Chapter 1: Number Systems – Important Questions with Solutions
1. Is 0 a rational number? Express it in p/q form.
Solution:
Yes, 0 is rational because it can be expressed as 0/1, where p = 0 and q = 1 (q ≠ 0).
2. Find six rational numbers between 3 and 4.
Solution:
We can find rationals by dividing the interval into equal parts:
3.1, 3.2, 3.3, 3.4, 3.5, 3.6 (all rational).
3. Find five rational numbers between 3/5 and 4/5.
Solution:
Convert to decimals: 3/5 = 0.6 and 4/5 = 0.8
Five rationals: 0.62, 0.65, 0.7, 0.75, 0.77 (or fractions: 31/50, 13/20, 7/10, 15/20, 77/100)
4. True or False: Natural numbers include zero.
Solution:
False. Natural numbers start from 1. Zero is a whole number but not natural.
5. Is √4 irrational? Explain.
Solution:
No, √4 = 2, which is rational because it can be expressed as 2/1.
6. Represent √5 on the number line.
Solution:
- Draw line segment of length 2 units (AB=2).
- At B, draw a perpendicular of 1 unit (BC=1).
- Join AC and draw the hypotenuse (AC) which equals √5.
- Use AC as radius, draw an arc from A on the number line to mark √5.
7. Classify 0.75, 0.666…, and √3.
Solution:
- 0.75: Terminating decimal (rational)
- 0.666…: Non-terminating recurring decimal (rational)
- √3: Non-terminating non-recurring decimal (irrational)
8. Show why a rational number’s denominator in simplest form can only have 2 and/or 5 as prime factors if its decimal is terminating.
Solution:
In simplest form, denominator q’s prime factors determine decimal nature:
- If q contains only 2’s and/or 5’s, decimal terminates.
- Else, decimal repeats.
9. Find three irrational numbers between 1 and 2.
Solution:
Examples: √2 ≈ 1.414, √3 ≈ 1.732, 1 + π/4 ≈ 1.785
10. Rationalize the denominator of 1/√3.
Solution:
Multiply numerator and denominator by √3:
(1/√3) × (√3/√3) = √3 / 3
11. Simplify (√5 + √2)².
Solution:
(√5)² + 2√5√2 + (√2)² = 5 + 2√10 + 2 = 7 + 2√10
12. Is sum of rational 2/3 and irrational √2 rational or irrational?
Solution:
Irrational. Rational + Irrational = Irrational.
13. Simplify: (81)^(-1/2).
Solution:
81 = 9², so (81)^(-1/2) = (9²)^(-1/2) = 9^{-1} = 1/9
14. Write √7 as a decimal up to 4 decimal places.
Solution:
√7 ≈ 2.6457
15. Find a number x such that x² is irrational but x⁴ is rational.
Solution:
Let x = √(√2) = 2^{1/4}
Then x² = √2 (irrational), x⁴ = 2 (rational)
16. Find five rational numbers between 0 and 1.
Solution:
0.1, 0.25, 0.5, 0.75, 0.9 (all rational).
17. Express 0.363636… as a rational number.
Solution:
Let x = 0.363636…
Then 100x = 36.363636…
Subtract: 100x – x = 36.3636… – 0.3636… = 36
=> 99x = 36
=> x = 36/99 = 4/11
18. Simplify: (16)^{3/4}.
Solution:
16 = 2^4
(2^4)^{3/4} = 2^{4 × 3/4} = 2^3 = 8
19. Find the product of (3 + √2)(3 − √2).
Solution:
(3)^2 − (√2)^2 = 9 − 2 = 7
20. Explain why π is irrational.
Solution:
π cannot be expressed as a ratio of two integers; its decimal expansion is infinite, non-repeating. This is proven using calculus and transcendental number theory.
21. Express 0.125 as a fraction.
Solution:
0.125 = 125/1000 = 1/8
22. Find the sixth root of 64.
Solution:
64 = 2^6
Sixth root = 2
23. Is −7 a rational number?
Solution:
Yes, −7 = −7/1 (ratio of integers).
24. Simplify (√3 + √2)(√3 − √2).
Solution:
= (√3)^2 − (√2)^2 = 3 − 2 = 1
25. Show that √2 is irrational.
Solution:
Assume √2 = p/q, with p, q integers, no common factors.
Then 2q² = p² → p² even → p even → p=2k.
Substitute back, leads to q even → both even contradicting no common factors. Hence √2 irrational.
26. Find four rational numbers between 1/4 and 1/2.
Solution:
Convert to decimals: 1/4=0.25, 1/2=0.5
Numbers: 0.3, 0.35, 0.4, 0.45
27. Write √50 in simplified surd form.
Solution:
√50 = √(25 × 2) = 5√2
28. Find (2/3)^0.
Solution:
Any nonzero number to power 0 is 1.
29. Find the decimal form of 7/11.
Solution:
7 ÷ 11 = 0.636363… (repeating)
30. Express 0.1̅6 (where 6 repeats) as a fraction.
Solution:
Let x = 0.166666…
10x = 1.66666…
Subtract: 10x − x = 1.6666 − 0.1666 = 1.5
=> 9x = 1.5
=> x = 1.5/9 = 1/6
31. Simplify (√3 + 1)^2.
Solution:
= (√3)^2 + 2×√3×1 + 1² = 3 + 2√3 + 1 = 4 + 2√3
32. Write −0.333… as a fraction.
Solution:
−0.333… = −1/3
33. Find five rational numbers between 2 and 3.
Solution:
2.2, 2.4, 2.6, 2.8, 2.9
34. Express 1.272727… as a fraction.
Solution:
Let x = 1.272727…
100x = 127.272727…
Subtract: 100x − x = 127.2727… − 1.2727… = 126
=> 99x = 126
=> x = 126/99 = 14/11
35. Simplify (16)^{-3/4}.
Solution:
16 = 2^4
(2^4)^{-3/4} = 2^{-3} = 1/8
36. Write √18 in simplest form.
Solution:
√18 = √(9 × 2) = 3√2
37. Find the cube root of 125.
Solution:
Cube root of 125 = 5
38. Express 0.0833… (3 repeats) as a fraction.
Solution:
Let x = 0.083333…
1000x = 83.3333…
Subtract: 1000x − 10x = 83.3333… − 0.8333… = 82.5
=> 990x = 82.5
=> x = 82.5 / 990 = 11/132
39. Simplify (√7 − √5)(√7 + √5).
Solution:
= 7 − 5 = 2
40. Show that 0.101001000100001… is irrational.
Solution:
Its decimal expansion is non-terminating and non-repeating, so it is irrational.
41. Find five rational numbers between 5/6 and 1.
Solution:
5/6=0.8333…
Numbers: 0.85, 0.87, 0.9, 0.95, 0.99
42. Simplify (3/4)^2 × (4/3)^2.
Solution:
= (3/4 × 4/3)^2 = 1^2 = 1
43. Rationalize denominator of 1/(2 + √3).
Solution:
Multiply numerator and denominator by (2 − √3):
[1 × (2 − √3)] / [(2 + √3)(2 − √3)] = (2 − √3) / (4 − 3) = 2 − √3
44. Is 0.123456789101112… rational?
Solution:
No, it is non-terminating and non-repeating, so irrational.
45. Simplify (√2)^6.
Solution:
= (√2)^6 = (2^{1/2})^6 = 2^{3} = 8
46. Express 0.45̅ (5 repeats) as a fraction.
Solution:
Let x = 0.4555…
10x = 4.555…
100x = 45.555…
Subtract: 100x − 10x = 45.555… − 4.555… = 41
=> 90x = 41
=> x = 41/90
47. Find five irrational numbers between 1 and 3.
Solution:
√2 ≈1.414, √5 ≈2.236, π ≈3.141 (exclude 3.141 since >3)
So pick √3 ≈1.732, √7 ≈2.645, √8 ≈2.828
48. Is the product of two irrational numbers always irrational?
Solution:
No. For example, (√2) × (√2) = 2 (rational).
49. Write 1.08̅3 (8 repeats) as a fraction.
Solution:
Let x = 1.088888…
10x = 10.88888…
100x = 108.8888…
Subtract: 100x − 10x = 108.888… − 10.888… = 98
=> 90x = 98
=> x = 98/90 = 49/45
50. Find (√5 − √3)^2.
Solution:
= (√5)^2 − 2√5√3 + (√3)^2 = 5 − 2√15 + 3 = 8 − 2√15