Negative numbers
Zero is not the end of the number line. Keep counting backwards past it and the numbers go below zero: −4 means four below zero, and it is a smaller number than −1, not a bigger one.
The negative numbers tool
Both sides of zero
Drag the marker across zero
−4 4 below zero
Zero sits in the middle.
How negative numbers works
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Put zero in the middle
The number line does not stop at zero, it carries on to the left. Zero is the point where counting changes direction: numbers to its right are above zero, numbers to its left are below it. The marker on this page can sit on either side.
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Count left for numbers below zero
Minus four sits four steps to the left of zero, and it is written −4. The minus sign is part of the number rather than an instruction to take something away: −4 is a place on the line, the place four below zero.
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Compare by position, not by digits
The further left a number is, the smaller it is. So −4 is smaller than −1, and −1 is smaller than 0, and 0 is smaller than 1. Four is further below zero than one is, and below zero means less — that is the whole of it.
Worked examples
−4 and −1
Minus one is bigger. Both sit below zero, but −1 is only one step down while −4 is four steps down, so −1 is the larger number. Reading the digits as a size is the mistake here: 4 is bigger than 1, but −4 is not bigger than −1.
2 take away 5
Minus three. Count back from 2: one, zero, and then keep going past zero — minus one, minus two, minus three. Counting back past zero is the everyday way people first meet negative numbers, and the answer is −3.
A temperature of −4°C
Four degrees below freezing. The minus sign says which side of zero the temperature is on, and 4°C is warmer than −4°C by eight degrees. Thermometers use negative numbers because there is real weather below zero, not just above it.
Where it goes wrong
- Thinking −4 is bigger because 4 is bigger The sign flips the meaning. Below zero, further from zero means less: −4 is four steps below and −1 is one step below, so −1 is the bigger number. Reading the line from left to right is the reliable check.
- Treating the minus as a subtraction to perform Sometimes a minus sign does mean "take away", as in 7 − 3, and sometimes it belongs to a number, as in −3. The context tells you which: a sign in front of a number at the start of a sentence is part of the number.
- Leaving zero out of the ordering Zero is a number like any other and it sits between the positives and negatives. Writing −3 < 2 is right; forgetting that 0 comes between them leads to mistakes such as sorting −2 next to 2.
Questions
What does a negative number mean in real life?
It means a measurement that has gone below a zero point we chose. A temperature of −4°C is four degrees below freezing, a balance of −20 is twenty in debt, and a floor at −2 is two storeys below ground. The negative sign records the direction as well as the size.
Is −4 bigger than −1?
No, and this is the classic trap. On the number line −4 is further left than −1, and further left means smaller. Think of a bank balance: being £4 overdrawn is worse than being £1 overdrawn, so −4 is the smaller amount.
What happens when you count back past zero?
You carry on into the negative numbers rather than stopping. Counting back five from 2 gives 1, 0, −1, −2, −3. Nothing special happens at zero; the counting just continues to the left, which is why the line is drawn in both directions.
Is zero negative?
No. Zero is neither positive nor negative: it is the point they are measured from, and it sits exactly between them. It is worth being precise about this, because a long list from −3 to 3 has zero in the middle rather than at either end.
How do negative numbers appear in temperature and money?
As a way of recording how far below a reference point you are. Freezing and "no money" are both zero points, so weather below freezing and an overdrawn account are recorded with minus signs. Once the reference point is agreed, one number carries both the size and which side of it you are on.