Math worksheet for classifying rational and irrational numbers, with exercises to convert decimals to fractions and identify decimal types.

Let's solve the problem step by step.

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Understanding Rational and Irrational Numbers

- Rational numbers can be expressed as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$. They include:
- Terminating decimals (e.g., 0.5)
- Repeating decimals (e.g., 0.333...)
- Integers
- Fractions

- Irrational numbers cannot be written as a simple fraction. Their decimal expansions are non-terminating and non-repeating (e.g., $\sqrt{2}, \pi$).

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## ✔ Part 1: Classify each number and convert if rational

| Number | Rational or Irrational? | Convert dec./fraction |
|--------|-------------------------|------------------------|
| 1) 0.8 | Rational | $\frac{8}{10} = \frac{4}{5}$ |
| 2) $\frac{-3}{10}$ | Rational | Already a fraction: $\frac{-3}{10}$ |
| 3) $\sqrt{40}$ | Irrational | Cannot simplify to a rational number. $\sqrt{40} = 2\sqrt{10}$, which is irrational. |
| 4) $\sqrt{81}$ | Rational | $\sqrt{81} = 9 = \frac{9}{1}$ |
| 5) $2\frac{1}{3}$ | Rational | $= \frac{7}{3}$ |
| 6) 0.35 | Rational | $= \frac{35}{100} = \frac{7}{20}$ |
| 7) 0.3333... | Rational | Repeating decimal → $\frac{1}{3}$ |
| 8) $-9$ | Rational | $= \frac{-9}{1}$ |
| 9) 3.4 | Rational | $= \frac{34}{10} = \frac{17}{5}$ |
| 10) $\sqrt{2}$ | Irrational | Cannot be written as a fraction; known to be irrational |

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✔ Part 2: Directions – Classify and describe decimal type

#### 11) $-0.6$

- Classification: Rational
(It’s a terminating decimal and can be written as a fraction.)

- Convert to fraction:
$-0.6 = -\frac{6}{10} = -\frac{3}{5}$

- Decimal type: Terminating

✔ Answer:
rational, terminating

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#### 12) $\sqrt{100}$

- First, simplify: $\sqrt{100} = 10$

- 10 is an integer → Rational

- As a decimal: 10.0 → terminating

✔ Answer:
rational, terminating

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✔ Final Answers:

#### Table:

| Number | Rational or Irrational? | Convert dec./fraction |
|--------|-------------------------|------------------------|
| 1) 0.8 | Rational | $\frac{4}{5}$ |
| 2) $\frac{-3}{10}$ | Rational | $\frac{-3}{10}$ |
| 3) $\sqrt{40}$ | Irrational | — |
| 4) $\sqrt{81}$ | Rational | $\frac{9}{1}$ |
| 5) $2\frac{1}{3}$ | Rational | $\frac{7}{3}$ |
| 6) 0.35 | Rational | $\frac{7}{20}$ |
| 7) 0.3333... | Rational | $\frac{1}{3}$ |
| 8) $-9$ | Rational | $\frac{-9}{1}$ |
| 9) 3.4 | Rational | $\frac{17}{5}$ |
| 10) $\sqrt{2}$ | Irrational | — |

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#### Directions:

11) $-0.6$:
→ rational, terminating

12) $\sqrt{100}$:
→ rational, terminating

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✔ Summary of Key Concepts:

- All integers, fractions, terminating, and repeating decimals are rational.
- Square roots of perfect squares (like $\sqrt{81} = 9$, $\sqrt{100} = 10$) are rational.
- Square roots of non-perfect squares (like $\sqrt{40}$, $\sqrt{2}$) are irrational.
- Repeating decimals like $0.\overline{3}$ are rational because they can be converted to fractions.

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