🎯 By the end of this lesson...
You should be able to:
- Identify natural numbers.
- Identify whole numbers.
- Identify integers.
- Identify rational numbers.
- Identify irrational numbers.
- Explain what real numbers are.
- Recognise terminating, recurring and non-recurring decimals.
- Classify a number into all the families to which it belongs.
🤔 Think About This...
Imagine your friend says:
Do you agree?
Many students make this mistake because they look only at how a number appears instead of asking which number family it belongs to.
📖 What Are Types of Numbers?
Numbers can be placed into different groups according to their properties. These groups are called number families.
Some smaller number families belong inside larger number families.
🌳 The Real Number Family
Follow the arrows from top to bottom.
1️⃣ Natural Numbers
Natural numbers are the counting numbers beginning at 1.
Natural numbers do not include:
- Zero
- Negative numbers
- Fractions
- Decimals between whole numbers
✅ Natural Numbers
❌ Not Natural Numbers
2️⃣ Whole Numbers
Whole numbers are the natural numbers together with zero.
Whole numbers do not include:
- Negative numbers
- Fractions
- Decimals between whole numbers
✅ Whole Numbers
❌ Not Whole Numbers
3️⃣ Integers
Integers are positive and negative whole numbers, including zero.
✅ Integers
❌ Not Integers
4️⃣ Rational Numbers
A rational number is any number that can be written exactly in the form:
where:
- a is an integer;
- b is an integer;
- b ≠ 0.
Rational numbers include:
- Integers
- Ordinary fractions
- Terminating decimals
- Recurring decimals
✅ Rational Numbers
❌ Not Rational Numbers
🔢 Types of Decimals
The way a decimal behaves can help us decide whether it is rational or irrational.
1. Terminating Decimals
A terminating decimal is a decimal that ends.
Every terminating decimal can be written as a fraction.
2. Recurring or Repeating Decimals
A recurring decimal does not end, but one digit or a group of digits repeats in a regular pattern.
Recurring decimals can also be written as fractions.
5️⃣ Irrational Numbers
Irrational numbers cannot be written exactly in the form:
where a and b are integers and b ≠ 0.
Their decimal expansions:
- do not end;
- do not repeat in a regular pattern.
Examples
The decimal continues forever without repeating.
The decimal continues forever without repeating.
The decimal continues forever without repeating.
√2 = a/b
Therefore, √2 is irrational.
Always simplify the square root first.
√16 = 4
Since 4 is an integer, √16 is rational.
📊 Rational and Irrational Decimals
| Decimal type | Does it end? | Does it repeat? | Number family |
|---|---|---|---|
| Terminating | Yes | Not needed | Rational |
| Recurring | No | Yes | Rational |
| Non-terminating and non-recurring | No | No | Irrational |
A decimal is irrational if it continues forever without repeating.
6️⃣ Real Numbers
Real numbers include all rational numbers and all irrational numbers.
Every number that can be placed on an ordinary number line is a real number.
🧠 The StudyNest Thinking Method
When a question asks you to classify a number, do not decide based only on how it looks.
Step 1: Simplify the number first, if possible.
For example:
√16 = 4
Step 2: Is it a positive counting number beginning at 1?
If yes, it is natural.
Step 3: Is it zero or a positive whole-valued number?
If yes, it is whole.
Step 4: Is it a positive or negative whole-valued number?
If yes, it is an integer.
Step 5: Can it be written exactly as a/b, where a and b are integers and b ≠ 0?
If yes, it is rational.
Step 6: If it is a decimal, does it end or repeat?
If it ends or repeats, it is rational.
Step 7: Does the decimal continue forever without repeating?
If yes, it is irrational.
Step 8: Every number classified above is also a real number.
📝 Worked Examples
Example 1: Classify −7
−7 is not natural or whole because it is negative.
It is an integer.
It can be written as a fraction of two integers, so it is rational.
Example 2: Classify 0
0 is not a natural number because natural numbers begin at 1.
0 is a whole number and an integer.
Example 3: Classify √3
√3 cannot be simplified to a whole-valued number.
Its decimal continues forever without repeating.
Example 4: Classify √16
Simplify first:
4 is a positive counting number.
Example 5: Classify 0.75
0.75 is a terminating decimal.
Example 6: Classify 0.333…
The digit 3 repeats forever.
⚠ Common Mistakes
Thinking −5 is a whole number because it has no decimal part.
Negative integers are not whole numbers.
Thinking every decimal is irrational.
Terminating and recurring decimals are rational.
Forgetting that every integer can be written over 1.
For example, 6 = 6/1, so 6 is rational.
Assuming every square root is irrational.
Always simplify the square root first.
Thinking a decimal is rational only when it ends.
Recurring decimals are also rational.
🎮 Your Turn!
Classify each number into all the number families to which it belongs.
- 8
- 0
- −15
- ⅝
- π
- √16
- 0.75
- √7
- 0.444…
- −2.5
✅ Show Answers
- Natural, whole, integer, rational and real
- Whole, integer, rational and real
- Integer, rational and real
- Rational and real
- Irrational and real
- Natural, whole, integer, rational and real
- Rational and real
- Irrational and real
- Rational and real
- Rational and real
🧠 Remember This
Whole numbers begin at 0.
Integers include negative and positive whole-valued numbers.
Rational decimals end or repeat.
Irrational decimals never end and never repeat.
Always simplify a square root before classifying it.
🎯 Before Moving On...
- ✅ Can I explain the difference between natural and whole numbers?
- ✅ Can I identify an integer?
- ✅ Can I explain why every integer is rational?
- ✅ Can I recognise terminating and recurring decimals?
- ✅ Can I identify an irrational decimal?
- ✅ Can I simplify a square root before classifying it?
- ✅ Can I classify one number into several families?
🎉 Well Done!
You now understand the main families within the real number system.
Simplify → Test the number → Check its decimal → List every family
📈 Your Progress
Lesson 1 of 17