Rational vs. Irrational Numbers | Worksheet - Free Printable
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Step-by-step solution for: Rational vs. Irrational Numbers | Worksheet
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Step-by-step solution for: Rational vs. Irrational Numbers | Worksheet
Absolutely! Let’s solve the problem step by step.
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## 🎯 Task:
> Circle the rational numbers and draw a square around the irrational numbers in the given list.
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## ✔ First, let’s recall the definitions:
- Can be written as a fraction of two integers (e.g., `a/b`, where `a` and `b` are integers, `b ≠ 0`).
- In decimal form, it either terminates (like 0.5) or repeats (like 0.333...).
- Cannot be written as a fraction of two integers.
- In decimal form, it is non-terminating and non-repeating (like π = 3.14159265...).
---
## 🔍 Now, let’s go through each number one by one:
We’ll classify each and explain why.
---
→ It’s already a fraction of two integers.
→ 13 is not a perfect square → √13 is irrational.
→ Can be written as –19/2 → terminating decimal.
→ π is irrational; multiplying by –1 doesn’t make it rational.
→ √16 = 4 → integer → rational.
→ Integer → can be written as 1000/1.
→ Already a fraction.
→ Terminating decimal → 272/100.
→ Terminating → 46/10.
→ √(1/10) = 1/√10 → irrational because √10 is irrational.
→ Fraction of integers.
→ 30 is not a perfect square → √30 irrational → negative still irrational.
→ √9 = 3 → so –√9 = –3 → integer → rational.
→ Fraction.
→ Fraction.
→ √25 = 5 → integer.
→ Fraction.
→ π is irrational; subtracting a rational number (5) doesn’t make it rational.
→ Fraction.
→ √12 = 2√3 → irrational → adding rational (–1) to irrational → still irrational.
→ Integer.
→ 90 is not a perfect square → √90 = 3√10 → irrational.
→ Fraction.
→ 5 not a perfect square.
→ Can be written as 0/1.
→ Terminating decimal → 104/10.
→ Integer.
→ 1000 is not a perfect square → √1000 = 10√10 → irrational.
→ Terminating → 36/10.
→ Terminating → –212/10.
→ π is irrational → multiplying by 3 doesn’t make it rational.
→ √4 = 2 (rational), √5 = irrational → rational + irrational = irrational.
→ Fraction.
→ 14 not a perfect square.
→ √1 = 1 → –1 → integer → rational.
→ Not a perfect square.
→ Terminating → 17/100.
→ Fraction (simplifies to –1/7).
→ Terminating → 83/10.
→ √64 = 8 → integer.
→ Fraction.
→ Terminating → 136/100.
---
## ✔ Final Answer Summary:
¾, –9.5, √16, 1000, 1/12, 2.72, 4.6, 2/7, –√9, –1/5, 14/3, √25, 1/30, π – 5 ✘ *(Wait — correction!)*
→ Actually, π – 5 is IRRATIONAL — we made a mistake earlier!
Let me correct that:
✔ Rational Numbers (Circle these):
¾, –9.5, √16, 1000, 1/12, 2.72, 4.6, 2/7, –√9, –1/5, 14/3, √25, 1/30, –4/5, –7, √90 ✘ *(No! √90 is irrational!)*
Wait — let’s re-check carefully and list only the correct rationals:
---
## ✔ CORRECT LIST OF RATIONAL NUMBERS (Circle These):
1. ¾
2. –9.5
3. √16 → 4
4. 1000
5. 1/12
6. 2.72
7. 4.6
8. 2/7
9. –√9 → –3
10. –1/5
11. 14/3
12. √25 → 5
13. 1/30
14. –4/5
15. –7
16. 0
17. 10.4
18. 13
19. 3.6
20. –21.2
21. –15/100
22. –√1 → –1
23. 0.17
24. –2/14
25. 8.3
26. √64 → 8
27. 2/25
28. 1.36
---
## ⬛ IRRATIONAL NUMBERS (Draw Square Around These):
1. √13
2. –π
3. √0.1
4. –√30
5. √90
6. √5
7. √1000
8. 3π
9. √4 + √5 → 2 + √5 → irrational
10. √14
11. –√1 → Wait, this is –1 → RATIONAL!
→ So remove from irrational list
12. √2
13. π – 5 → irrational
14. –1 + √12 → irrational
---
## ✔ Final Correct Classification:
¾, –9.5, √16, 1000, 1/12, 2.72, 4.6, 2/7, –√9, –1/5, 14/3, √25, 1/30, –4/5, –7, 0, 10.4, 13, 3.6, –21.2, –15/100, –√1, 0.17, –2/14, 8.3, √64, 2/25, 1.36
*(Total: 28 numbers)*
---
√13, –π, √0.1, –√30, √90, √5, √1000, 3π, √4 + √5, √14, √2, π – 5, –1 + √12
*(Total: 13 numbers)*
---
## 📝 How to Mark on Worksheet:
- Use a circle (○) around each rational number.
- Use a square (□) around each irrational number.
---
## 💡 Pro Tip:
When in doubt, ask:
- Is it a perfect square root? → If yes, likely rational.
- Is it π or e or √non-perfect-square? → Likely irrational.
- Does the decimal terminate or repeat? → Rational.
- Otherwise → Probably irrational.
---
✔ You’re all set! Just mark the worksheet accordingly.
Let me know if you’d like a printable answer key or visual guide!
---
## 🎯 Task:
> Circle the rational numbers and draw a square around the irrational numbers in the given list.
---
## ✔ First, let’s recall the definitions:
➤ Rational Number:
- Can be written as a fraction of two integers (e.g., `a/b`, where `a` and `b` are integers, `b ≠ 0`).
- In decimal form, it either terminates (like 0.5) or repeats (like 0.333...).
➤ Irrational Number:
- Cannot be written as a fraction of two integers.
- In decimal form, it is non-terminating and non-repeating (like π = 3.14159265...).
---
## 🔍 Now, let’s go through each number one by one:
We’ll classify each and explain why.
---
1. ¾ → Rational ✔
→ It’s already a fraction of two integers.
2. √13 → Irrational ⬛
→ 13 is not a perfect square → √13 is irrational.
3. –9.5 → Rational ✔
→ Can be written as –19/2 → terminating decimal.
4. –π → Irrational ⬛
→ π is irrational; multiplying by –1 doesn’t make it rational.
5. √16 → Rational ✔
→ √16 = 4 → integer → rational.
6. 1000 → Rational ✔
→ Integer → can be written as 1000/1.
7. 1/12 → Rational ✔
→ Already a fraction.
8. 2.72 → Rational ✔
→ Terminating decimal → 272/100.
9. 4.6 → Rational ✔
→ Terminating → 46/10.
10. √0.1 → Irrational ⬛
→ √(1/10) = 1/√10 → irrational because √10 is irrational.
11. 2/7 → Rational ✔
→ Fraction of integers.
12. –√30 → Irrational ⬛
→ 30 is not a perfect square → √30 irrational → negative still irrational.
13. –√9 → Rational ✔
→ √9 = 3 → so –√9 = –3 → integer → rational.
14. –1/5 → Rational ✔
→ Fraction.
15. 14/3 → Rational ✔
→ Fraction.
16. √25 → Rational ✔
→ √25 = 5 → integer.
17. 1/30 → Rational ✔
→ Fraction.
18. π – 5 → Irrational ⬛
→ π is irrational; subtracting a rational number (5) doesn’t make it rational.
19. –4/5 → Rational ✔
→ Fraction.
20. –1 + √12 → Irrational ⬛
→ √12 = 2√3 → irrational → adding rational (–1) to irrational → still irrational.
21. –7 → Rational ✔
→ Integer.
22. √90 → Irrational ⬛
→ 90 is not a perfect square → √90 = 3√10 → irrational.
23. 3/11 → Rational ✔
→ Fraction.
24. √5 → Irrational ⬛
→ 5 not a perfect square.
25. 0 → Rational ✔
→ Can be written as 0/1.
26. 10.4 → Rational ✔
→ Terminating decimal → 104/10.
27. 13 → Rational ✔
→ Integer.
28. √1000 → Irrational ⬛
→ 1000 is not a perfect square → √1000 = 10√10 → irrational.
29. 3.6 → Rational ✔
→ Terminating → 36/10.
30. –21.2 → Rational ✔
→ Terminating → –212/10.
31. 3π → Irrational ⬛
→ π is irrational → multiplying by 3 doesn’t make it rational.
32. √4 + √5 → Irrational ⬛
→ √4 = 2 (rational), √5 = irrational → rational + irrational = irrational.
33. –15/100 → Rational ✔
→ Fraction.
34. √14 → Irrational ⬛
→ 14 not a perfect square.
35. –√1 → Rational ✔
→ √1 = 1 → –1 → integer → rational.
36. √2 → Irrational ⬛
→ Not a perfect square.
37. 0.17 → Rational ✔
→ Terminating → 17/100.
38. –2/14 → Rational ✔
→ Fraction (simplifies to –1/7).
39. 8.3 → Rational ✔
→ Terminating → 83/10.
40. √64 → Rational ✔
→ √64 = 8 → integer.
41. 2/25 → Rational ✔
→ Fraction.
42. 1.36 → Rational ✔
→ Terminating → 136/100.
---
## ✔ Final Answer Summary:
🔵 CIRCLE (Rational Numbers):
¾, –9.5, √16, 1000, 1/12, 2.72, 4.6, 2/7, –√9, –1/5, 14/3, √25, 1/30, π – 5 ✘ *(Wait — correction!)*
→ Actually, π – 5 is IRRATIONAL — we made a mistake earlier!
Let me correct that:
✔ Rational Numbers (Circle these):
¾, –9.5, √16, 1000, 1/12, 2.72, 4.6, 2/7, –√9, –1/5, 14/3, √25, 1/30, –4/5, –7, √90 ✘ *(No! √90 is irrational!)*
Wait — let’s re-check carefully and list only the correct rationals:
---
## ✔ CORRECT LIST OF RATIONAL NUMBERS (Circle These):
1. ¾
2. –9.5
3. √16 → 4
4. 1000
5. 1/12
6. 2.72
7. 4.6
8. 2/7
9. –√9 → –3
10. –1/5
11. 14/3
12. √25 → 5
13. 1/30
14. –4/5
15. –7
16. 0
17. 10.4
18. 13
19. 3.6
20. –21.2
21. –15/100
22. –√1 → –1
23. 0.17
24. –2/14
25. 8.3
26. √64 → 8
27. 2/25
28. 1.36
---
## ⬛ IRRATIONAL NUMBERS (Draw Square Around These):
1. √13
2. –π
3. √0.1
4. –√30
5. √90
6. √5
7. √1000
8. 3π
9. √4 + √5 → 2 + √5 → irrational
10. √14
11. –√1 → Wait, this is –1 → RATIONAL!
→ So remove from irrational list
12. √2
13. π – 5 → irrational
14. –1 + √12 → irrational
---
## ✔ Final Correct Classification:
🔵 RATIONAL (Circle):
¾, –9.5, √16, 1000, 1/12, 2.72, 4.6, 2/7, –√9, –1/5, 14/3, √25, 1/30, –4/5, –7, 0, 10.4, 13, 3.6, –21.2, –15/100, –√1, 0.17, –2/14, 8.3, √64, 2/25, 1.36
*(Total: 28 numbers)*
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⬛ IRRATIONAL (Square):
√13, –π, √0.1, –√30, √90, √5, √1000, 3π, √4 + √5, √14, √2, π – 5, –1 + √12
*(Total: 13 numbers)*
---
## 📝 How to Mark on Worksheet:
- Use a circle (○) around each rational number.
- Use a square (□) around each irrational number.
---
## 💡 Pro Tip:
When in doubt, ask:
- Is it a perfect square root? → If yes, likely rational.
- Is it π or e or √non-perfect-square? → Likely irrational.
- Does the decimal terminate or repeat? → Rational.
- Otherwise → Probably irrational.
---
✔ You’re all set! Just mark the worksheet accordingly.
Let me know if you’d like a printable answer key or visual guide!
Parent Tip: Review the logic above to help your child master the concept of rational or irrational worksheet.
