## Introduction

### The Number Line

- Integers are whole numbers, negative whole numbers and zero.
- Integers are always represented on the number line at equal intervals which are equal to one unit.

## Operations on Integers

### Addition of Integers

- Addition of integers can be represented on a number line .
- For example, to add +3 to 0 , we begin at 0 and move 3 units to the right as shown below in red to get +3,
- Also to add + 4 to +3 we move 4 units to the right as shown in blue to get +7.
- To add -3 to zero we move 3 units to the left as shown in red below to get -3 while to add -2 to -3 we move 2 steps to the left as shown in blue to get -5.
**Note;** - When adding positive numbers we move to the right.
- When dealing with negative we move to the left.

### Subtraction of Integers.

**Example**

(+7) – (0) = (+7)

To subtract +7 from 0 ,we find a number n which when added to get 0 we get +7 and in this case n = +7 as shown above in red.

**Example**

(+2) – (+7) = (-5)

Start at +7 and move to +2. 5 steps will be made towards the left. The answer is therefore -5.

**Example**

-3 – (+6) = -9

|__**|_←|__|__|__|__|__| | ←|**__|__|__|__|__|

-4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

We start at +6 and moves to -3. 9 steps to the left, the answer is -9.

**Note:**

- In general positives signs can be ignored when writing positive numbers i.e. +2 can be written as 2 but negative signs cannot be ignored when writing negative numbers -4 can only be written s -4.

4 – (+3) = 4 -3

= 1

-3- (+6) =3 – 6

= -3 - Positive integers are also referred to as natural numbers. The result of subtracting the negative of a number is the same as adding that number.

2 – (- 4) = 2 + 4

= 6

(-5) – (- 1 ) = -5 + 2

= -3 - In mathematics it is assumed that that the number with no sign before it has appositive sign.

### Multiplication

- In general

- (a negative number) x (appositive number ) = (a negative number)
- (a positive number) x (a negative number ) = (a negative number)
- (a negative number) x (a negative number ) = (a positive number)

**Examples**

-6 x 5 = -30

7 x -4 = - 28

-3 x -3 = 9

-2 x -9 = 18

### Division

- Division is the inverse of multiplication. In general

- (a positive number ) ÷ (a positive number ) = (a positive number)
- (a positive number ) ÷ (a negative number ) = (a negative number)
- (a negative number ) ÷ (a negative number ) = (a positive number)
- (a negative number ) ÷ (appositive number ) = (a negative number)

- For multiplication and division of integer:

- Two like signs gives positive sign.
- Two unlike signs gives negative sign
- Multiplication by zero is always zero and division by zero is always zero.

### Order of Operations

- BODMAS is always used to show as the order of operations.

B – Bracket first.

O – Of is second.

D – Division is third.

M – Multiplication is fourth.

A – Addition is fifth.

S – Subtraction is considered last.**Example**

6 x 3 – 4 ÷ 2 + 5 + (2-1) =**Solution**

Use BODMAS

(2 – 1 ) = 1 we solve brackets first

(4÷ 2) = 2 we then solve division

(6 x 3) = 1 8 next is multiplication

Bring them together

18 – 2 +5 +1 = 22 we solve addition first and lastly subtraction

18 + 6 – 2 = 22

## Past KCSE Questions on the Topic

- The sum of two numbers exceeds their product by one. Their difference is equal to their product less five.Find the two numbers. (3mks)
- 3x – 1 > -4

2x + 1 ≤ 7 - Evaluate

-12 ÷ (-3) x 4 – (-15)

-5 x 6 ÷ 2 + (-5) - Without using a calculator/mathematical tables, evaluate leaving your answer as a simple fraction

(-4)(-2) + (-12) ÷ (+3) + -20 + (+4) + -6)

-9 – (15) 46- (8+2)-3 - Evaluate -8 ÷ 2 + 12 x 9 – 4 x 6

56 ÷7 x 2 - Evaluate without using mathematical tables or the calculator

1.9 x 0.032

20 x 0.0038

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