Number Systems Class 9 Maths Worksheet with Answers Chapter 1

The first chapter of the class 9 mathematics NCERT book. The worksheet below is divided into 4 sections: Basic, Standard, Advance, and HOTS. Attempt the questions only after you complete the chapter and revise the examples. Try each question first, and then check your answers using the Answer Key given below the questions. Below are top sections in the chapter –

• Real number types
• Rational vs irrational
• Decimal expansion rules
• Exponents and surds
• Rationalisation basics

Class 9 Maths Worksheet – Chapter 1: Number Systems

Basic

1. Classify each number as Natural / Whole / Integer / Rational / Irrational:
   • -7
   • 0
   • 5/8
   • √2
   • 0.375
   • 3.12112111211112…
2. Write six rational numbers between 3 and 4.
3. Find five rational numbers between 3/5 and 4/5.
4. Find the decimal expansion of:
   • 10/3
   • 7/8
   • 1/7

   Also write which are terminating and which are non-terminating recurring.

5. Express 0.6 in the form p/q (p, q are integers, q ≠ 0).
6. True/False (write one-line reason):
   • Every integer is a rational number.
   • Every rational number is an integer.
   • Every irrational number is a real number.

Standard

1. Convert to fraction (in simplest form):
   • 0.3333…
   • 1.272727…
2. Decide whether the decimal expansion is terminating or non-terminating recurring (no long division):
   • 13/40
   • 7/125
   • 17/60
   • 11/24
3. Write any two irrational numbers between 2 and 3.
4. Simplify:
   • (√11 – √7)(√11 + √7)
   • (5 + √7)(2 + √5)
5. Rationalise the denominator:
   • 1/√2
   • 1/(2 + √3)
6. Simplify using laws of exponents:
   • 17² · 17⁵
   • (5²)⁷
   • (7³ · 9³)

Advance

1. Show that 3.142678 is a rational number by writing it in p/q form.
2. Convert to fraction:
   • 0.47 (bar on 47)
   • 0.001 (bar on 001)
3. Simplify (write as “rational + irrational”):
   • (√5 + √7)²
   • (√3 + √7)(√3 – √7)
4. Rationalise the denominator:
   • 5/(3 – √5)
   • 1/(7 – √6)
5. Simplify using rational exponents:
   • 64^(1/2)
   • 32^(1/5)
   • 9^(3/2)
6. If r is rational and s is irrational, decide whether each is rational or irrational:
   • r + s
   • r – s
   • rs (where r ≠ 0)
   • r/s (where r ≠ 0)

HOTS

1. Prove that 0.9999… = 1 (use an algebra method).
2. Someone says: “π = c/d, so π is rational.”
   Explain why this is NOT a contradiction.
3. Without long division, predict the decimal expansions of:
   • 2/7
   • 3/7
   • 6/7

   Use the repeating block of 1/7.

4. What is the maximum possible number of digits in the repeating block of 1/17?
   (Answer + one-line reason. No need to do full division.)
5. Find three different irrational numbers between 5/7 and 9/11.
6. A number has a decimal expansion that is non-terminating non-recurring.
   What type of number is it? Write one example of such a decimal (you may invent it).

Answer Key

Basic – Answers

1. • -7: Integer, Rational Hint: -7 = -7/1.
   • 0: Whole, Integer, Rational Hint: 0 = 0/1.
   • 5/8: Rational Hint: p/q form.
   • √2: Irrational Hint: √2 cannot be written as p/q.
   • 0.375: Rational Hint: 0.375 = 375/1000 = 3/8.
   • 3.12112111211112…: Irrational Hint: non-terminating, non-recurring.
2. Example answers: 3.1, 3.2, 3.25, 3.5, 3.75, 3.9
   Hint: Write 3 = 30/10 and 4 = 40/10, choose any 6 fractions between.
3. Example answers: 31/50, 16/25, 13/20, 7/10, 19/25
   Hint: Convert to common denominator: 3/5 = 30/50, 4/5 = 40/50.
4. • 10/3 = 3.3333… (non-terminating recurring) Hint: 3 repeats.
   • 7/8 = 0.875 (terminating) Hint: denominator has only 2s.
   • 1/7 = 0.142857142857… (non-terminating recurring) Hint: 142857 repeats.
5. 0.6 = 6/10 = 3/5
   Hint: Multiply by 10 and simplify.
6. • True Hint: m = m/1.
   • False Hint: 3/5 is rational but not integer.
   • True Hint: real numbers = rationals ∪ irrationals.

Standard – Answers

1. • 0.3333… = 1/3 Hint: Let x=0.333…, then 10x=3.333…, subtract.
   • 1.272727… = 14/11 Hint: Let x=1.2727…, then 100x=127.2727…, subtract.
2. • 13/40: terminating Hint: 40 = 2³·5.
   • 7/125: terminating Hint: 125 = 5³.
   • 17/60: non-terminating recurring Hint: 60 has factor 3.
   • 11/24: non-terminating recurring Hint: 24 has factor 3.

   Rule Hint: In lowest form, denominator must be only 2ᵃ5ᵇ for terminating.

3. Example answers: √5, √6
   Hint: Any √n where n is not a perfect square and lies between 4 and 9 will work.
4. • (√11 – √7)(√11 + √7) = 11 – 7 = 4 Hint: (a-b)(a+b)=a²-b².
   • (5 + √7)(2 + √5) = 10 + 5√5 + 2√7 + √35 Hint: Use distributive law.
5. • 1/√2 = √2/2 Hint: Multiply by √2/√2.
   • 1/(2 + √3) = (2 – √3)/(4 – 3) = 2 – √3 Hint: Multiply by conjugate.
6. • 17² · 17⁵ = 17⁷ Hint: aᵐ·aⁿ = aᵐ⁺ⁿ.
   • (5²)⁷ = 5¹⁴ Hint: (aᵐ)ⁿ = aᵐⁿ.
   • (7³ · 9³) = (63)³ Hint: aᵖbᵖ = (ab)ᵖ.

Advance – Answers

1. 3.142678 = 3142678/1000000 = 1571339/500000
   Hint: Move decimal 6 places.
2. • 0.47 (recurring) = 47/99 Hint: x=0.4747…, 100x=47.47…, subtract.
   • 0.001 (recurring) = 1/999 Hint: x=0.001001…, 1000x=1.001…, subtract.
3. • (√5 + √7)² = 5 + 7 + 2√35 = 12 + 2√35 Hint: (a+b)² = a²+2ab+b².
   • (√3 + √7)(√3 – √7) = 3 – 7 = -4 Hint: (a+b)(a-b)=a²-b².
4. • 5/(3 – √5) = 5(3 + √5)/(9 – 5) = (15 + 5√5)/4 Hint: Use conjugate.
   • 1/(7 – √6) = (7 + √6)/(49 – 6) = (7 + √6)/43 Hint: Use conjugate.
5. • 64^(1/2) = √64 = 8 Hint: a^(1/2)=√a.
   • 32^(1/5) = 2 Hint: 2⁵=32.
   • 9^(3/2) = (√9)³ = 3³ = 27 Hint: a^(m/n)=(ⁿ√a)ᵐ.
6. • r + s: irrational Hint: rational ± irrational is irrational.
   • r – s: irrational Hint: same reason.
   • rs: irrational (r ≠ 0) Hint: non-zero rational × irrational is irrational.
   • r/s: irrational (r ≠ 0) Hint: non-zero rational ÷ irrational is irrational.

HOTS – Answers

1. 0.9999… = 1
   Hint: Let x=0.999…, then 10x=9.999…, subtract: 9x=9 ⇒ x=1.
2. No contradiction
   Hint: π = c/d is true for a specific circle, but c and d are real measurements, not necessarily integers. Rational means “ratio of integers”.
3. Using 1/7 = 0.142857 repeating:
   • 2/7 = 0.285714 repeating
   • 3/7 = 0.428571 repeating
   • 6/7 = 0.857142 repeating

   Hint: Multiply 0.142857 by 2, 3, 6 and observe cyclic shift.

4. Maximum repeating block length is 16
   Hint: For 1/q, repeating block length is less than q, so at most (17−1).
5. Example answers:
   • √0.6
   • √0.65
   • √0.7

   Hint: 5/7 ≈ 0.714…, 9/11 ≈ 0.818…, choose any non-square decimals between and take square roots.

6. It is irrational
   Example: 0.101001000100001…
   Hint: Non-terminating + non-recurring ⇒ irrational.

Worksheet for Other Class 9 Maths Chapters

• Polynomials Class 9 Maths Worksheet Chapter 2
• Linear Equations in Two Variables Class 9 Maths Worksheet Chapter 4
• Introduction to Euclid’s Geometry Class 9 Maths Worksheet Chapter 5
• Heron’s Formula Class 9 Maths Worksheet Chapter 10
• Surface Areas and Volumes Class 9 Maths Worksheet Chapter 11
• Statistics Class 9 Maths Worksheet Chapter 12