Class 9 Maths - Number Systems NCERT Solutions

Chapter 1: Number Systems (NCERT Solutions)

Exercise 1.1

Q1. Is zero a rational number? Can you write it in the form p/q, where p and q are integers and q ≠ 0?

Yes, zero is a rational number. It can be written as 0 = 0/1 = 0/2 = 0/(-5) etc., where p = 0 (integer) and q is any non-zero integer.

Q2. Find six rational numbers between 3 and 4.

Multiply 3 and 4 by 7: 3 = 21/7, 4 = 28/7.
Six rationals: 22/7, 23/7, 24/7, 25/7, 26/7, 27/7.

Q3. Find five rational numbers between 3/5 and 4/5.

3/5 = 18/30, 4/5 = 24/30.
Five rationals: 19/30, 20/30, 21/30, 22/30, 23/30.

Q4. State whether the following are true or false. Give reasons.
(i) Every natural number is a whole number.
(ii) Every integer is a whole number.
(iii) Every rational number is a whole number.

(i) True. N = {1,2,3,...} ⊂ W = {0,1,2,...}.

(ii) False. Negative integers (e.g., -1, -2) are integers but not whole numbers.

(iii) False. 1/2 is rational but not a whole number.

Exercise 1.2

Q1. State whether the following are true or false. Justify.
(i) Every irrational is a real number.
(ii) Every point on number line = √m (m natural).
(iii) Every real number is irrational.

(i) True. Irrational numbers are a subset of real numbers.

(ii) False. Negative real numbers cannot be expressed as √m.

(iii) False. Rational numbers such as 1, 2, 3/4 are also real numbers.

Q2. Are the square roots of all positive integers irrational? Give an example.

No. √4 = 2 and √9 = 3 are rational. Only square roots of non-perfect-square integers are irrational.

Q3. Show how √5 can be represented on the number line.

1. On number line, mark O at 0 and A at 2 (OA = 2 units).
2. At A, draw AB ⊥ OA where AB = 1 unit.
3. OB = √(4 + 1) = √5 (by Pythagoras).
4. With O as centre and OB as radius, draw an arc meeting the number line at C.
5. OC = √5.

Exercise 1.3

Q1. Write in decimal form and state kind of expansion:
(i) 36/100 (ii) 1/11 (iii) 4⅛ (iv) 3/13 (v) 2/11 (vi) 329/400

(i) 0.36 — Terminating.

(ii) 0.09̅ — Non-terminating repeating.

(iii) 4.125 — Terminating.

(iv) 0.230769̅ — Non-terminating repeating.

(v) 0.18̅ — Non-terminating repeating.

(vi) 0.8225 — Terminating.

Q3. Express in the form p/q: (i) 0.6̅ (ii) 0.47̅ (iii) 0.001̅

(i) Let x = 0.666...; 10x = 6.666...; 9x = 6; x = 2/3.

(ii) Let x = 0.4777...; 10x = 4.777..., 100x = 47.777...; 90x = 43; x = 43/90.

(iii) Let x = 0.001001...; 1000x = 1.001...; 999x = 1; x = 1/999.

Q4. Express 0.99999... in p/q form. Are you surprised?

x = 0.9999...; 10x = 9.999...; 9x = 9; x = 1. Yes, 0.9999... = 1 exactly — they represent the same real number.

Q9. Classify as rational or irrational:
(i) √23 (ii) √225 (iii) 0.3796 (iv) 7.478478... (v) 1.101001000100001...

(i) Irrational (23 not a perfect square).

(ii) Rational (√225 = 15).

(iii) Rational (terminating decimal).

(iv) Rational (non-terminating repeating: 7.478̅).

(v) Irrational (non-terminating, non-repeating).

Exercise 1.5

Q1. Classify as rational or irrational:
(i) 2 - √5 (ii) (3+√23) - √23 (iii) 2√7/7√7 (iv) 1/√2 (v) 2π

(i) Irrational. (ii) = 3 ⇒ Rational. (iii) = 2/7 ⇒ Rational. (iv) Irrational. (v) Irrational.

Q2. Simplify:
(i) (3+√3)(2+√2) (ii) (3+√3)(3-√3) (iii) (√5+√2)² (iv) (√5-√2)(√5+√2)

(i) 6 + 3√2 + 2√3 + √6.

(ii) 9 - 3 = 6.

(iii) 5 + 2√10 + 2 = 7 + 2√10.

(iv) 5 - 2 = 3.

Q4. Rationalise the denominators:
(i) 1/√7 (ii) 1/(√7-√6) (iii) 1/(√5+√2) (iv) 1/(√7-2)

(i) Multiply by √7/√7: √7/7.

(ii) Multiply by (√7+√6)/(√7+√6): √7 + √6.

(iii) Multiply by (√5-√2)/(√5-√2): (√5-√2)/3.

(iv) Multiply by (√7+2)/(√7+2): (√7+2)/3.

Exercise 1.6 — Laws of Exponents

Q1. Find: (i) 64^1/2 (ii) 32^1/5 (iii) 125^1/3

(i) √64 = 8. (ii) ⁵√32 = 2. (iii) ³√125 = 5.

Q2. Find: (i) 9^3/2 (ii) 32^2/5 (iii) 16^3/4 (iv) 125^-1/3

(i) (9^1/2)³ = 3³ = 27.

(ii) (32^1/5)² = 2² = 4.

(iii) (16^1/4)³ = 2³ = 8.

(iv) 1/125^1/3 = 1/5 = 0.2.

Q3. Simplify: (i) 2^2/3×2^1/5 (ii) (1/3³)⁷ (iii) 11^1/2/11^1/4 (iv) 7^1/2×8^1/2

(i) 2^10/15+3/15 = 2^13/15.

(ii) 3^-21.

(iii) 11^1/2-1/4 = 11^1/4.

(iv) (7×8)^1/2 = √56 = 2√14.

Class 9 Maths - Number Systems Practice

Chapter 1: Number Systems (HOT / Practice Questions)

Higher Order Thinking / RD Sharma

Q1. Prove that √3 is irrational.

Proof by contradiction: Assume √3 = p/q (lowest terms, q ≠ 0).
Squaring: 3 = p²/q² ⇒ p² = 3q².
⇒ 3 divides p² ⇒ 3 divides p. Let p = 3m.
(3m)² = 3q² ⇒ 9m² = 3q² ⇒ q² = 3m² ⇒ 3 divides q.
Both p and q are divisible by 3 — contradiction. Hence √3 is irrational.

Q2. Prove that √5 + √3 is irrational.

Suppose √5 + √3 = r (rational).
⇒ √5 = r - √3.
Squaring: 5 = r² - 2r√3 + 3 ⇒ 2r√3 = r² - 2 ⇒ √3 = (r² - 2)/2r.
Since r is rational, (r²-2)/2r is rational, but √3 is irrational. Contradiction.
Hence √5 + √3 is irrational.

Q3. Find the value of: (6 + √5)(6 - √5).

Using (a+b)(a-b) = a² - b²:
= 36 - 5 = 31.

Q4. Rationalise: 4/(3√5 + √3).

Multiply by (3√5 - √3)/(3√5 - √3):
= 4(3√5 - √3) / (45 - 3) = 4(3√5 - √3)/42
= 2(3√5 - √3)/21.

Q5. If x = 5 - 2√6, find √x + 1/√x.

x = 5 - 2√6 = 3 + 2 - 2√6 = (√3 - √2)².
√x = √3 - √2 (since √3 > √2).
1/√x = 1/(√3 - √2) = (√3 + √2)/(3 - 2) = √3 + √2.
√x + 1/√x = (√3 - √2) + (√3 + √2) = 2√3.

Q6. Write 3 rational numbers between √2 and √3.

√2 ≈ 1.414 and √3 ≈ 1.732. Any rational between them works.
Three examples: 1.5, 1.6, 1.7 (these are rational and lie between √2 and √3).

Q7. Simplify: (2 + √3)/(2 - √3) + (2 - √3)/(2 + √3).

First term: (2+√3)² / (4-3) = (4 + 4√3 + 3)/1 = 7 + 4√3.
Second term: (2-√3)² / (4-3) = (4 - 4√3 + 3)/1 = 7 - 4√3.
Sum = (7 + 4√3) + (7 - 4√3) = 14.

Q8. Find the four rational numbers between 1/5 and 1/4.

1/5 = 4/20 and 1/4 = 5/20. Need more range: 1/5 = 40/200 and 1/4 = 50/200.
Four rationals: 41/200, 43/200, 45/200, 47/200.

Q9. Which is greater: 7^1/2 or 5^1/3?

LCM of 2 and 3 = 6.
7^1/2 = 7^3/6 = (7³)^1/6 = 343^1/6.
5^1/3 = 5^2/6 = (5²)^1/6 = 25^1/6.
Since 343 > 25, 7^1/2 > 5^1/3.

Q10. Evaluate: [5^1/3 × 25^1/3] / 125^1/3.

= [5^1/3 × (5²)^1/3] / (5³)^1/3
= [5^1/3 × 5^2/3] / 5
= 5¹ / 5 = 1.

Q11. If 2^x = 3^y = 6^z, show that z = xy/(x+y).

Let 2^x = 3^y = 6^z = k.
2 = k^1/x, 3 = k^1/y, 6 = k^1/z.
Since 6 = 2 × 3: k^1/z = k^1/x × k^1/y = k^{1/x +
1/y}.
1/z = 1/x + 1/y = (x+y)/(xy).
⇒ z = xy/(x+y). (Proved)

Q12. Simplify: (81)^3/4 ÷ (81)^1/4.

= 81^{3/4 - 1/4} = 81^2/4 = 81^1/2 = √81 = 9.

Class 9 Maths - Number Systems Summary

Chapter 1: Number Systems (Concepts & Formulas)

1. Number Hierarchy

N ⊂ W ⊂ Z ⊂ Q ⊂ R

N = {1, 2, 3, ...} — Natural numbers
W = {0, 1, 2, ...} — Whole numbers
Z = {..., -2, -1, 0, 1, 2, ...} — Integers
Q = {p/q : p, q ∈ Z, q ≠ 0} — Rational numbers
Irrational = Non-terminating, non-repeating (e.g., √2, π)
R = Q ∪ Irrationals — Real numbers

2. Decimal Expansions

Type	Category	Example
Terminating	Rational	0.375, 2.5
Non-term. Repeating	Rational	0.333..., 1.285714̅
Non-term. Non-Repeating	Irrational	√2 = 1.41421..., π

Terminating condition: p/q terminates iff q = 2^m × 5ⁿ.

3. Irrational Number Operations

√a × √b = √(ab)
√a / √b = √(a/b)
(√a + √b)(√a - √b) = a - b
(√a + √b)² = a + 2√(ab) + b

4. Rationalisation of Denominator

1/√a ⇒ multiply by √a/√a ⇒ √a/a
1/(a+√b) ⇒ multiply by (a-√b)/(a-√b)
1/(√a+√b) ⇒ multiply by (√a-√b)/(√a-√b)

5. Laws of Exponents

a^m × aⁿ = a^m+n
a^m ÷ aⁿ = a^m-n
(a^m)ⁿ = a^mn
a^p/q = (^q√a)^p
a⁰ = 1 (a ≠ 0)
a^-n = 1/aⁿ

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