Negative numbers

Order, add, subtract, multiply and divide negative numbers, using the number line and the sign rules.

Also known as: directed numbers, integers, adding and subtracting negative numbers, multiplying negative numbers.

GCSE Maths · Number IGCSE Maths · Number

Key ideas

On a number line, numbers get smaller as you move left: $-7 < -2 < 0 < 3$ .
Subtracting a negative is the same as adding: $4 - (-6) = 4 + 6 = 10$ .
Multiplying or dividing two numbers with the same sign gives a positive answer; different signs give a negative answer.
$(-2)^2 = 4$ but $(-2)^3 = -8$ : an even power of a negative is positive, an odd power is negative.

The number line

Negative numbers are numbers less than zero. They're used for temperatures below freezing, bank balances in debt, and floors below ground level.

On a number line, numbers increase to the right and decrease to the left. So $-7$ is smaller than $-2$ , even though 7 is bigger than 2: $-7$ is further below zero.

Differences work the same way as with positive numbers. If the temperature goes from $-3\,°\text{C}$ to $5\,°\text{C}$ , it has risen by $5 - (-3) = 8$ degrees: 3 degrees to reach zero, then 5 more.

Adding and subtracting

Adding a positive moves right; subtracting a positive moves left: $-3 + 8 = 5$ , and $2 - 9 = -7$ .
Adding a negative is the same as subtracting: $6 + (-4) = 6 - 4 = 2$ .
Subtracting a negative is the same as adding: $-4 - (-9) = -4 + 9 = 5$ .

A useful way to think about the last rule: taking away a debt leaves you better off.

Multiplying and dividing

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

Signs	Result	Examples
Same signs	Positive	$(-5) \times (-4) = 20$ , and $(-12) \div (-3) = 4$
Different signs	Negative	$(-5) \times 4 = -20$ , and $24 \div (-3) = -8$

With several numbers, count the negatives: an even number of negative factors gives a positive answer, an odd number gives a negative one. That's why $(-2)^2 = 4$ and $(-2)^3 = -8$ . Watch the brackets: $-2^2 = -4$ , because the power applies only to the 2.

Worked examples

Example 1 · Subtracting a negative

Work out $-4 - (-9)$ .

Step 1.Subtracting a negative is the same as adding: $-4 - (-9) = -4 + 9$ .
Step 2.Start at $-4$ and move 9 to the right: $5$ .

Answer: $5$

Example 2 · Multiplying and dividing

Work out $(-6) \times 3 \div (-2)$ .

Step 1.Left to right: $(-6) \times 3 = -18$ (different signs, so negative).
Step 2. $-18 \div (-2) = 9$ (same signs, so positive).

Answer: $9$

Example 3 · Temperature change

At 6 am the temperature is $-5\,°\text{C}$ . By noon it has risen by 12 degrees, and by midnight it has fallen by 9 degrees from the noon temperature. What is the temperature at midnight?

Step 1.Noon: $-5 + 12 = 7\,°\text{C}$ .
Step 2.Midnight: $7 - 9 = -2\,°\text{C}$ .

Answer: $-2\,°\text{C}$

Common mistakes

Thinking $-7$ is bigger than $-2$ because 7 is bigger than 2.
Treating $4 - (-6)$ as $4 - 6$ : subtracting a negative means adding.
Applying the "two negatives make a positive" rule to addition: $-3 + (-5) = -8$ , not $8$ .
Writing $-3^2 = 9$ : without brackets, the power applies only to 3, so $-3^2 = -9$ .

Try it yourself

Original practice questions. Your answer is checked instantly.

Question 1Negative numbersEasier

Work out $-3 + 8$ .

Question 2Negative numbersEasier

Which of these numbers is the smallest?