GCSE Maths Number

Negative Numbers

# Negative Numbers

Here we will learn about negative numbers including how to add, subtract, multiply, and divide negative numbers.

There are also negative numbers worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

## What are negative numbers?

Negative numbers are any numbers less than zero and have a negative or minus sign () in front of them.

Numbers greater than zero are referred to as positive numbers. If there is no sign in front of a number the number is positive.

On the number line below we can see some positive and negative integers (whole numbers):

The numbers in orange are negative values and the blue numbers are positive values. Zero is neither positive or negative.

Just like you can add, subtract, multiply and divide positive numbers, you can do the same with negative numbers whether they are integers, decimals or fractions.

### What are negative numbers? ### Adding and subtracting negative numbers

When adding and subtracting negative numbers use a number line:

If you are adding, move to the right of the number line.

If you are subtracting, move to the left of the number line.

Sometimes a question might have two operations next to each other:

E.g. 4 + (−2)

If the signs are the same replace with a positive sign.

If the signs are different, replace with a negative sign.

The chart below summarises this:

As a rule of thumb: same signs add, different signs subtract.

### What do you need to remember when adding and subtracting negative numbers? ### Multiplying and dividing negative numbers

Similar rules apply for multiplying and dividing negative numbers:

If the signs are the same, the answer is positive.

If the signs are different, the answer is negative.

When multiplying negative numbers:

The same rules apply for dividing negative numbers:

### What do you need to remember when multiplying and dividing negative numbers? ## How to add and subtract negative numbers

In order to add and subtract negative numbers:

1. If you have two signs next to each other, change them to a single sign.
If the signs are the same, replace with a positive sign (+).
If the signs are different, replace with a negative sign ().
2. Circle the first number on the number line.
If you are adding, move to the right of the number in step 2 ().
If you are subtracting, move to the left of the number in step 2 ().

### Explain how to add and subtract negative numbers in 4 steps ## How to multiply and divide negative numbers

In order to multiply and divide negative numbers:

1. Multiply or divide the numbers normally.
2. Change the sign as necessary using the rules of multiplying and dividing negative numbers:
If the signs are the same, the answer is positive.
If the signs are different, the answer is negative.

### Explain how to multiply and divide negative numbers in 2 steps ## Negative numbers examples

### Example 1: adding a positive number

$−5 + 8$

1. If you have two signs next to each other, change them to a single sign.

If the signs are the same in the middle of the calculation, replace with a positive sign (+).

If the signs are different, replace with a negative sign ().

In this case you do not have two signs next to each other.

2Circle the first number on the number line.

The first number in the question is (−5)

If you are adding, move to the right of the number in step 2 (→).

If you are subtracting, move to the left of the number in step 2 (←).

In this case we are adding the 8 so move 8 spaces right from the (-5) on the number line:

$-5 + 8 = 3$

### Example 2: subtracting a negative number (two operations)

$−7 − (−5)$

In this case you have a minus and a minus next to each other.

Since the signs are the same, replace with a plus sign (+).

$-7 \color{violet}+ 5$

The first number in the question is (-7)

In this case we are adding 5 so move 5 spaces right from the (-7) on the number line:

$-7 – (-5) = -2$

### Example 3: multiplying negative numbers

$−6 × (−7)$

$6 × 7 = 42$

$\color{orange}{-6} × \color{orange}{-7}$

In this case we have a number that is negative times a negative number.

The signs are the same so we must have a positive answer.

$= 42$

### Example 4: dividing negative numbers

$-25 ÷ 5$

$25 ÷ 5 = 5$

$\color{orange}{-25} ÷ \color{blue}5$

In this case we have a negative number divided by a positive number.

The signs are the same so we must have a negative quotient.

$= -5$

### Example 5: mixed operations (Order of operations)

$-18 ÷ (-6) – 3 × (-2)^2=$

In this case we are dealing with three different operations (+, × and ÷) and an index. We need to use BIDMAS to determine the order to calculate.

First thing we calculate is the indices: BIDMAS

(−2)2 is the same as −2 × (−2)

We know 2 × 2 = 4

If the signs are the same ((−2 × (−2)) , the answer is positive.
Therefore

$(-2)^2 =4$

For the first part 18 ÷ 6 = 3

For the second part 3 × 4 = 12

For the first part, −18 ÷ (−6),
the signs are the same so the answer is positive 3.

For the second part -3 × 4,
the signs are different so the answer is negative (−12).

We are left with:

$3-12$

The first number in the question is 3.

In this case we are subtracting the 12 so move 12 spaces left from the 3 on the number line:

$3 – 12 = -9$

### Example 6: real life word problem

The table below shows the temperatures recorded in Leicester at different times of the day.

a) What is the product of the highest and lowest temperatures?

b) What is the difference between the temperatures at 2am and 1pm?

Time of Day

2 am

7 am

1 pm

6 pm

Temperature ( ^{\circ}C )

−8

−5

3

−1

To calculate a):

The highest temperature is 3ºC.

The lowest temperature is −8ºC.

$8 × 3 = 24$

$\color{blue}3 × \color{orange}{-8}$

In this case we have a number that is positive times a negative number.

The signs are the same so we must have a negative answer.

$= -24^{\circ}C$

To calculate b):

The temperature at 2am was -8°C and the temperature at 1pm was 3°C.

To find the difference we must work out -8 – 3.

In this case you do not have two signs next to each other.

The first number in the question is (-8):

In this case we are subtracting the 3 so move 3 spaces right from the (-8) on the number line:

$-8 – 3 = -11$

### Common misconceptions

• Greater negative does mean a larger number

Students sometimes assume that the larger a negative number the greater it is. For example, students might incorrectly assume -3 is greater than 2 because 3 is a larger number.

• Raising a negative number to a power greater than one

Remember when raising a negative number to a power greater than 1 the resulting answer could be positive or negative. When you raise a negative number to an odd exponent the resulting answer is negative, when you raise a negative number to an even exponent the resulting answer is positive.

• Changing signs when adding and subtracting
When adding or subtracting with negative numbers the signs change only if they are next to each other in the middle of the calculation and are different

### Practice negative numbers questions

1. Work out: -8+10

-2 2 -18 18  2. Work out: -11-(-8)

-3 3 -19 19 We have – \; – together so change it to a + .

-11+8 = -3

3. Work out: -8 \times (-9)

-72 72 17 -17 8 \times 9 = 72

The signs are the same so the answer is positive.

-8 \times (-9)=72

4. Work out: -144 \div 3

147 -147 -48 48 -144 \div 3 = 48

The signs are different so the answer is negative.

-144 \div 3= -48

5. Calculate: 18 \div (-9) +5 \times (-3)^2=

-47 -43 43 27 We need to follow BIDMAS here:

(-3)^2=-3 \times -3=9

5 \times 9=45

18 \div (-9)=-2

So we have

-2+45=43

6. The table shows the temperature in Manchester at different times of the day.

Time of the Day          Temperature   (^{\circ}C)

What is the difference between the highest and lowest temperatures?

1^{\circ}C 7^{\circ}C 10^{\circ}C 11^{\circ}C The calculation we need to do here is 2 – – 9. There is a – \; – together so this becomes a +.

2–9 = 2 + 9 = 11

### Negative numbers GCSE questions

1.   Tilly has the following 6 cards: She is going to choose 3 cards and multiply them.

(a)   What is the largest possible number she can make?

(b)   What is the smallest possible number she can make?

(6 Marks)

(a)

For identifying 2 of the following: -9 or -8 or 7 .

(1)

Multiply -9, and -8 to get 72 .

(1)

Correctly multiplying -9, -8 and 7 or 504 seen.

(1)

(b)

For identifying 2 of the following: -9 or 7 or 3 .

(1)

Shows working how to multiply.

(1)

Correctly multiplying -9, 7 and 3 or -189 seen.

(1)

2.  Mr. and Mrs. Brown had \pounds 198.78 in their bank account. At the end of the month they had to pay 4 bills

They paid the tv license of \pounds 57.20 , utilities of \pounds 134.78 , car insurance \pounds 38.25 , and credit card bill of \pounds 94.

How much were the Browns overdrawn?

(3 Marks)

Finding the sum of the 4 bills.

(1)

Subtracting the cost of the four bills from the balance in the bank account.

(1)

cao

-\pounds 125.45

(1)

3.  Below is a list of cities and their elevations:

City                         Elevation

Helsinki                  25m

New Orleans          -2m

Buenos Aires          10m

Amsterdam              -7m

Baku                        -28m

(a)   Which cities are below sea level?

(b)   What is the difference between the elevation of Buenos Aires and Baku?

(5 Marks)

(a)

Baku

(1)

Amsterdam

(1)

New Orleans

(1)

(b)

10 – (-28)

(1)

38m

(1)

## Learning checklist

You have now learned how to:

• Add and subtract integers both positive and negative
• Multiply and divide integers both positive and negative
• Use negative numbers in context and calculate intervals across 0

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