Subsets of Set of Real Numbers - Subsets of Real Numbers and Examples - Free Printable

The image you've uploaded is a visual representation of the definition and classification of real numbers, as presented by BYJU'S. Let's break down the content and explain the solution to understanding this diagram.

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🔷 Overview: Real Numbers

Real numbers include all numbers that can be represented on the number line. They are divided into two main categories:

1. Rational Numbers
2. Irrational Numbers

These two together make up the set of real numbers.

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## 🟡 1. Rational Numbers

> Definition: A number is rational if it can be expressed as a fraction $ \frac{p}{q} $, where $ p $ and $ q $ are integers and $ q \neq 0 $. Rational numbers have decimal expansions that either terminate or repeat.

Examples shown:
- $ \frac{5}{3} $ → This is a fraction (rational).
- $ 0.63 $ → Terminating decimal → rational.
- $ 0.0\overline{12} $ → Repeating decimal (the "12" repeats) → rational.

✔ All these are rational because they can be written as fractions.

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## 🔵 Nested Subsets Within Rational Numbers

The diagram shows that rational numbers contain several subsets:

➤ Integers

> Whole numbers and their negatives:
$ \{ ..., -2, -1, 0, 1, 2, ... \} $

- Examples: $ -3, 0, 5 $
- All integers are rational because they can be written as $ \frac{n}{1} $

➤ Whole Numbers

> Non-negative integers:
$ \{ 0, 1, 2, 3, ... \} $

- Includes zero and positive integers
- Subset of integers

➤ Natural Numbers

> Positive integers starting from 1:
$ \{ 1, 2, 3, ... \} $

- Sometimes includes 0 depending on context, but here it starts at 1
- Subset of whole numbers

> So the hierarchy is:
> Natural ⊂ Whole ⊂ Integers ⊂ Rational

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## 🟣 2. Irrational Numbers

> Definition: Numbers that cannot be written as a fraction $ \frac{p}{q} $. Their decimal expansions are non-terminating and non-repeating.

Examples shown:
- $ \sqrt{3} $ → Cannot be simplified to a fraction; decimal goes on forever without repeating.
- $ \pi $ → Famous irrational number (~3.14159...), non-repeating, non-terminating.
- $ 0.10100110... $ → The pattern doesn’t repeat and isn't predictable — so irrational.

⚠️ Note: Even though it looks like a pattern, if it doesn't repeat in a cyclic way, it’s still irrational.

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## ✔ Summary: Classification of Real Numbers

| Category | Description | Examples |
|----------------|---------------------------------------------------|-------------------------------------------|
| Rational | Can be written as $ \frac{p}{q} $; terminating or repeating decimals | $ \frac{5}{3}, 0.63, 0.\overline{12} $ |
| Irrational | Cannot be written as a fraction; non-repeating, non-terminating decimals | $ \sqrt{3}, \pi, 0.10100110... $ |
| Integers | Negative, zero, positive whole numbers | $ ..., -2, -1, 0, 1, 2, ... $ |
| Whole | Zero and positive integers | $ 0, 1, 2, 3, ... $ |
| Natural | Positive counting numbers | $ 1, 2, 3, ... $ |

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## 🔍 Key Takeaways

- Every integer is a rational number.
- Every whole number is an integer.
- Every natural number is a whole number.
- Rational + Irrational = Real Numbers.
- Not all decimals are rational — only those that terminate or repeat.

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💡 Example Problem (Based on Diagram)

Question: Is $ 0.10100110... $ rational or irrational?

Answer: It is irrational, because although it has a pattern, it does not repeat in a fixed cycle. Its decimal expansion is non-repeating and non-terminating → therefore, irrational.

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This diagram is a great tool for understanding how different types of numbers fit together within the broader category of real numbers.