Specific Objectives
By the end of this topic, the learner should be able to:
- Define integers.
- Identify integers on a number line.
- Perform the four basic operations on integers using the number line.
- Work out combined operations on integers in the correct order.
- Apply knowledge of integers to real-life situations.
Content
- Integers
- The number line
- Operations on integers
- Order of operations
- Application to real-life situations
Introduction
The Number Line
Integers include whole numbers, negative whole numbers, and zero. They are always represented on the number line at equal intervals, each interval representing 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, start at 0 and move 3 units to the right, as shown below in red, to reach +3. Similarly, to add +4 to +3, move 4 units to the right, shown in blue, to get +7.
To add -3 to zero, move 3 units to the left, shown in red below, to get -3. To add -2 to -3, move 2 steps to the left, shown in blue, to get -5.
Note:
Subtraction of Integers |
Example
(+7) – (0) = (+7)
To subtract +7 from 0, find a number n which when added to +7 gives 0; here, n = -7, as shown above in red.
Example
(+2) – (+7) = (-5)
Start at +7 and move to +2. Five steps are 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 move to -3. Nine steps to the left, the answer is -9.
Note:
- Positive signs can generally be omitted when writing positive numbers, e.g., +2 can be written as 2. However, negative signs must always be included for negative numbers, e.g., -4 cannot be written as 4.
4 – (+3) = 4 – 3 = 1
-3 – (+6) = -3 – 6 = -9
- Positive integers are also called natural numbers. Subtracting a negative number is equivalent to adding its positive counterpart.
2 – (-4) = 2 + 4 = 6
(-5) – (-1) = -5 + 1 = -4
- In mathematics, a number without a sign is assumed to be positive.
Multiplication
In general:
- (a negative number) × (a positive number) = (a negative number)
- (a positive number) × (a negative number) = (a negative number)
- (a negative number) × (a negative number) = (a positive number)
Examples
-6 × 5 = -30
7 × -4 = -28
-3 × -3 = 9
-2 × -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) ÷ (a positive number) = (a negative number)
Note:
For multiplication and division of integers:
- Two like signs result in a positive sign.
- Two unlike signs result in a negative sign.
- Multiplication by zero always results in zero, and division by zero is undefined.
Order of Operations
The order of operations is always shown using BODMAS:
- B – Brackets first.
- O – Of (powers and roots) second.
- D – Division third.
- M – Multiplication fourth.
- A – Addition fifth.
- S – Subtraction last.
Example
Evaluate: 6 × 3 – 4
Solution
Use BODMAS:
(2 – 1) = 1 – solve brackets first
(4) = 2 – solve division next
(6 × 3) = 18 – then multiplication
Bring them together:
18 – 2 + 5 + 1 = 22 – solve addition first, then subtraction
18 + 6 – 2 = 22
End of Topic
Did you understand everything?
If not, ask a teacher, friends, or anybody else, and make sure you understand before going to sleep!
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. (3 marks)
- 3x – 1 > -4
- 2x + 1 ≤ 7
- Evaluate -12 ÷ (-3) × 4 – (-15)
- -5 × 6 ÷ 2 + (-5)
- Without using a calculator or 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 × 9 – 4 × 6
- 7 × 2
Evaluate without using mathematical tables or a calculator:
1.9 × 0.032
20 × 0.0038
