Classifying Rational and Irrational Worksheet | Download Free PDF ... - Free Printable

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 form is non-repeating and non-terminating (e.g., $\sqrt{2}$, $\pi$).

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Now, let’s go through each number:

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Part 1: Classify and Convert

| Number | Rational or Irrational? | Convert dec./fraction |
|--------|--------------------------|------------------------|
| 1) 0.8 | Rational | $\frac{8}{10} = \frac{4}{5}$ ✔ (already given) |
| 2) $\frac{-3}{10}$ | Rational | Already a fraction: $\frac{-3}{10}$ |
| 3) $\sqrt{40}$ | Irrational | $\sqrt{40} = \sqrt{4 \cdot 10} = 2\sqrt{10}$ → not a perfect square → irrational |
| 4) $\sqrt{81}$ | Rational | $\sqrt{81} = 9 = \frac{9}{1}$ |
| 5) $2\frac{1}{3}$ | Rational | $2 + \frac{1}{3} = \frac{7}{3}$ |
| 6) 0.35 | Rational | $= \frac{35}{100} = \frac{7}{20}$ |
| 7) 0.3333... | Rational | This is $0.\overline{3} = \frac{1}{3}$ |
| 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; decimal is non-repeating, non-terminating |

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Part 2: Directions – Classify and Tell Type

#### 11) -0.6

- Is it rational or irrational?
- It is a terminating decimal.
- Can be written as $-\frac{6}{10} = -\frac{3}{5}$
- So, rational
- Is it terminating or repeating?
- It terminates after one decimal place.

✔ Answer:
- Rational
- Terminating

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

- $\sqrt{100} = 10$, which is an integer.
- Integers are rational.
- 10 can be written as $\frac{10}{1}$ → rational
- As a decimal: 10.0 → terminates

✔ 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 Part Answers

11) $-0.6$
→ Rational, Terminating

12) $\sqrt{100}$
→ Rational, Terminating

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✔ Summary of Key Concepts Used:

- All fractions (including negative ones) are rational.
- Square roots of perfect squares are rational.
- Square roots of non-perfect squares are irrational.
- Terminating and repeating decimals are rational.
- Non-repeating, non-terminating decimals are irrational.

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