Rational and Irrational Numbers exercise - Free Printable

Let's solve this step by step.

We are asked to sort the given numbers into Rational and Irrational categories.

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🔹 Definitions:

- Rational Numbers: Can be written as a fraction (ratio of two integers).
They include:
- Terminating decimals (e.g., 0.5)
- Repeating decimals (e.g., 0.333... = 1/3)
- Integers
- Roots that simplify to whole or fractional numbers

- Irrational Numbers: Cannot be written as a simple fraction.
Their decimal expansions are non-repeating and non-terminating.
Examples: π, e, √2, etc.

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🔍 Let’s analyze each number:

| Number | Analysis | Category |
|-------|--------|---------|
| 3.14 | This is a terminating decimal, so it can be written as a fraction (314/100). Even though it looks like π, it's just an approximation. | ✔ Rational |
| 0.444 | Terminating decimal → 444/1000 = 111/250 | ✔ Rational |
| √121 | √121 = 11 → integer | ✔ Rational |
| √1000 | √1000 ≈ 31.62..., not a perfect square → non-repeating, non-terminating → irrational | ✘ Irrational |
| π | Famous irrational number | ✘ Irrational |
| π/7 | π is irrational; dividing by 7 still gives an irrational number (rational × irrational = irrational) | ✘ Irrational |
| 0.333... | Repeating decimal → 1/3 → rational | ✔ Rational |
| 1.12313... | The dots suggest non-repeating pattern (no bar or repetition shown), so likely non-repeating and non-terminating → irrational | ✘ Irrational |
| 0.12̄ | Repeating decimal (0.121212...) → rational | ✔ Rational |
| e | Euler's number — irrational | ✘ Irrational |
| ∛9 | Cube root of 9 → not a perfect cube → irrational | ✘ Irrational |
| 2.9̄ | Repeating decimal (2.999...) → actually equals 3 → rational | ✔ Rational |
| 2e | 2 × e → e is irrational → multiplying by 2 (rational) keeps it irrational | ✘ Irrational |
| 1.234̄ | Repeating decimal → rational | ✔ Rational |
| ∛8 | ∛8 = 2 → perfect cube → rational | ✔ Rational |

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✔ Final Sorting:

#### 🟩 Rational Numbers:
- 3.14
- 0.444
- √121
- 0.333...
- 0.12̄
- 2.9̄
- 1.234̄
- ∛8

> These are all either terminating, repeating decimals, or exact roots.

#### 🔴 Irrational Numbers:
- √1000
- π
- π/7
- 1.12313...
- e
- ∛9
- 2e

> These are either known irrationals (π, e), non-repeating decimals, or roots of non-perfect powers.

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✔ Answer Table:

#### Rational:
```
3.14 0.444 √121
0.333... 0.12̄ 2.9̄
1.234̄ ∛8
```

#### Irrational:
```
√1000 π π/7
1.12313... e ∛9
2e
```

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✔ Summary:

| Rational | Irrational |
|--------|------------|
| 3.14 | √1000 |
| 0.444 | π |
| √121 | π/7 |
| 0.333... | 1.12313... |
| 0.12̄ | e |
| 2.9̄ | ∛9 |
| 1.234̄ | 2e |
| ∛8 | |

> Note: You can arrange them in any order within the columns.

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✔ Final Answer:
Place the numbers accordingly in the table based on the above classification.