Negative Numbers: Adding, Subtracting, Multiplying and Dividing

Q: Why does subtracting a negative number give a bigger answer?

Subtraction undoes addition. Adding -5 moves you 5 to the left, so subtracting -5 must move you 5 to the right, which is the same as adding 5.

Negative numbers on the number line, why subtracting a negative means adding, and why a negative times a negative is positive, shown with jumps, patterns and reflections.

4 min read Supports 3 lessons

Negative numbers describe amounts below a starting point: $-5\,°\text{C}$ is five degrees below freezing, a bank balance of $-£20$ is a debt, and $-30$ m is thirty metres below sea level.

On the number line, positive numbers run to the right of zero and negative numbers mirror them on the left.

$5$ and $-5$ are the same distance from zero, on opposite sides. They are opposites.

Ordering negative numbers

Further right always means bigger. So $-2 > -7$ , even though $7 > 2$ . A temperature of $-2$ degrees is warmer than $-7$ .

Adding and subtracting

Think of the number line as a road. Adding a positive number moves you right; adding a negative number moves you left.

$3 + (-8) = -5$ : start at 3 and move 8 to the left.

Subtracting is the reverse of adding, so subtracting a number moves you the opposite way to adding it. That gives the key rule:

Subtracting is adding the opposite

a - b = a + (-b) \qquad a - (-b) = a + b

$-4 - (-9)$
Subtracting a negative looks awkward.

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Tip

Taking away a debt makes you richer. Removing $-£10$ from what you owe leaves you £10 better off: $x - (-10) = x + 10$ .

Multiplying and dividing

Multiplying by $-1$ reflects a number to the other side of zero. Multiplying by a negative does this reflection as well as any stretching. Reflect twice and you are back where you started, which is why a negative times a negative is positive.

You can also see it in a pattern. Each line below adds $2$ to the answer, and the pattern has no reason to stop at zero:

$3 \times (-2) = -6$

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Sign rules

(+)(+) = + \quad (-)(-) = + \quad (+)(-) = - \quad (-)(+) = -

Division follows exactly the same rules.

Tip

For a long product, count the negatives. An even number of them gives a positive answer; an odd number gives a negative one.

Worked example

Compute $(-6)(-4) \div (-3)$ .

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Negative times negative is positive.
$(-6)(-4) = 24$

Watch out

The sign rules are for multiplying and dividing. For adding, $-3 + -4$ is $-7$ , not $+7$ : two debts make a bigger debt.

Common questions

Why does subtracting a negative number give a bigger answer?: Subtraction undoes addition. Adding $-5$ moves you 5 to the left, so subtracting $-5$ must move you 5 to the right, which is the same as adding $5$ .
Why is a negative times a negative positive?: Multiplying by a negative reflects a number across zero. Doing that twice returns it to the positive side. The pattern $-1 \times (-2)$ , $0 \times (-2)$ , $1 \times (-2)$ rising in steps of 2 shows the same thing.

Practise negative numbers

The The Number Line & Negatives lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Common questions

Why does subtracting a negative number give a bigger answer?

Subtraction undoes addition. Adding $-5$ moves you 5 to the left, so subtracting $-5$ must move you 5 to the right, which is the same as adding $5$ .

Why is a negative times a negative positive?

Multiplying by a negative reflects a number across zero. Doing that twice returns it to the positive side. The pattern $-1 \times (-2)$ , $0 \times (-2)$ , $1 \times (-2)$ rising in steps of 2 shows the same thing.