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Negative Numbers

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How to put negative numbers in order, add, subtract, multiply and divide with them, and use them to solve problems about temperatures and missing numbers.

Numbers below zero

A negative number is a number less than zero, such as -5. Negative numbers are used for temperatures below zero, money owed and heights below sea level.

On a number line, numbers get greater as you move right and smaller as you move left. So -7 is smaller than -2, because it is further left, even though 7 is greater than 2.

An integer is a whole number. Integers can be positive, negative or zero, such as -3, 0 and 8.

In the exam you are usually asked to put numbers or temperatures in order, to work out a calculation such as -6+2 or (-20)\div(-5), or to find a temperature after it rises or falls. These questions are worth 1 to 3 marks.

Worked example 1: Putting numbers in order

Write -4, 2, -7, 0 and -1 in order of size. Start with the smallest number.

  1. The smallest number is the one furthest left on a number line. The negative numbers come first.

  2. Your turn. Which of the five numbers is the smallest?

    -7 is the smallest, because it is furthest left.

  3. Next come -4 and then -1, because -4 is further left than -1. Then 0, then 2.
  4. Answer: -7,\ -4,\ -1,\ 0,\ 2

Adding and subtracting on a number line

Adding a positive number moves you right on the number line. Subtracting a positive number moves you left. Always start at the first number, even if it is negative.

For example, to work out 3-8, start at 3 and move 8 places left. Three places take you to 0, and five more take you to -5. So 3-8=-5

Adding a negative number moves you left, so it is the same as subtracting: 5+(-2)=5-2=3

Worked example 2: Adding across zero

Work out -4+7

  1. Start at -4. Adding 7 means moving 7 places right.
  2. Moving 4 places right takes you from -4 to 0.

  3. Your turn. How many more places right do you still need to move?

    7-4=3 more places, so you finish at 3

  4. Answer: -4+7=3

Subtracting a negative number

Subtracting is the opposite of adding. Adding -3 moves you left by 3, so subtracting -3 moves you right by 3.

So subtracting a negative number is the same as adding the positive number:

7-(-3)=7+3=10

This is only for subtracting a negative number. It does not mean that any two minus signs make a positive answer.

Worked example 3: Subtracting a negative number

Work out -6-(-9)

  1. Start at -6. The number being subtracted is -9.
  2. Subtracting -9 is the same as adding 9: -6-(-9)=-6+9

  3. Your turn. Work out -6+9

    -6+9=3

  4. Check on a number line: from -6 to 0 is 6 places right, and 3 more places take you to 3. That is 9 places right, the same as adding 9.
  5. Answer: 3

Multiplying and dividing negative numbers

To multiply or divide, first ignore the signs and work out the answer with the numbers. Then decide the sign of the answer.

  • If the two signs are the same, the answer is positive.
  • If the two signs are different, the answer is negative.

For example, (-4)\times(-5)=20 but (-4)\times5=-20. These sign rules are only for multiplying and dividing. When you add, -4+(-5)=-9

You can see why from a pattern: 2\times(-3)=-6, 1\times(-3)=-3, 0\times(-3)=0. Each answer is 3 more than the one before, so next comes (-1)\times(-3)=3

Worked example 4: Multiplying and dividing

Work out (-8)\times5 and (-27)\div(-3)

  1. 8\times5=40. The signs are different, so (-8)\times5=-40
  2. For the division, 27\div3=9. Both numbers are negative, so the signs are the same.

  3. Your turn. Work out (-27)\div(-3)

    (-27)\div(-3)=9

  4. Check by multiplying: 9\times(-3)=-27
  5. Answer: -40 and 9

Temperatures and missing numbers

When a temperature rises, add. When it falls, subtract. For example, -4^\circC rising by 6^\circC gives -4+6=2^\circC.

The difference between two temperatures is the higher one subtract the lower one. From -3^\circC to 5^\circC the difference is 5-(-3)=8^\circC. You can also count: 3 up to 0, then 5 more.

To find a missing starting number, work backwards. Undo each step in reverse order: undo adding by subtracting, and undo multiplying by dividing.

Worked example 5: Working backwards

Amira thinks of a number. She multiplies it by -3 and then adds 5. Her answer is -7. What number did Amira think of?

  1. Work backwards from -7. Her last step was adding 5, so subtract 5: -7-5

  2. Your turn. Work out -7-5

    -7-5=-12

  3. Before that she multiplied by -3, so divide by -3: (-12)\div(-3)=4
  4. Check: 4\times(-3)=-12, and -12+5=-7
  5. Answer: 4

Worked example 6: The difference between two temperatures

At 6 am the temperature in Ottawa is -9^\circC and the temperature in Seville is 4^\circC. By noon, the temperature in Ottawa has risen by 6^\circC and the temperature in Seville has risen by 8^\circC. Work out the difference between the two temperatures at noon.

  1. Ottawa at noon: -9+6=-3^\circC
  2. Seville at noon: 4+8=12^\circC
  3. The difference is the higher temperature subtract the lower one: 12-(-3)

  4. Your turn. Work out 12-(-3)

    12-(-3)=12+3=15

  5. Answer: 15^\circC

Common mistakes

  • Using "two negatives make a positive" when adding or subtracting. That rule is only for multiplying and dividing: -9-2=-11, not 11.
  • Getting the sign wrong when dividing. Same signs give a positive answer, so (-20)\div(-5)=4, not -4.
  • Ordering negative numbers by their digits. The further left a number is, the smaller it is, so -9 is smaller than -2.
  • Missing out zero when counting across it. Count the steps up to 0, then the steps after it: from -2 to 5 is 2+5=7, not 6.
  • Ignoring the minus sign when finding a difference. Write the higher temperature subtract the lower one, with brackets: 6-(-2)=8, not 4.
  • Stopping before the end of the question. Check what the question asks for: a difference is not the same as a temperature.

Practice questions

More on this topic: Negative Numbers worksheet with full solutions