Negative numbers: counting through zero and the rules

What are negative numbers?

Negative numbers are numbers below zero. We write them with a minus sign in front: −3 is "negative three", and many people say "minus three". Numbers above zero are positive numbers. Positive numbers are usually written without a sign, so 3 means positive 3.

Children meet negative numbers in real life before they meet them in maths. A thermometer on a frosty morning might show −2 °C, which is 2 degrees below freezing. A lift button marked −1 goes to the floor below the ground floor. The national curriculum asks children to use negative numbers "in context" like this in Years 5 and 6.

Counting through zero on a negative number line

A negative number line carries on to the left of zero, with the same equal steps. Counting back from 3 in ones goes 3, 2, 1, 0, −1, −2, −3. Year 4 children learn to count backwards through zero like this.

Here's what happens when the temperature is 3 °C and falls by 7 degrees. Counting back 7 goes 3 steps to zero, then 4 more steps past it:

So the temperature ends at −4 °C. Splitting the jump at zero makes it easier: 3 to get to zero, then the 4 that are left.

Counting forwards works the same way. If it's −5 °C and the temperature rises by 8 degrees, count on 5 to get to zero, then 3 more:

It finishes at 3 °C. In Year 5 children count forwards and backwards through zero in bigger steps too, and with decimals and fractions, for example 0.5, 0, −0.5, −1.

Ordering positive and negative numbers

On a number line, numbers get bigger going right and smaller going left. That means −8 is smaller than −2, even though 8 is bigger than 2. The bigger the number after the minus sign, the further below zero it is.

Put in order from smallest to largest, −6, 4, −1, 0 and 2 go: −6, −1, 0, 2, 4. A temperature of −6 °C is colder than −1 °C.

Is zero a positive or a negative number?

Zero is neither positive nor negative. It's the point on the number line that separates the positive numbers on its right from the negative numbers on its left. That's why we write 0 with no sign.

Adding and subtracting negative numbers at primary school

At primary school, adding and subtracting with negative numbers means moving along the number line. Adding moves right, and taking away moves left. The national curriculum notes that Year 6 children add and subtract positive and negative numbers on a number line "for measures such as temperature".

Worked example: the gap across zero

In Year 6, children "calculate intervals across zero": the difference between a negative number and a positive one. Say it's −4 °C at night and 6 °C by lunchtime. How much warmer is it?

Count from −4 up to zero, which is 4 degrees. Then count from zero up to 6, which is 6 more. Add the two parts: 4 + 6 = 10. It's 10 degrees warmer.

Breaking the jump at zero is the key step. It turns one tricky question into two easy ones.

Worked example: going down past zero

A diver is 2 metres above the water on a diving board and goes 5 metres down. Take away 2 to reach the water (zero), then 3 more below it. The diver ends at −3 metres, 3 metres under the surface.

Coordinates use negative numbers too. In Year 6, children plot points in all four quadrants of a grid, where the axes carry on past zero into negative numbers.

At secondary school: the rules for negative numbers

The rules for calculating with negative numbers are secondary school maths, from Year 7. They're here because they're the part parents most often want to check. Brackets keep the signs apart.

Adding and subtracting negative numbers

• Adding a negative number is the same as taking away. 5 + (−2) is the same as 5 take away 2, which is 3.
• Subtracting a negative number is the same as adding. 5 − (−2) is the same as 5 add 2, which is 7.

Multiplying negative numbers and dividing negative numbers

• Two signs the same give a positive answer. (−3) × (−4) is positive 12, and (−12) ÷ (−3) is positive 4.
• Two different signs give a negative answer. (−3) × 4 is −12, and (−12) ÷ 3 is −4.

A way to see why a negative times a negative is positive: look at the pattern 3 × (−4) is −12, 2 × (−4) is −8, 1 × (−4) is −4, 0 × (−4) is 0. Each answer goes up by 4. Carry the pattern on, and (−1) × (−4) is 4.

Are negative numbers integers and real numbers?

Whole negative numbers, like −1, −7 and −100, are integers. Integers are all the whole numbers, positive and negative, and zero. A negative number that isn't whole, like −2.5 or −¹⁄₂, is not an integer.

Negative numbers are real numbers. Every point on the number line is a real number, on either side of zero. At secondary school, children order positive and negative integers, decimals and fractions using the number line.

Common mistakes with negative numbers

• Thinking −8 is bigger than −2. It's further below zero, so it's smaller. A thermometer helps: −8 °C is colder.
• Missing out zero when counting. Counting back from 2 goes 2, 1, 0, −1. Children sometimes jump from 1 straight to −1.
• Counting the gap wrongly across zero. From −4 to 6 is 10, not 2. Split the jump at zero and add the two parts.
• Mixing up the minus sign and the take away sign. In −3, the sign is part of the number. In 7 − 3, it means take away.
• At secondary school, getting the sign wrong. Two signs the same make a positive, and two different signs make a negative. Check whether the answer should be above or below zero.

How to help at home

• Read a thermometer together. Weather forecasts in winter are full of negative numbers. Ask how much warmer tomorrow will be than today.
• Count back past zero. Count down from 5 to −5, then try counting back in twos or tens.
• Draw a number line. A line from −10 to 10 on paper makes every question easier to see. The number line guide shows how children use them.
• Use lifts and car parks. "We're on floor 2 and the car is on floor −1. How many floors down is that?"

When do children learn negative numbers?

Children count back through zero in Year 4 maths. In Year 5 maths they use negative numbers in real situations and count forwards and backwards through zero. In Year 6 maths they work out the gap across zero and use negative numbers on coordinate grids. The rules for calculating with negative numbers come at secondary school. A secure sense of place value and rounding on a number line helps too.

Negative number questions to try

1. Count back in ones from 2 to −3. Write down every number.
2. Put these in order, smallest first: −3, 2, −8, 0, −1.
3. It's 4 °C and the temperature falls by 10 degrees. What is it now?
4. It's −2 °C and the temperature rises by 9 degrees. What is it now?
5. What is the difference between −7 °C and 5 °C?
6. Secondary school challenge: what is (−4) × (−5)?

Answers

1. 2, 1, 0, −1, −2, −3
2. −8, −3, −1, 0, 2
3. −6 °C (4 down to zero, then 6 more)
4. 7 °C (2 up to zero, then 7 more)
5. 12 degrees (7 up to zero, then 5 more: 7 + 5 = 12)
6. Positive 20, because two negative numbers multiplied give a positive answer.

For the rest of what children learn in these years, see the KS2 maths guide.