Negative Numbers Explained — KS3 Maths

Negative numbers are numbers less than zero, written with a minus sign, for example −3 or −15. At KS3 you need to add, subtract, multiply and divide them. The key rules: adding a negative is the same as subtracting, subtracting a negative is the same as adding, and two negatives multiplied (or divided) together make a positive.

What are negative numbers?

Negative numbers sit to the left of zero on the number line. They appear in everyday life: temperatures below zero (−5°C), bank overdrafts (−£200), and floors below ground in lifts.

Number line to remember:

... −5 −4 −3 −2 −1 0 1 2 3 4 5 ...

Moving right increases the value; moving left decreases it. So −1 is larger than −5 (because it is further right).

How do you order and compare negative numbers?

The further left on the number line, the smaller the number.

Example: Order these from smallest to largest: 3, −7, 0, −2, 5

Smallest to largest: −7, −2, 0, 3, 5

A quick check: −7 < −2 because −7 is further left.

How do you add and subtract negative numbers?

Two adjacent signs can be simplified:

Signs | Becomes | Example
+ + | + | 5 + (+3) = 5 + 3 = 8
+ − | − | 5 + (−3) = 5 − 3 = 2
− + | − | 5 − (+3) = 5 − 3 = 2
− − | + | 5 − (−3) = 5 + 3 = 8

The rule is: two signs the same make a positive; two different signs make a negative.

Worked example 1 — adding a negative

Calculate 7 + (−4).

Two different signs (+ and −) → becomes subtraction: 7 − 4 = 3

Answer: 3

Worked example 2 — subtracting a negative

Calculate 3 − (−5).

Two same signs (− and −) → becomes addition: 3 + 5 = 8

Answer: 8

Worked example 3 — crossing zero

Calculate −6 + 10.

On the number line, start at −6 and move 10 to the right: land on 4.

Answer: 4

Alternatively: 10 − 6 = 4 (the positive is larger, so the answer is positive).

Worked example 4 — both negative

Calculate −4 + (−3).

+ (−3) becomes −3: so −4 − 3 = −7

Answer: −7

How do you multiply negative numbers?

For multiplication and division, ignore the signs first, multiply (or divide) the numbers, then apply the sign rule.

Sign rule:

First number | Second number | Result
Positive | Positive | Positive
Positive | Negative | Negative
Negative | Positive | Negative
Negative | Negative | Positive

Short version: same signs → positive; different signs → negative.

Worked example 5 — multiplying negatives

Calculate (−4) × (−5).

Ignore signs: 4 × 5 = 20 Same signs (both negative): result is positive

Answer: 20

Worked example 6 — mixed signs

Calculate 6 × (−3).

Ignore signs: 6 × 3 = 18 Different signs (positive × negative): result is negative

Answer: −18

How do you divide negative numbers?

The same sign rule applies to division.

Worked example 7 — dividing negatives

Calculate (−20) ÷ (−4).

Ignore signs: 20 ÷ 4 = 5 Same signs (both negative): result is positive

Answer: 5

Worked example 8 — different signs

Calculate (−15) ÷ 3.

Ignore signs: 15 ÷ 3 = 5 Different signs (negative ÷ positive): result is negative

Answer: −5

Using a number line to add and subtract

A number line is the most reliable visual tool for addition and subtraction involving negatives.

Adding a positive number: move right

Adding a negative number: move left

Subtracting a positive number: move left

Subtracting a negative number: move right

Example: −3 − (−8) The double negative becomes addition: −3 + 8 = 5 On the number line: start at −3, move 8 places right → land on 5. ✓

Common mistakes with negative numbers

Mistake | Error | Correct
Confusing subtraction and negative signs | −4 − −2 = −6 | −4 + 2 = −2 (double negative → +)
Thinking −10 > −2 | Ordering by size of digits | −2 > −10 (−2 is to the right on the number line)
Forgetting sign rule in multiplication | (−3) × (−4) = −12 | Same signs → positive: 12
Applying sign rule to addition | (−3) + (−4) = 12 | Sign rule is for × and ÷ only: (−3) + (−4) = −7

Why do two negatives make a positive when multiplied?

A simple way to see this: multiplying by −1 reverses direction on the number line. If 3 × (−1) = −3 (one reversal), then (−3) × (−1) = +3 (reversing the reversal). Two reversals bring you back to where you started. This pattern is consistent throughout all of maths — you will see it again in algebra, coordinate geometry and trigonometry.

Negative numbers in the KS3 national curriculum

The DfE's KS3 mathematics programme of study lists "use directed number in context" as a required skill for all pupils, including ordering, adding, subtracting, multiplying and dividing negative integers. The AQA GCSE Mathematics specification requires directed number fluency as a prerequisite for algebra — reinforcing the number line approach alongside the sign rules is far more effective than memorising the rules alone.

Frequently asked questions

What is a negative number?

A negative number is any number less than zero, written with a minus sign in front of it, such as −1, −12, or −0.5. On the number line, negative numbers are to the left of zero.

How do you add and subtract negative numbers?

Simplify the two adjacent signs first: two like signs (+ + or − −) become a positive; two unlike signs (+ − or − +) become a negative. Then carry out the resulting addition or subtraction. Using a number line makes this very clear.

Why does a negative times a negative equal a positive?

Multiplying by a negative number reverses direction on the number line. Applying that reversal twice — as in negative × negative — takes you back to a positive value. This is consistent with all other rules in maths and can be proved from the properties of arithmetic.

How do you divide negative numbers?

Divide as normal, ignoring the signs, then apply the sign rule: if both numbers have the same sign, the answer is positive; if they have different signs, the answer is negative.

What is the biggest mistake students make with negative numbers?

The most common error is applying the multiplication sign rule to addition. The rule "same signs give positive" applies only to multiplication and division. When adding, (−3) + (−4) = −7, not +7.

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