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

Absolutely! Let's solve this worksheet step by step, classifying each number as rational or irrational, and for rational numbers, we’ll convert them to fractions (or show why they are repeating/terminating decimals).

---

🔍 Definitions Recap:

- Rational Number: Any number that can be written as a fraction `a/b` where `a` and `b` are integers and `b ≠ 0`. This includes integers, terminating decimals, and repeating decimals.
- Irrational Number: A number that cannot be written as a simple fraction. Its decimal goes on forever without repeating. Examples: √2, π, etc.

---

## ✔ Worksheet Solutions:

---

1) 0.8

✔ Already done:
Rational → `8/10` or `4/5`

---

2) -3/10

➡️ This is already a fraction of two integers.
✔ Rational
→ Fraction: `-3/10` (already in simplest form)

---

3) √40

Let’s simplify:
√40 = √(4×10) = 2√10
√10 is irrational → so 2√10 is also irrational.
✔ Irrational
→ *Cannot be written as a fraction*

---

4) √81

√81 = 9 (since 9×9=81)
9 is an integer → all integers are rational.
✔ Rational
→ Fraction: `9/1`

---

5) 2 1/3

This is a mixed number. Convert to improper fraction:
2 1/3 = (2×3 + 1)/3 = 7/3
✔ Rational
→ Fraction: `7/3`

---

6) 0.35

This is a terminating decimal → always rational.
0.35 = 35/100 = 7/20 (simplified)
✔ Rational
→ Fraction: `7/20`

---

7) 0.3333...

This is a repeating decimal (0.̅3). Repeating decimals are rational.
Let x = 0.3333...
Then 10x = 3.3333...
Subtract: 10x - x = 3.333... - 0.333... → 9x = 3 → x = 3/9 = 1/3
✔ Rational
→ Fraction: `1/3`

---

8) -9

Integer → all integers are rational.
✔ Rational
→ Fraction: `-9/1`

---

9) 3.4

Terminating decimal → rational.
3.4 = 34/10 = 17/5
✔ Rational
→ Fraction: `17/5`

---

10) √2

This is the classic irrational number. It cannot be expressed as a fraction.
✔ Irrational
→ *Cannot be written as a fraction*

---

## 📝 Directions Section:

---

11) -0.6

- Terminating decimal → rational.
- -0.6 = -6/10 = -3/5
✔ Rational | Terminating

---

12) √100

√100 = 10 (since 10×10=100)
10 is an integer → rational.
✔ Rational | Terminating (since 10.0 is terminating)

---

## 🧾 Final Answer Table:

| # | Number | Rational or Irrational? | Convert dec./fraction |
|---|----------------|--------------------------|------------------------------|
| 1 | 0.8 | Rational | 8/10 or 4/5 |
| 2 | -3/10 | Rational | -3/10 |
| 3 | √40 | Irrational | — |
| 4 | √81 | Rational | 9/1 |
| 5 | 2 1/3 | Rational | 7/3 |
| 6 | 0.35 | Rational | 7/20 |
| 7 | 0.3333... | Rational | 1/3 |
| 8 | -9 | Rational | -9/1 |
| 9 | 3.4 | Rational | 17/5 |
|10 | √2 | Irrational | — |

---

## ✔ Directions Answers:

11) -0.6 → Rational, Terminating
12) √100 → Rational, Terminating

---

Let me know if you’d like a printable version or explanation for any specific one! 😊