How negative numbers work, and the rules for their signs

What a negative number is

Negative numbers describe amounts below a starting point: a temperature of −5 °C, a bank balance that is $20 overdrawn, a depth of 30 m below sea level, a loss of 4 yards in football. Zero is the starting point, and it is neither positive nor negative.

On a number line, positive numbers go to the right of zero and negative numbers go to the left. Every number has an opposite at the same distance on the other side: the opposite of 7 is −7, and the opposite of −7 is 7.

The distance from zero is called the absolute value, written with bars. Both |−7| and |7| equal 7. Absolute value is the size of a number without its sign, and it is the tool behind all of the sign rules below.

Comparing and ordering negative numbers

Farther left on the number line means smaller. So −8 is less than −3, even though 8 is bigger than 3. A temperature of −8 °C is colder than −3 °C, and owing $8 is worse than owing $3.

To order a list such as 4, −6, 0, −1 and 2 from least to greatest, place them on a number line and read left to right: −6, −1, 0, 2, 4.

Adding negative numbers

Picture moving along the number line: adding a positive number moves right, adding a negative number moves left. That gives two rules:

• Same signs: add the absolute values and keep the shared sign. −4 + (−9) = −13.
• Different signs: subtract the smaller absolute value from the larger one and keep the sign of the number with the larger absolute value. −11 + 4 = −7, and 11 + (−4) = 7.
• A number plus its opposite is always zero: −6 + 6 = 0.

Subtracting: add the opposite

Subtracting a number is the same as adding its opposite. So 5 − 8 becomes 5 + (−8), which is −3, and 5 − (−8) becomes 5 + 8, which is 13. Once a subtraction is rewritten as an addition, the adding rules do the rest.

Why does subtracting a negative increase the result? Subtraction measures a difference. The gap between 5 °C and −8 °C on a thermometer is 13 degrees, so 5 − (−8) must be 13. Another way to see it: taking away a debt leaves you better off.

Multiplying and dividing: the sign rules

Multiply or divide the absolute values as usual, then decide the sign:

• Positive times or divided by positive is positive: 6 × 3 = 18.
• Negative times or divided by positive is negative: −6 × 3 = −18, and −18 ÷ 3 = −6.
• Negative times or divided by negative is positive: −6 × (−3) = 18, and −18 ÷ (−3) = 6.
• With several factors, count the negatives: an even number of them gives a positive product, an odd number gives a negative one.

Why a negative times a negative is positive

Look at a pattern. 3 × (−2) = −6, 2 × (−2) = −4, 1 × (−2) = −2, 0 × (−2) = 0. Each time the first number drops by 1, the answer goes up by 2. Continuing the pattern, −1 × (−2) must be 2, and −2 × (−2) must be 4.

There is also an algebra reason. We know −2 × (3 + (−3)) = −2 × 0 = 0. Distributing gives −2 × 3 plus −2 × (−3), which is −6 plus something. For the total to be 0, that something must be +6. Any other answer would break the distributive property that the rest of arithmetic depends on.

Worked example: a temperature problem

Word problems with negatives are easier if you write each change as a signed number: rises and gains are positive, drops and losses are negative.

Common sign mistakes

Watch for these:

• Using the multiplication rule for addition: −3 + (−5) is −8, not 8. Two negatives make a positive only when multiplying or dividing.
• Forgetting to change both signs when subtracting a negative: 4 − (−6) is 4 + 6.
• Thinking −10 is greater than −2 because 10 is greater than 2.
• Confusing −3² (which is −9) with (−3)² (which is 9).
• Losing a negative sign when copying a line. Circle every minus sign before you start.
• Writing two signs side by side, as in 5 + −3. Put the negative number in parentheses: 5 + (−3).

How to practise and check your work

Sketch a quick number line for addition and subtraction until the moves feel automatic. For multiplication and division, decide the sign first, write it down, and only then work out the size.

To check a subtraction, add back: if 5 − (−8) = 13, then 13 + (−8) should give 5, and it does. To check a division, multiply: if −18 ÷ (−3) = 6, then 6 × (−3) should be −18.