# Arithmetic with Positive and Negative Integers

## Topic Overview

A **positive integer** is a number which is more than 0.

A **negative integer** is a number which is less than 0.

There are a few simple rules worth remembering when adding and subtracting a mix of positive and negative integers:

**Positive + Positive = Positive**

For example: 4 + 2 = 6

**Negative + Negative = Negative**

For example: (-9) + (-1) = -10

**Negative - Positive = Negative**

For example : (- 5) - 3 = -5 + (-3) = -8

**Positive - Negative = Positive + Positive = Positive**

For example: 5 - (-3) = 5 + 3 = 8

**Negative - Negative = Negative + Positive**

For example: (-4) - (-2) = ( -4) + 2 = -2

## Key Concepts

In the new linear GCSE Maths paper, you will be required to work out several problems involving positive and negative integers. According to the Edexcel Revision Checklist for the linear GCSE Maths paper, you will be required to:

- Order integers and decimals
- Add, subtract, multiply and divide whole number integers and decimals

Listed below are a series of summaries and worked examples to help you solidify your knowledge about positive and negative integers.

## Worked Examples

**1 - Recognising negative numbers**

During your GCSE Maths exam, you may be asked to place a series of positive and negative numbers in order.

*Example*

(a) - Order the numbers **- 5, −2, 3, −9, −2.5, 0.5**.

*Solution*

(a) - The lowest number in this list is **- 9**. The highest number is 3. By using the number line below, you can order the numbers from lowest to highest: the order is **−9, −5, −2.5, −2, 0.5, 3**.

**2 - Negative numbers and temperature**

During your exam, you also need to be able to work out the increase and decrease in temperature using negative numbers.

*Example*

(a) - At 6pm, the temperature was 3°C. By midnight, it had dropped to −5°C. How great was the fall in temperature?

*Solution*

(a) - In order to work out the decrease in temperature, you must work out the difference between the positive value and 0°C, the difference between 0°C and the negative value, and then add these two numbers together.

For example, the difference between -3°C and 0°C is 3 and the difference between 0°C and -5°C is 5.

Therefore you add 3 and 5, so the fall in temperature is 8°C.

**3 - Simple negative number arithmetic**

Throughout your GCSE Maths exam, you will be required to add, subtract, multiply and divide a mix of positive and negative numbers. In order to work out these calculations correctly, you must follow the basic rules for **adding** and **subtracting**:

**Adding**a negative number is the same as subtracting:

*Example*

(a) - Calculate 7 + (−3)

*Solution*

(a) -

7 + (−3) is the same as 7 − 3

Therefore 7 + (-3) = 4

**Subtracting**a negative number is the same as adding:

*Example*

(b) - Calculate (−5) − (−2)

*Solution*

(b) -

(−5) − (−2) is the same as (−5) + 2

Therefore (-5) - (-2) = −3

When multiplying positive and negative values, you adhere to the following rules:

- Positive × positive =
**positive** - Positive × negative =
**negative** - Negative × positive =
**negative** - Negative × negative =
**positive**

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

**Example**

(c) - Calculate -5 x -4

*Solution*

(c) - Negative x negative = positive

Therefore -5 x -4 = 20

When dividing positive and negative values, you follow the same rules as multiplication:

- Positive ÷ positive =
**positive** - Positive ÷ negative =
**negative** - Negative ÷ positive =
**negative** - Negative ÷ negative =
**positive**

*Example*

(d) - Calculate 24÷ -6

*Solution*

(d) - Positive ÷ negative = negative

Therefore 24 ÷ -6 = -4

## Exam Tips

- When calculating negative values, it is helpful to draw a number line in order to provide yourself with a clear visual aid
- In order to work out the increase or decrease of negative values, you work out the difference between the positive value and 0, the difference between 0 and the negative value and then add these two numbers together
- When multiplying positive and negative values, you follow the rules: Positive x positive = positive, Positive x negative = negative, Negative x positive = negative, Negative x negative = positive
- When dividing positive and negative values, you follow the rules: Positive ÷ positive = positive, Positive ÷ negative = negative, Negative ÷ positive = negative, Negative ÷ negative = positive

## Topic Summary

When calculating a mix of positive and negative integers, it is helpful to remember the basic principles mentioned above. Furthermore, it is often helpful to draw a basic number line. By doing so you provide yourself with a clear visual aid through which you can easily add and subtract various positive and negative values. As long as you remember the rules of positive and negatives arithmetic, and double check your working, you will soon be able to calculate arithmetic with positive and negative integers with ease!

## Related Topics

- Arithmetic with Fractions
- Indices
- Problem Solving with Decimals and Percentages
- Accuracy of Measurement Problems
- Standard Form
- Rational and Irrational Numbers
- Combinations