3 leaves you 3 better off — so the value goes up. Watch the signs carefully:

Expression	Rewrite	Answer
4 + (−2)	4 − 2	2
4 − (−2)	4 + 2	6
−3 + (−5)	−3 − 5	−8
−3 − (−5)	−3 + 5	2

Two different signs together become a minus; two same signs together become a plus.

Multiplying and dividing

For multiplication and division there is one tidy rule about the signs:

Same signs → positive answer. Different signs → negative answer.

Work out the size of the answer as normal, then decide the sign.

Calculation	Signs	Answer
(+6) × (+2)	same	+12
(−6) × (+2)	different	−12
(+6) × (−2)	different	−12
(−6) × (−2)	same	+12

Example 3 — Calculate (−8) × (−4).

1. Multiply the numbers: 8 × 4 = 32.
2. The signs are the same (both negative), so the answer is positive.
3. Result: +32.

Example 4 — Calculate 15 ÷ (−3).

1. Divide: 15 ÷ 3 = 5.
2. The signs are different, so the answer is negative.
3. Result: −5.

A common trap

Be careful: the sign rules for multiplying and dividing are not the same as for adding. For example, −2 + (−3) = −5 (you move further left), but (−2) × (−3) = +6 (same signs give positive). Always check whether you are adding or multiplying before applying a rule.

A practice activity

Play "temperature drop":

1. Write a start temperature on paper, e.g. 3°C.
2. Roll a dice. An even roll means the temperature rises by that many degrees; an odd roll means it falls.
3. Keep a running total, writing negatives whenever you drop below zero.
4. Challenge: after five rolls, work out the difference between your highest and lowest temperatures using subtraction of negatives.

Where this leads

Confidence with negative numbers is essential for algebra, coordinates and exponents and powers, where signs appear constantly. Whenever you are unsure, slow down, draw a quick number line, and remember the golden rule: same signs positive, different signs negative.