Rational vs. Irrational Numbers | Worksheet - Free Printable

You're working on a worksheet titled “Rational vs. Irrational Numbers”.

Let’s break down the concepts first, then solve the practice section step by step.

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## ✔ Key Definitions:

🟢 Rational Number:

- Can be written as a fraction of two integers (like `a/b`, where `b ≠ 0`).
- When written as a decimal, it either:
- Terminates (e.g., 0.5, 3.75), or
- Repeats (e.g., 0.333..., 0.142857142857...).

Examples from your sheet:
- `0.9 = 9/10`
- `-8.13 = -813/100`
- `4.3 = 43/10`

🔴 Irrational Number:

- Cannot be written as a fraction of two integers.
- As a decimal, it is non-terminating and non-repeating.
- Often involves square roots of non-perfect squares, π, e, etc.

Examples from your sheet:
- `√2 = 1.414213562...` → irrational
- `π = 3.14159265...` → irrational
- `√10 = 3.16227766...` → irrational

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## 📝 Practice Section: Circle Rational Numbers, Draw Square Around Irrational Numbers

We’ll go row by row and classify each number.

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Row 1:

> `3/4`, `√13`, `-9.5`, `-π`, `√16`, `1000`, `1 1/12`

- `3/4` → Fraction → ✔ Rational → CIRCLE
- `√13` → 13 not perfect square → ✘ Irrational → SQUARE
- `-9.5` → Terminating decimal → = `-19/2` → ✔ Rational → CIRCLE
- `-π` → π is irrational → ✘ Irrational → SQUARE
- `√16` → = 4 → integer → ✔ Rational → CIRCLE
- `1000` → Integer → ✔ Rational → CIRCLE
- `1 1/12` → Mixed number = `13/12` → ✔ Rational → CIRCLE

✔ Circles: `3/4`, `-9.5`, `√16`, `1000`, `1 1/12`
🔴 Squares: `√13`, `-π`

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Row 2:

> `2.72`, `4.6`, `√0.1`, `2/3`, `-√25`, `√9`, `-1/5`

- `2.72` → Terminating → = `272/100` → ✔ Rational → CIRCLE
- `4.6` → Terminating → = `46/10` → ✔ Rational → CIRCLE
- `√0.1` → = √(1/10) = 1/√10 → irrational → ✘ Irrational → SQUARE
- `2/3` → Fraction → ✔ Rational → CIRCLE
- `-√25` → = -5 → integer → ✔ Rational → CIRCLE
- `√9` → = 3 → integer → ✔ Rational → CIRCLE
- `-1/5` → Fraction → ✔ Rational → CIRCLE

✔ Circles: `2.72`, `4.6`, `2/3`, `-√25`, `√9`, `-1/5`
🔴 Square: `√0.1`

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Row 3:

> `14/2`, `√25`, `1/100`, `π`, `n = 5`, `-4/7`, `1 - √2`, `-7`

- `14/2` → = 7 → integer → ✔ Rational → CIRCLE
- `√25` → = 5 → integer → ✔ Rational → CIRCLE
- `1/100` → Fraction → ✔ Rational → CIRCLE
- `π` → irrational → ✘ Irrational → SQUARE
- `n = 5` → Assuming this means the number 5 → integer → ✔ Rational → CIRCLE
- `-4/7` → Fraction → ✔ Rational → CIRCLE
- `1 - √2` → √2 irrational → subtracting irrational from rational → still irrational → ✘ Irrational → SQUARE
- `-7` → Integer → ✔ Rational → CIRCLE

✔ Circles: `14/2`, `√25`, `1/100`, `n=5`, `-4/7`, `-7`
🔴 Squares: `π`, `1 - √2`

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Row 4:

> `√90`, `3/11`, `√5`, `0`, `10.4`, `13`, `√100`

- `√90` → 90 not perfect square → ✘ Irrational → SQUARE
- `3/11` → Fraction → ✔ Rational → CIRCLE
- `√5` → Not perfect square → ✘ Irrational → SQUARE
- `0` → Integer → ✔ Rational → CIRCLE
- `10.4` → Terminating → = `104/10` → ✔ Rational → CIRCLE
- `13` → Integer → ✔ Rational → CIRCLE
- `√100` → = 10 → integer → ✔ Rational → CIRCLE

✔ Circles: `3/11`, `0`, `10.4`, `13`, `√100`
🔴 Squares: `√90`, `√5`

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Row 5:

> `3.6`, `-21.2`, `3π`, `√4 + √5`, `-3/10`, `√14`, `-√1`

- `3.6` → Terminating → = `36/10` → ✔ Rational → CIRCLE
- `-21.2` → Terminating → = `-212/10` → ✔ Rational → CIRCLE
- `3π` → π irrational → multiplying by 3 doesn’t make it rational → ✘ Irrational → SQUARE
- `√4 + √5` → = `2 + √5` → √5 irrational → sum is irrational → ✘ Irrational → SQUARE
- `-3/10` → Fraction → ✔ Rational → CIRCLE
- `√14` → Not perfect square → ✘ Irrational → SQUARE
- `-√1` → = -1 → integer → ✔ Rational → CIRCLE

✔ Circles: `3.6`, `-21.2`, `-3/10`, `-√1`
🔴 Squares: `3π`, `√4 + √5`, `√14`

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Row 6:

> `√2`, `0.17`, `-2/14`, `8.3`, `√64`, `2/25`, `1.36`

- `√2` → Irrational → ✘ Irrational → SQUARE
- `0.17` → Terminating → = `17/100` → ✔ Rational → CIRCLE
- `-2/14` → Simplifies to `-1/7` → Fraction → ✔ Rational → CIRCLE
- `8.3` → Terminating → = `83/10` → ✔ Rational → CIRCLE
- `√64` → = 8 → integer → ✔ Rational → CIRCLE
- `2/25` → Fraction → ✔ Rational → CIRCLE
- `1.36` → Terminating → = `136/100` → ✔ Rational → CIRCLE

✔ Circles: `0.17`, `-2/14`, `8.3`, `√64`, `2/25`, `1.36`
🔴 Square: `√2`

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## 🎯 Final Answer Summary:

You should CIRCLE these numbers (Rational):

```
3/4, -9.5, √16, 1000, 1 1/12,
2.72, 4.6, 2/3, -√25, √9, -1/5,
14/2, √25, 1/100, n=5, -4/7, -7,
3/11, 0, 10.4, 13, √100,
3.6, -21.2, -3/10, -√1,
0.17, -2/14, 8.3, √64, 2/25, 1.36
```

You should DRAW A SQUARE around these numbers (Irrational):

```
√13, -π,
√0.1,
π, 1 - √2,
√90, √5,
3π, √4 + √5, √14,
√2
```

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## 💡 Pro Tip:
To quickly check if a square root is rational — ask: “Is the number under the root a perfect square?” If yes → rational. If no → irrational.

Also remember: π and any multiple of π (like 3π) are always irrational.

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Let me know if you’d like this formatted as an answer key for printing or coloring! 😊