Understanding Negative Numbers

What are Negative Numbers?

Negative numbers are numbers that are less than zero. They are written with a minus sign (-) in front of them, like -1, -2, -3, and so on.

Think of a number line: positive numbers are to the right of zero, and negative numbers are to the left. On a vertical number line, positive numbers are above zero, and negative numbers are below zero.

Negative numbers are used in many real-life situations, such as:

Vertical Number Line

Negative Number Calculations: Examples

1. Adding a Negative Number

Problem: Calculate 5 + (-3).

Adding a negative number is the same as subtracting the positive version of that number.

So, 5 + (-3) is the same as 5 - 3.

On the number line: Start at 5. Adding -3 means moving 3 units downwards (or to the left on a horizontal line).

Answer: 5 + (-3) = 2

2. Subtracting a Negative Number

Problem: Calculate 4 - (-2).

Subtracting a negative number is the same as adding the positive version of that number. Think of it as "taking away a debt" which makes you richer.

So, 4 - (-2) is the same as 4 + 2.

On the number line: Start at 4. Subtracting -2 means moving 2 units upwards (or to the right on a horizontal line).

Answer: 4 - (-2) = 6

3. Starting with a Negative and Adding a Positive

Problem: Calculate -3 + 7.

On the number line: Start at -3. Adding 7 means moving 7 units upwards (or to the right).

You will pass 0: -3 to 0 is 3 units. You have 7 - 3 = 4 more units to move upwards from 0.

Answer: -3 + 7 = 4

4. Starting with a Positive and Subtracting a Larger Positive

Problem: Calculate 2 - 5.

On the number line: Start at 2. Subtracting 5 means moving 5 units downwards (or to the left).

You will pass 0: 2 to 0 is 2 units. You need to move 5 - 2 = 3 more units downwards from 0.

Answer: 2 - 5 = -3

5. Multiplying Negative Numbers

Rules:

  • Positive × Positive = Positive (e.g., 3 × 4 = 12)
  • Positive × Negative = Negative (e.g., 3 × (-4) = -12)
  • Negative × Positive = Negative (e.g., -3 × 4 = -12)
  • Negative × Negative = Positive (e.g., -3 × (-4) = 12)

Problem: Calculate (-5) × (-2).

A negative multiplied by a negative gives a positive.

Answer: (-5) × (-2) = 10

6. Dividing Negative Numbers

The rules for division are the same as for multiplication:

  • Positive ÷ Positive = Positive (e.g., 12 ÷ 4 = 3)
  • Positive ÷ Negative = Negative (e.g., 12 ÷ (-4) = -3)
  • Negative ÷ Positive = Negative (e.g., -12 ÷ 4 = -3)
  • Negative ÷ Negative = Positive (e.g., -12 ÷ (-4) = 3)

Problem: Calculate -15 ÷ 3.

A negative divided by a positive gives a negative.

Answer: -15 ÷ 3 = -5

Exam-Style Negative Number Problems

1. Temperature Changes

The temperature in a town at midday was 4°C. By midnight, the temperature had dropped by 7°C. What was the temperature at midnight?

Answer: -3°C

Explanation:

  1. Start with the initial temperature: 4°C.
  2. The temperature dropped, which means subtraction: 4°C - 7°C.
  3. Visualise on a number line: Start at 4. Move 7 units down (colder).
    • Moving 4 units down from 4°C gets you to 0°C.
    • You still need to move 7 - 4 = 3 more units down.
    • Moving 3 units down from 0°C gets you to -3°C.
  4. Calculation: 4 - 7 = -3.

2. Bank Account Balance

Sarah has £25 in her bank account. She spends £40 on groceries. What is her new bank balance?

Answer: -£15 (or an overdraft of £15)

Explanation:

  1. Start with the initial balance: £25.
  2. She spends money, which means subtraction: £25 - £40.
  3. Visualise on a number line: Start at 25. Move 40 units to the left (spending).
    • Moving 25 units left from 25 gets you to 0.
    • You still need to move 40 - 25 = 15 more units left.
    • Moving 15 units left from 0 gets you to -15.
  4. Calculation: 25 - 40 = -15.

3. Debt Repayment

John owes his friend £12 (his balance is -£12 with his friend). He then gives his friend £5. What is his new balance with his friend?

Answer: -£7 (he still owes £7)

Explanation:

  1. Start with John's current balance (debt): -£12.
  2. He gives money, which means he is paying back part of the debt. This moves his balance towards zero (less negative). So, we add the amount he gives: -12 + 5.
  3. Visualise on a number line: Start at -12. Move 5 units to the right (paying back).
  4. Calculation: -12 + 5 = -7.

4. Elevation Change

A submarine is at a depth of -150 metres relative to sea level. It then descends a further 80 metres. What is its new depth?

Answer: -230 metres

Explanation:

  1. Initial depth: -150 metres.
  2. Descending further means going more negative (or subtracting a positive distance from its current negative position, which makes it more negative). So, -150 - 80 or -150 + (-80).
  3. Visualise on a number line (imagine it vertical, sea level is 0): Start at -150. Move 80 units further down.
  4. Calculation: -150 - 80 = -230.

Key Points for Negative Numbers