Negative Numbers
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Revision notes for Edexcel GCSE Maths Negative Numbers. Open the guide for explanations and worked examples. Written against the Edexcel GCSE Maths (1MA1) specification, so the content matches what's examinable rather than general Maths background.

Negative Numbers

What you'll learn

  • What negative numbers mean on a number line.
  • How to add and subtract with negative numbers.
  • How to multiply and divide with negative signs.
  • How to use negative numbers in sequences and temperature questions.

Numbers below zero

A number line is a straight line with numbers in order. Numbers get bigger as you move right and smaller as you move left.

Definition

Negative number

A negative number is less than zero. It has a minus sign in front, like -3. A positive number is greater than zero, like 3.

A number line shows negative numbers to the left of 0 and positive numbers to the right.

Key Idea

Left means smaller

On a number line, the further left a number is, the smaller it is. So -6 is smaller than -2 because -6 is further left.

The point -6 is further left than -2, so -6 is the smaller number.

Example

Finding the lowest temperature

Four towns have temperatures of 4°C, -2°C, -7°C and 1°C. Which temperature is lowest?

Placing the temperatures on a number line makes the lowest temperature the leftmost point.

  1. Lowest means the smallest number, or the coldest temperature.

  2. Negative temperatures are lower than positive temperatures.

  3. Compare the negative temperatures: -7 is further left than -2, so -7°C is the lowest temperature.

Adding and subtracting negative numbers

For Grade 1 questions, it helps to think about moving on a number line.

  • Adding a positive number moves you right.
  • Subtracting a positive number moves you left.
  • Adding a negative number also moves you left.
  • Subtracting a negative number moves you right.
Example

Subtracting past zero

Work out 6 subtract 11.

Subtracting 11 from 6 means moving 11 steps left, crossing 0 and landing at -5.

  1. Start at 6 on the number line.

  2. Subtract 11, so move 11 places left. Moving 6 places reaches zero, then 5 more reaches -5.

  3. Write the calculation:

    6−11=−56 - 11 = -56−11=−5
Example

Adding a negative number

Work out negative 4 plus negative 6.

Adding a negative number moves left on the number line.

  1. Start at -4.

  2. Adding -6 means move 6 places left, because you are adding a negative number.

  3. The calculation is:

    −4+(−6)=−10-4 + (-6) = -10−4+(−6)=−10

Subtracting a negative

This is the bit that often catches people out. Subtracting a negative is the same as adding the positive version.

Example

Subtracting a negative number

Work out 9 subtract negative 4.

Subtracting a negative is shown as moving right, the same direction as adding 4.

  1. Notice the two signs: you are subtracting a negative number.

  2. Change “subtract negative 4” into “add 4”.

  3. Calculate:

    9−(−4)=9+4=13\begin{aligned} 9 - (-4) &= 9 + 4 \\ &= 13 \end{aligned}9−(−4)​=9+4=13​
Common Mistake

Dropping the second minus

In a calculation like 9 - -4, the second minus belongs to the negative 4. Do not just ignore it. Subtracting a negative changes to addition.

Filling in missing boxes

Sometimes you are given a few numbers and need to choose which ones make a correct calculation. Try to think about the sign of the answer first.

Example

Choosing numbers for an addition

Choose from -8, -3, 3 and 8 to make an addition with answer -11.

  1. The answer is negative, so try adding two negative numbers.

  2. Choose -8 and -3.

  3. Check the calculation:

    −8+(−3)=−11-8 + (-3) = -11−8+(−3)=−11
Tip

Quick sign check

If two numbers add to a negative answer, the negative number or numbers must have the bigger effect.

Multiplying and dividing negative numbers

For multiplying and dividing, work out the size of the answer first, then decide whether it should be positive or negative.

Key Idea

Sign rules for multiplying and dividing

  • Same signs give a positive answer.

The sign-rule grid summarises when multiplication or division answers are positive or negative.

  • Different signs give a negative answer.
Example

Multiplying with one negative sign

Work out negative 7 times 5.

  1. Ignore the sign first: 7 times 5 is 35.

  2. There is one negative sign, so the answer is negative.

  3. Write the answer:

    −7×5=−35-7 \times 5 = -35−7×5=−35
Example

Dividing with two negative signs

Work out negative 42 divided by negative 6.

  1. Ignore the signs first: 42 divided by 6 is 7.

  2. Both numbers are negative, so the signs are the same.

  3. Same signs give a positive answer:

    −42÷(−6)=7-42 \div (-6) = 7−42÷(−6)=7
Common Mistake

Adding rules are not multiplying rules

Two negatives multiplied or divided give a positive answer. But two negatives added together stay negative.

Missing numbers in multiplication and division

An inverse operation is an operation that undoes another operation. Division undoes multiplication, and multiplication undoes division.

Example

Finding a missing multiplier

A number is multiplied by 6 to give -18. Find the missing number.

  1. Use the inverse operation: divide -18 by 6.

  2. Work out the sign. Negative divided by positive gives negative.

  3. Calculate:

    −18÷6=−3-18 \div 6 = -3−18÷6=−3

  4. Check:

    6×(−3)=−186 \times (-3) = -186×(−3)=−18
Example

Finding a missing number before division

A missing number divided by -4 gives 5. Find the missing number.

  1. Undo the division by multiplying 5 by -4.

  2. Positive times negative gives negative.

  3. Calculate:

    5×(−4)=−205 \times (-4) = -205×(−4)=−20

Number sequences with negatives

A sequence is a list of numbers in order. Each number in the sequence is called a term.

To continue a simple sequence, find the change from one term to the next.

Example

Continuing a decreasing sequence

Continue this sequence: 13, 8, 3, …

  1. Find the change: 13 to 8 is subtract 5, and 8 to 3 is subtract 5.

  2. Keep subtracting 5.

  3. The next two terms are found by:

    3−5=−2,−2−5=−73 - 5 = -2,\quad -2 - 5 = -73−5=−2,−2−5=−7
Tip

Crossing zero

When a sequence goes below zero, keep counting through zero carefully: 3, 2, 1, 0, -1, -2.

Temperatures and differences

Temperature questions often use negative numbers because temperatures can go below 0°C.

  • “Risen” or “increased” means add.
  • “Lower” or “fallen” means subtract.
  • “Difference” means the distance between two values.
Definition

Difference

The difference between two numbers is the distance between them. In temperature questions, a difference is written as a positive number of degrees.

Example

Temperature change and difference

At midnight, a temperature is -5°C. By morning, it has risen by 8°C. Another city is 6°C. Find the new temperature and the difference between -5°C and 6°C.

The temperature rise moves from -5°C to 3°C, and the difference from -5°C to 6°C is split at 0°C.

  1. Risen by 8°C means add 8.

  2. Calculate the new temperature:

    −5+8=3-5 + 8 = 3−5+8=3

  3. To find the difference between -5°C and 6°C, count from -5 to 0, then from 0 to 6.

  4. Add the two parts:

    5+6=115 + 6 = 115+6=11
Exam technique

In the exam

  1. Look carefully at each sign before you start, especially when you see two signs next to each other.

  2. For multiplying or dividing, decide the sign first: same signs positive, different signs negative.

  3. In temperature questions, underline words like “risen”, “lower” and “difference”.

Self review

Check yourself

  • If you start at -3 and subtract 5, where do you land?
  • What is the sign of a negative number multiplied by a negative number?
  • What does “difference” mean in a temperature question?

Recap questions

Test yourself with 5 quick questions on this guide. Answer them all correctly to complete it.

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Negative Numbers Revision Guide

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