1. What Is Mathematics?
Mathematics is a subject that helps us work with numbers.
It helps us do things like:
- Counting people
- Counting money
- Measuring distance
- Solving problems
- How many students are in the classroom?
- How many pens do you have?
- How much money is needed to buy 3 books?
Mathematics helps us answer such questions.
2. What Is a Number?
A number is something we use to tell how many things there are.
1 book → we write 1
5 books → we write 5
20 desks → we write 20
Numbers help us count things.
3. Counting
Counting means saying numbers one after another in order.
1, 2, 3, 4, 5, 6, 7, 8, 9, 10
We can continue counting forever:
11, 12, 13, 14, 15 ...
Numbers never end.
4. Natural Numbers
Natural numbers are the numbers we use when counting things.
Natural numbers are counting numbers.
If a teacher counts students:
- 1 student
- 2 students
- 3 students
- 4 students
Those numbers are natural numbers.
5. Numbers That Are Not Natural Numbers
Some numbers are not natural numbers.
Fractions
1/2, 3/4
You cannot say: “There are 1/2 students.”
Decimals
2.5
You cannot say: “There are 2.5 desks.”
Important: Natural numbers are whole counting numbers only.
Short History of Natural Numbers
Long ago, people did not have written numbers like we do today.
If someone had animals, they would count using:
- Fingers
- Stones
- Sticks
If a person had five cows, they might put five stones in a small pile.
Each stone represented one cow.
Later, humans developed symbols to represent numbers.
The number system we use today was improved by different civilizations such as:
- Indians
- Arabs
- Europeans
Today we use digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Using these digits we can write any number in the world.
Subtopic: Place Value
6. Understanding Place Value
Place value tells us the value of a digit because of where it is placed in a number.
Example: 23
This number has two digits: 2 and 3
The 3 means 3 ones
The 2 means 2 tens
23 = 20 + 3
Place Value Table
| Place | Value |
|---|---|
| Ones | 1 |
| Tens | 10 |
| Hundreds | 100 |
| Thousands | 1,000 |
| Ten Thousands | 10,000 |
| Hundred Thousands | 100,000 |
| Millions | 1,000,000 |
| Billions | 1,000,000,000 |
Example: Number 345
| Digit | Place | Value |
|---|---|---|
| 5 | Ones | 5 |
| 4 | Tens | 40 |
| 3 | Hundreds | 300 |
345 = 300 + 40 + 5
Face Value: Face value means the digit itself.
In 346, the face value of 4 is 4, but the place value of 4 is 40.
Subtopic: Billions
7. Very Large Numbers
Some numbers are very big.
1,000 = one thousand
1,000,000 = one million
1,000,000,000 = one billion
Writing Large Numbers
We separate large numbers using commas.
1000000 → difficult to read
1,000,000 → easy to read
Each comma separates three digits.
345,678,912
We read it as:
Three hundred forty-five million, six hundred seventy-eight thousand, nine hundred twelve.
Subtopic: Rounding Off
8. What Is Rounding Off?
Rounding off means making a number simpler or easier to use.
If a shop has 48 customers, we may say about 50 customers.
48 → 50
Rounding to the Nearest Ten
Round 34 to the nearest ten
Look at the ones digit: 4
If the digit is 0, 1, 2, 3, 4 → round down
34 → 30
Round 37 to the nearest ten
Look at the ones digit: 7
If the digit is 5, 6, 7, 8, 9 → round up
37 → 40
Rounding to the Nearest Hundred
Round 243 to nearest hundred
Look at the tens digit: 4
Since it is less than 5:
243 → 200
Round 278 to nearest hundred
Look at the tens digit: 7
Since it is 5 or more:
278 → 300
Subtopic: Operations on Whole Numbers
Operations are actions we do with numbers. There are four main operations:
Addition
Addition means putting numbers together.
3 + 2 = 5
23 + 14
Step 1: add ones → 3 + 4 = 7
Step 2: add tens → 2 + 1 = 3
Answer: 37
Subtraction
Subtraction means taking away.
7 − 3 = 4
15 − 6 = 9
Multiplication
Multiplication means repeated addition.
3 × 4 means 4 + 4 + 4 = 12
5 × 3 means 3 + 3 + 3 + 3 + 3 = 15
Division
Division means sharing equally.
12 ÷ 3 = 4
If 12 sweets are shared among 3 children, each child gets 4.
Subtopic: Word Problems
Word problems are questions written as stories or sentences.
A farmer has 10 cows.
He buys 5 more cows.
How many cows does he have now?
Step 1: Identify operation → “more” means addition
Step 2: 10 + 5 = 15
Answer: 15 cows
Subtopic: Even Numbers
Even numbers are numbers that can be shared equally between two people.
8 sweets shared between 2 children gives 4 each.
Nothing remains, so 8 is even.
Examples of even numbers:
Easy trick: If a number ends in 0, 2, 4, 6, or 8, it is even.
Subtopic: Odd Numbers
Odd numbers cannot be shared equally between two people.
5 sweets shared between 2 children gives 2 each and 1 remains.
So 5 is odd.
Examples of odd numbers:
Odd numbers end in: 1, 3, 5, 7, or 9.
Subtopic: Prime Numbers
First understand factors.
A factor is a number that divides another number exactly.
Factors of 8: 1, 2, 4, 8
Prime numbers have only two factors:
- 1
- And itself
Examples of prime numbers:
Is 11 prime?
Factors of 11 are 1 and 11 only.
So 11 is prime.
Is 12 prime?
Factors of 12 are 1, 2, 3, 4, 6, 12.
It has many factors, so it is not prime.
Common Mistakes Students Make
- Thinking 1 is a prime number — it is not. Prime numbers start from 2.
- Confusing odd and even numbers — always check the last digit.
- Forgetting place value — in 234, the digit 2 means 200, not just 2.
Practice Questions
Easy
- Write the first 10 natural numbers.
- State the place value of 5 in 352.
- Write 456 in expanded form.
- Round 37 to the nearest ten.
- Round 264 to the nearest hundred.
Medium
- Add: 345 + 128
- Subtract: 500 − 276
- Multiply: 14 × 5
- Divide: 96 ÷ 8
- List all even numbers between 10 and 20.
KCSE Style
- Write 5,304,210 in words.
- State the place value of digit 6 in 3,642,178.
- Round 45,768 to the nearest thousand.
- Identify prime numbers between 10 and 30.
Lesson Summary
In this lesson we learned that natural numbers are counting numbers.
- Place value
- Large numbers up to billions
- Rounding numbers
- Addition, subtraction, multiplication, and division
- Word problems
- Even numbers
- Odd numbers
- Prime numbers
These ideas form the foundation of all mathematics. When you understand them well, many other mathematics topics become easier.