Rational and Irrational Numbers exercise - Free Printable
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Step-by-step solution for: Rational and Irrational Numbers exercise
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Show Answer Key & Explanations
Step-by-step solution for: Rational and Irrational Numbers exercise
Let’s solve this step by step.
---
- Rational Number: A real number that can be written as a simple fraction (i.e., ratio of two integers). This includes:
- Terminating decimals (e.g., 0.5, 3.14)
- Repeating decimals (e.g., 0.333..., 2.9̅, 0.12̅)
- Perfect squares/cubes under roots if they simplify to integers or fractions
- Any number that can be expressed as `a/b` where `a` and `b` are integers and `b ≠ 0`
- Irrational Number: A real number that cannot be written as a simple fraction. These have non-repeating, non-terminating decimal expansions. Examples given:
- π (Pi)
- e (Euler’s number)
- Many square/cube roots like √3, ∛9, etc.
---
## ✔ Step-by-step Classification:
We’ll go through each number in the grid and classify it.
---
- This is a terminating decimal → can be written as 314/100 → Rational
---
- Terminating decimal → 444/1000 → Rational
---
- √121 = 11 → integer → Rational
---
- √1000 = √(100 × 10) = 10√10
- √10 is irrational → so 10√10 is Irrational
---
- Given as famous irrational number → Irrational
---
- π is irrational; dividing an irrational number by a rational number (7) still gives irrational → Irrational
---
- Repeating decimal → = 1/3 → Rational
---
- The “...” suggests it continues without repeating pattern (not marked with bar) → Irrational
> ⚠️ Important: If it were 1.123̅13̅ or something with a repeating block, it would be rational. But here, no bar → assumed non-repeating → irrational.
---
- The bar is over “2”, meaning 0.12222... → repeating decimal → Rational
---
- Euler’s number — given as irrational → Irrational
---
- Cube root of 9 — 9 is not a perfect cube → cannot be simplified to a fraction → Irrational
---
- Bar over 9 → 2.999... = 3 → which is rational → Rational
> 💡 Fun fact: 2.9̅ = 3 exactly! So definitely rational.
---
- e is irrational → multiplying by 2 (rational) → still irrational → Irrational
---
- Bar over “234” → repeating block → 1.234234234... → Rational
---
- ∛8 = 2 → integer → Rational
---
## ✔ Final Sorting:
- 3.14
- 0.444
- √121
- 0.333...
- 0.12̅
- 2.9̅
- 1.234̅
- ∛8
→ Total: 8 numbers
---
- √1000
- π
- π/7
- 1.12313...
- e
- ∛9
- 2e
→ Total: 7 numbers
---
## 📋 Answer Grid:
```
3.14 0.444 √121
0.333... 0.12̅ 2.9̅
1.234̅ ∛8
```
*(You can arrange them in any order in the boxes — just fill all 8)*
---
```
√1000 π π/7
1.12313... e ∛9
2e
```
*(Fill these 7 into the boxes — again, order doesn’t matter)*
---
✔ Done! You’ve correctly sorted all 15 numbers.
Let me know if you’d like this formatted for printing or pasting into a worksheet!
---
📌 Definitions Recap (from the worksheet):
- Rational Number: A real number that can be written as a simple fraction (i.e., ratio of two integers). This includes:
- Terminating decimals (e.g., 0.5, 3.14)
- Repeating decimals (e.g., 0.333..., 2.9̅, 0.12̅)
- Perfect squares/cubes under roots if they simplify to integers or fractions
- Any number that can be expressed as `a/b` where `a` and `b` are integers and `b ≠ 0`
- Irrational Number: A real number that cannot be written as a simple fraction. These have non-repeating, non-terminating decimal expansions. Examples given:
- π (Pi)
- e (Euler’s number)
- Many square/cube roots like √3, ∛9, etc.
---
## ✔ Step-by-step Classification:
We’ll go through each number in the grid and classify it.
---
1. 3.14
- This is a terminating decimal → can be written as 314/100 → Rational
---
2. 0.444
- Terminating decimal → 444/1000 → Rational
---
3. √121
- √121 = 11 → integer → Rational
---
4. √1000
- √1000 = √(100 × 10) = 10√10
- √10 is irrational → so 10√10 is Irrational
---
5. π
- Given as famous irrational number → Irrational
---
6. π/7
- π is irrational; dividing an irrational number by a rational number (7) still gives irrational → Irrational
---
7. 0.333...
- Repeating decimal → = 1/3 → Rational
---
8. 1.12313...
- The “...” suggests it continues without repeating pattern (not marked with bar) → Irrational
> ⚠️ Important: If it were 1.123̅13̅ or something with a repeating block, it would be rational. But here, no bar → assumed non-repeating → irrational.
---
9. 0.12̅
- The bar is over “2”, meaning 0.12222... → repeating decimal → Rational
---
10. e
- Euler’s number — given as irrational → Irrational
---
11. ∛9
- Cube root of 9 — 9 is not a perfect cube → cannot be simplified to a fraction → Irrational
---
12. 2.9̅
- Bar over 9 → 2.999... = 3 → which is rational → Rational
> 💡 Fun fact: 2.9̅ = 3 exactly! So definitely rational.
---
13. 2e
- e is irrational → multiplying by 2 (rational) → still irrational → Irrational
---
14. 1.234̅
- Bar over “234” → repeating block → 1.234234234... → Rational
---
15. ∛8
- ∛8 = 2 → integer → Rational
---
## ✔ Final Sorting:
🟩 RATIONAL NUMBERS (Can be written as fractions):
- 3.14
- 0.444
- √121
- 0.333...
- 0.12̅
- 2.9̅
- 1.234̅
- ∛8
→ Total: 8 numbers
---
🟧 IRRATIONAL NUMBERS (Cannot be written as fractions):
- √1000
- π
- π/7
- 1.12313...
- e
- ∛9
- 2e
→ Total: 7 numbers
---
## 📋 Answer Grid:
Rational Column (8 numbers):
```
3.14 0.444 √121
0.333... 0.12̅ 2.9̅
1.234̅ ∛8
```
*(You can arrange them in any order in the boxes — just fill all 8)*
---
Irrational Column (7 numbers):
```
√1000 π π/7
1.12313... e ∛9
2e
```
*(Fill these 7 into the boxes — again, order doesn’t matter)*
---
✔ Done! You’ve correctly sorted all 15 numbers.
Let me know if you’d like this formatted for printing or pasting into a worksheet!
Parent Tip: Review the logic above to help your child master the concept of identifying rational numbers worksheet.
