# Negative and Positive Numbers

> Use the number line to compare, add and subtract with numbers below zero, and find the difference between two numbers.

From Selective School Maths on WhizzNotes, free study notes for NSW students: https://whizznotes.com/selective-school-maths/negative-numbers/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: Night in the snow
Kai's family goes to Thredbo for a ski trip. At 3 pm the thermometer shows $4^\circ\text{C}$. By midnight it shows $-5^\circ\text{C}$, which means 5 degrees **below** zero.

"How much colder did it get?" asks Kai's dad. Kai pictures the thermometer. From 4 down to 0 is 4 degrees. From 0 down to $-5$ is another 5 degrees. Altogether it dropped $4 + 5 = 9$ degrees.

Numbers below zero are called **negative numbers**. They are used for cold temperatures, floors below the ground in a car park, and money you owe.

**In short:** Negative numbers are numbers less than zero, and they sit to the left of zero on a number line. The further right a number is, the bigger it is, so $-1$ is bigger than $-10$. Adding moves right and subtracting moves left.

## Before you start

- Counting forwards and backwards on a number line.
- **Difference** means how far apart two numbers are.

## The number line

*Diagram: Number line from negative 9 to 9 with zero in the middle; negative numbers on the left and positive numbers on the right*
*Numbers get bigger as you move right and smaller as you move left.*

### Key idea: Bigger and smaller
A number further **right** is **bigger**. So $-2$ is bigger than $-7$, even though 7 looks bigger. Think of temperature: $-2^\circ\text{C}$ is warmer than $-7^\circ\text{C}$.

Positive and negative numbers together are called **directed numbers**, because the sign shows a direction: right (positive) or left (negative).

## The difference between two numbers

The difference is the **distance** between them on the number line. It is never negative.

### Worked example: Same side of zero
Difference between $-9$ and $-5$: from $-9$ count right to $-5$. That is 4 steps. Difference $= 4$.

### Worked example: Opposite sides of zero
Difference between $-4$ and $3$: from $-4$ to 0 is 4 steps, and from 0 to 3 is 3 steps. Difference $= 4 + 3 = 7$.

Difference between $8$ and $-7$: $8 + 7 = 15$.

### Exam tip: Across zero, add the sizes
If one number is negative and the other positive, the difference is the two sizes added together, ignoring the signs.

## Adding

Start at the first number. **Adding** a positive number means moving **right**.

### Worked example: Moving right
$-4 + 8$: start at $-4$, move 8 right. You pass 0 after 4 steps and go 4 more. Answer: $4$.

$-9 + 3$: start at $-9$, move 3 right. Answer: $-6$.

$-6 + 6 = 0$. A number and its opposite cancel out.

## Subtracting

**Subtracting** a positive number means moving **left**.

### Worked example: Moving left
$5 - 9$: start at 5, move 9 left. You reach 0 after 5 steps and go 4 more. Answer: $-4$.

$-2 - 3$: start at $-2$, move 3 left. Answer: $-5$.

$-10 - 10 = -20$.

### Watch out: $-2 - 3$ is not $1$ or $-1$
Both numbers send you left. You go 2 left of zero and then 3 further left: $-5$.

## Longer chains

Work from left to right, one move at a time.

### Worked example: A chain
$11 - 20 + 4$

$11 - 20 = -9$, then $-9 + 4 = -5$.

### Challenge: Selective level: taking away a negative
Taking away a negative number is the same as **adding**.

The temperature is $-3^\circ\text{C}$. The forecast says tomorrow's high will be $5$ degrees more than the difference between $6$ and $-3$. The difference is $6 - (-3) = 6 + 3 = 9$. So the high is 14 degrees.

A lift is on floor $-2$ (two floors below the ground). It goes up 7 floors and then down 9. It ends on $-2 + 7 - 9 = -4$, four floors below the ground.

**Mind map**

```text
Negative numbers
  Number line
    Right is bigger
    $-2 > -7$
  Difference
    Distance apart
    Across zero: add the sizes
  Adding
    Move right
    $-4 + 8 = 4$
  Subtracting
    Move left
    $5 - 9 = -4$
  Real life
    Temperature
    Floors below ground
    Money owed
```

## Quick summary

- Negative numbers are less than zero and sit to the left of zero.
- Further right means bigger: $-1$ is bigger than $-10$.
- The difference between two numbers is how far apart they are. Across zero, add the sizes.
- Adding moves right. Subtracting moves left.
- For a chain like $3 - 12 + 8$, work left to right.

## Quiz

Try these questions from the chapter quiz.

1. What is the difference between $-8$ and $0$?
   - A. $-8$
   - B. 8
   - C. 0
   - D. 16
   - Answer: B. 8. From $-8$ to 0 is 8 steps.

2. $-3 + 5 = $
   - A. $-8$
   - B. $-2$
   - C. 2
   - D. 8
   - Answer: C. 2. Start at $-3$ and move 5 right: you land on 2.

3. $-2 - 3 = $
   - A. 1
   - B. $-1$
   - C. $-5$
   - D. 5
   - Answer: C. $-5$. Start at $-2$ and move 3 left: $-5$.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/selective-school-maths/negative-numbers/#quiz