Educational worksheet for classifying rational and irrational numbers, including examples and sorting exercise.

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 Recap:

- Rational Numbers: Can be written as a fraction (ratio of two integers), or have a terminating or repeating decimal.
- Examples: $ \frac{1}{2}, 0.5, 0.\overline{3}, 4.5 $

- Irrational Numbers: Cannot be written as a simple fraction; their decimal form is non-repeating and non-terminating.
- Examples: $ \pi, e, \sqrt{2}, \sqrt{3} $, etc.

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Now let’s analyze each number one by one.

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✔ Given Numbers:

| Number | Analysis |
|-------|---------|
| 3.14 | This is a terminating decimal, so it can be written as $ \frac{314}{100} $. → Rational |
| 0.444 | Terminating decimal → $ \frac{444}{1000} $ → Rational |
| √121 | $ \sqrt{121} = 11 $ → Integer → Rational |
| √1000 | $ \sqrt{1000} = \sqrt{100 \times 10} = 10\sqrt{10} $ → $ \sqrt{10} $ is irrational → So this is Irrational |
| π | Famous irrational number → Irrational |
| π/7 | $ \pi $ is irrational, divided by 7 (rational) → still irrational → Irrational |
| 0.333... | Repeating decimal → $ \frac{1}{3} $ → Rational |
| 1.12313... | The dots suggest non-repeating, non-terminating → If no pattern, likely Irrational → Assume Irrational |
| 0.12̄ | Repeating decimal: $ 0.121212... $ → Can be expressed as a fraction → Rational |
| e | Euler's number, known to be irrational → Irrational |
| ∛9 | Cube root of 9 → Not a perfect cube → Non-repeating, non-terminating → Irrational |
| 2.9̄ | Repeating decimal: $ 2.9999... $ → Actually equals 3 → But even if not, repeating decimals are rational → Rational |
| 2e | $ e $ is irrational, multiplied by 2 (rational) → still irrational → Irrational |
| 1.234̄ | Repeating decimal → $ 1.234234234... $ → Can be written as a fraction → Rational |
| ∛8 | $ \sqrt[3]{8} = 2 $ → Perfect cube → Integer → Rational |

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✔ Final Classification:

#### 🟩 Rational Numbers:
These can be written as fractions or have terminating/repeating decimals.

1. 3.14 → terminating
2. 0.444 → terminating
3. √121 = 11 → integer
4. 0.333... = 1/3 → repeating
5. 0.12̄ = 0.121212... → repeating
6. 2.9̄ = 2.999... = 3 → repeating
7. 1.234̄ = 1.234234... → repeating
8. ∛8 = 2 → integer

✔ Total: 8 rational numbers

#### 🔴 Irrational Numbers:
Non-repeating, non-terminating decimals; cannot be expressed as fractions.

1. √1000 = 10√10 → irrational
2. π → irrational
3. π/7 → irrational
4. 1.12313... → non-repeating (assumed)
5. e → irrational
6. 2e → irrational
7. ∛9 → irrational

✔ Total: 7 irrational numbers

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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
- 2e
- ∛9

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✔ Final Answer:

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

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

> ⚠️ Note: Some students might question 1.12313... — since it's written with "...", we assume it's non-repeating/non-terminating, hence irrational. In real problems, such notation implies an unknown or non-repeating pattern.

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✔ You're done! Just place these in the boxes accordingly.