Number Systems Class 9 – Study Material

Chapter 1: Number Systems

Introduction: The number system is the foundation of mathematics. It starts with natural numbers and extends to integers, rational numbers, and irrational numbers. Together, rational and irrational numbers make up the set of real numbers. This chapter will help you understand different kinds of numbers, their properties, and how they are represented and used in calculations.

Study Notes

Rational Numbers as Recurring/Terminating Decimals

Rational numbers are numbers that can be expressed in the form \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \ne 0\). When expressed as decimals, they can either terminate or repeat. If the denominator in the lowest form has only the prime factors 2 and/or 5, the decimal terminates. Otherwise, it repeats. For example, \(\frac{1}{4} = 0.25\) (terminating) and \(\frac{2}{3} = 0.666\ldots\) (repeating).

Solved Examples

• \(\frac{5}{8} = 0.625\) (Terminating)
• \(\frac{4}{11} = 0.363636\ldots\) (Repeating)
• \(\frac{6}{15} = 0.4\) (Terminating, after simplification)
• \(\frac{7}{6} = 1.1666\ldots\) (Repeating)

Operations on Real Numbers

Real numbers include both rational and irrational numbers. We can perform the four basic operations (addition, subtraction, multiplication, and division) on real numbers. These operations follow certain rules and properties such as closure, commutativity, associativity, and distributivity. For example, adding two real numbers always gives a real number (closure).

Solved Examples

• \(\sqrt{2} + \sqrt{3} = \sqrt{2} + \sqrt{3}\) (Cannot be simplified further)
• \((\sqrt{5})^2 = 5\)
• \(2 + 3.5 = 5.5\)
• \(\frac{1}{2} \times \sqrt{2} = \frac{\sqrt{2}}{2}\)

1. Problem: Identify which of these are real numbers: \(\sqrt{-1}, 0.5, \sqrt{7}, \frac{1}{0}\)
   Solution:
   • \(\sqrt{-1}\) is not real (imaginary number)
   • \(0.5\) is real (rational)
   • \(\sqrt{7}\) is real (irrational)
   • \(\frac{1}{0}\) is undefined (not real)
2. Problem: Find two irrational numbers between 0.5 and 0.6
   Solution: \(0.5010010001\ldots\) and \(0.5305300530005\ldots\) (non-terminating, non-repeating decimals)
3. Problem: Represent \(\sqrt{3}\) on the number line
   Solution:
   • Draw right triangle with base 1, height \(\sqrt{2}\) (constructed previously)
   • Hypotenuse will be \(\sqrt{3}\)
   • Use compass to mark this length on number line
4. Problem: Simplify \((\sqrt{5} + \sqrt{3})(\sqrt{5} – \sqrt{3})\)
   Solution:
   \[ \begin{align*} &= (\sqrt{5})^2 – (\sqrt{3})^2 \\ &= 5 – 3 \\ &= 2 \end{align*} \]

Irrational Numbers and Their Representation

Irrational numbers are numbers that cannot be written in the form \(\frac{p}{q}\). Their decimal expansions are non-terminating and non-repeating. Examples include \(\sqrt{2}\), \(\sqrt{3}\), and \(\pi\). These numbers cannot be exactly located on a number line, but can be approximately represented using geometric methods.

Solved Examples

• \(\sqrt{2} = 1.4142\ldots\) (non-terminating, non-repeating)
• \(\sqrt{5} = 2.2360\ldots\)
• \(\pi = 3.14159\ldots\)
• \(\sqrt{7} = 2.6457\ldots\)

Rationalization of Real Numbers

Rationalization is the process of removing irrationality from the denominator of a fraction. This is done by multiplying both numerator and denominator by the conjugate of the denominator if it contains a surd. This is useful in simplifying expressions.

Solved Examples

• \(\frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\)
• \(\frac{2}{1 + \sqrt{2}} = \frac{2(1 – \sqrt{2})}{(1 + \sqrt{2})(1 – \sqrt{2})} = \frac{2(1 – \sqrt{2})}{-1}\)
• \(\frac{4}{2 – \sqrt{5}} = \frac{4(2 + \sqrt{5})}{(2 – \sqrt{5})(2 + \sqrt{5})} = \frac{4(2 + \sqrt{5})}{-1}\)
• \(\frac{3}{\sqrt{2} + \sqrt{3}} = \frac{3(\sqrt{2} – \sqrt{3})}{(\sqrt{2} + \sqrt{3})(\sqrt{2} – \sqrt{3})} = \frac{3(\sqrt{2} – \sqrt{3})}{-1}\)

Laws of Exponents for Real Numbers

Exponents or powers are a way to express repeated multiplication. Real numbers follow certain exponent rules which help simplify calculations. These include:

• \(a^m \cdot a^n = a^{m+n}\)
• \(\frac{a^m}{a^n} = a^{m-n}\)
• \((a^m)^n = a^{mn}\)
• \(a^0 = 1\) (if \(a \ne 0\))
• \(a^{-m} = \frac{1}{a^m}\)

Solved Examples

• \(2^3 \cdot 2^2 = 2^5 = 32\)
• \(\frac{5^4}{5^2} = 5^{2} = 25\)
• \((3^2)^3 = 3^6 = 729\)
• \(4^{-1} = \frac{1}{4}\)

Solved Examples

Example 1 (NCERT): Represent \(\sqrt{2}\) on the number line

Solution:

1. Draw a number line and mark 0 and 1.
2. Draw a perpendicular of 1 unit at point 1.
3. Join (0,0) to (1,1). The length = \(\sqrt{2}\).
4. Use a compass to mark this length on the number line from 0.

Example 2: Rationalize \(\frac{1}{\sqrt{3}}\)

Additional Solved Problems

• Convert \(0.777\ldots\) into a rational number.
• Rationalize \(\frac{5}{2 + \sqrt{3}}\)
• Simplify: \((\sqrt{5} + \sqrt{2})^2\)

Practice Problems

Unsolved Questions

1. Convert \(0.272727\ldots\) into a rational number.
2. Simplify: \((2 + \sqrt{3})(2 – \sqrt{3})\)
3. Is \(\pi\) a rational number?
4. Represent \(\sqrt{5}\) on the number line.
5. Express \(0.16\overline{6}\) as a fraction.

MCQ Test – Chapter 1: Number Systems

1. Which of the following is an irrational number?
   (A) \(\frac{3}{7}\) (B) \(0.666\ldots\) (C) \(\sqrt{2}\) (D) \(2.5\)
   Answer: (C)
2. Which is not a real number?
   (A) \(0\) (B) \(\sqrt{4}\) (C) \(-5\) (D) \(\sqrt{-1}\)
   Answer: (D)
3. Rationalization of \(\frac{1}{\sqrt{2}}\) is:
   (A) \(\sqrt{2}\) (B) \(\frac{\sqrt{2}}{2}\) (C) \(\frac{2}{\sqrt{2}}\) (D) None
   Answer: (B)
4. Decimal expansion of \(\frac{7}{8}\) is:
   (A) Terminating (B) Non-terminating (C) Irrational (D) None
   Answer: (A)
5. \((\sqrt{5} + \sqrt{2})^2\) equals:
   (A) \(7\) (B) \(5 + 2\) (C) \(7 + 2\sqrt{10}\) (D) \(25\)
   Answer: (C)

Multiple Choice Questions

1. Which of the following numbers is irrational?
   (A) \(\dfrac{7}{8}\) (B) \(0.25\) (C) \(\sqrt{2}\) (D) \(3\)
   Answer: (C)
2. The decimal expansion of \(\dfrac{1}{3}\) is:
   (A) Terminating (B) Non-terminating and repeating (C) Non-terminating and non-repeating (D) Integer
   Answer: (B)
3. The product \((\sqrt{3} + \sqrt{2})(\sqrt{3} – \sqrt{2})\) is:
   (A) \(1\) (B) \(5\) (C) \(6\) (D) \(\sqrt{6}\)
   Answer: (B)
4. Rationalizing the denominator of \(\dfrac{1}{\sqrt{5}}\) gives:
   (A) \(\dfrac{\sqrt{5}}{5}\) (B) \(\sqrt{5}\) (C) \(5\) (D) \(\dfrac{1}{5}\)
   Answer: (A)
5. The number \(\sqrt{49}\) is:
   (A) Rational (B) Irrational (C) Complex (D) None
   Answer: (A)
6. The decimal expansion of \(\dfrac{13}{20}\) is:
   (A) Terminating (B) Non-terminating and repeating (C) Irrational (D) None
   Answer: (A)
7. Which of the following is not a real number?
   (A) \(\pi\) (B) \(-\sqrt{2}\) (C) \(\sqrt{-1}\) (D) \(0\)
   Answer: (C)
8. The number \(\sqrt{16} + \sqrt{25}\) is:
   (A) Irrational (B) \(9\) (C) \(41\) (D) Undefined
   Answer: (B)
9. Which of the following is a rational number?
   (A) \(\sqrt{7}\) (B) \(\dfrac{3}{5}\) (C) \(\pi\) (D) \(\sqrt{2}\)
   Answer: (B)
10. \((\sqrt{2} + 1)^2\) equals:
    (A) \(3\) (B) \(2 + 2\sqrt{2}\) (C) \(2\sqrt{2}\) (D) \(1 + \sqrt{2}\)
    Answer: (B)
11. Which of the following is a terminating decimal?
    (A) \(\dfrac{5}{6}\) (B) \(\dfrac{1}{7}\) (C) \(\dfrac{1}{4}\) (D) \(\dfrac{2}{9}\)
    Answer: (C)
12. \(\dfrac{1}{\sqrt{2} + 1}\) rationalizes to:
    (A) \(\dfrac{\sqrt{2} – 1}{2 – 1}\) (B) \(\sqrt{2} – 1\) (C) \(1 – \sqrt{2}\) (D) \(\sqrt{2} + 1\)
    Answer: (B)
13. Which law of exponent is used in: \(a^m \cdot a^n = a^{m+n}\)?
    (A) Division law (B) Power law (C) Product law (D) Zero exponent law
    Answer: (C)
14. \(2^3 \times 2^4\) equals:
    (A) \(2^7\) (B) \(16\) (C) \(2^{12}\) (D) \(8\)
    Answer: (A)
15. \((\sqrt{3} + 2)^2 =\)
    (A) \(3 + 4\sqrt{3} + 4\) (B) \(7 + 2\sqrt{3}\) (C) \(9\) (D) \(4 + \sqrt{3}\)
    Answer: (A)
16. The number \(0.1010010001\ldots\) is:
    (A) Rational (B) Irrational (C) Terminating (D) Repeating
    Answer: (B)
17. Which of the following is the correct value of \(a^0\) (where \(a \ne 0\))?
    (A) \(0\) (B) \(1\) (C) \(a\) (D) Undefined
    Answer: (B)
18. If \(x = \sqrt{3}\) and \(y = \sqrt{2}\), then \((x – y)(x + y)\) is:
    (A) \(\sqrt{6}\) (B) \(5\) (C) \(1\) (D) \(3 – 2 = 1\)
    Answer: (B)
19. Which of these numbers has a non-terminating, repeating decimal expansion?
    (A) \(\dfrac{3}{4}\) (B) \(\dfrac{1}{9}\) (C) \(\dfrac{1}{2}\) (D) \(\dfrac{5}{10}\)
    Answer: (B)
20. The number \(\sqrt{2}\sqrt{8}\) simplifies to:
    (A) \(2\) (B) \(4\) (C) \(\sqrt{16}\) (D) \(4\)
    Answer: (D)

Self Assessment Test – Class 9 – Number System

Instructions

• Attempt all questions section-wise.
• Choose the correct option in MCQs.
• Write your answers neatly for Short and Long answer questions.
• Check your answers with the key provided at the end.

Section A: Multiple Choice Questions (1 Mark each)

Total Questions: 8

1. Which of the following numbers is irrational?
   (A) \(0.25\) (B) \(\sqrt{7}\) (C) \(\dfrac{7}{2}\) (D) \(3\)
   Answer: (B)
2. The decimal expansion of \(\dfrac{5}{6}\) is:
   (A) Terminating (B) Non-terminating (C) Repeating (D) Both B and C
   Answer: (D)
3. \(\sqrt{49}\) equals:
   (A) \(6\) (B) \(8\) (C) \(7\) (D) \(9\)
   Answer: (C)
4. Which of the following is not a real number?
   (A) \(\sqrt{-1}\) (B) \(\sqrt{2}\) (C) \(-5\) (D) \(\pi\)
   Answer: (A)
5. Between two rational numbers:
   (A) No rational number exists (B) Infinite irrational numbers exist
   (C) Infinite rational numbers exist (D) None
   Answer: (C)
6. Which of the following is a rational number?
   (A) \(\sqrt{11}\) (B) \(3.14159\ldots\) (C) \(\dfrac{8}{11}\) (D) \(\sqrt{3}\)
   Answer: (C)
7. A rational number in the form of non-terminating, repeating decimal is:
   (A) \(0.333\ldots\) (B) \(1.010010001\ldots\) (C) \(\pi\) (D) \(\sqrt{3}\)
   Answer: (A)
8. The number \(0.121212\ldots\) is:
   (A) Irrational (B) Rational (C) Terminating (D) None
   Answer: (B)

Section B: Short Answer Questions (2 Marks each)

Total Questions: 6

9. Define rational and irrational numbers with one example each.
   Answer: Rational: Can be expressed as p/q (e.g., 1/2). Irrational: Cannot be expressed as p/q (e.g., √2).
10. Express \(0.777\ldots\) in the form \(\dfrac{p}{q}\).
    Answer: \(\dfrac{7}{9}\)
11. Is the number \(\dfrac{1}{7}\) a terminating or non-terminating decimal? Explain.
    Answer: Non-terminating repeating (denominator has prime factor other than 2 or 5).
12. Represent \(\sqrt{9.5}\) on a number line (conceptually).
    Answer: Use right triangle method with sides √9 and √0.5.
13. Find 3 rational numbers between \(-2\) and \(0\).
    Answer: -1.5, -1, -0.5 (any three between -2 and 0).
14. Classify the following as rational or irrational: \(\sqrt{25}\), \(0.1010010001\ldots\), \(\dfrac{4}{9}\)
    Answer: Rational, Irrational, Rational.

Section C: Long Answer Questions (4 Marks each)

Total Questions: 4

15. Show that \(\sqrt{5}\) is an irrational number.
    Answer: Proof by contradiction assuming √5 is rational leads to contradiction.
16. Prove that the sum of a rational and an irrational number is irrational.
    Answer: Proof by contradiction assuming sum is rational leads to contradiction.
17. Write five rational numbers between \(\dfrac{2}{3}\) and \(\dfrac{3}{4}\).
    Answer: \(\dfrac{17}{24}, \dfrac{35}{48}, \dfrac{3}{4}, \dfrac{37}{48}, \dfrac{19}{24}\) (any five between).
18. Represent \(\sqrt{2}\) on the number line using the geometrical method.
    Answer: Right triangle method with sides 1 unit each, hypotenuse √2.

Section D: Case Study Based Questions (1 Mark each)

Total Questions: 5

19. Neha saw the number \(1.4142135\ldots\) and guessed it is:
    (A) Rational (B) Irrational (C) Integer (D) Whole Number
    Answer: (B)
20. She had to convert \(0.\overline{16}\) to a rational form. What should it be?
    (A) \(\dfrac{16}{99}\) (B) \(\dfrac{4}{25}\) (C) \(\dfrac{8}{99}\) (D) \(\dfrac{32}{99}\)
    Answer: (A)
21. She found a number that repeats: \(0.123123\ldots\). It is:
    (A) Irrational (B) Rational (C) Terminating (D) Whole number
    Answer: (B)
22. Neha sees \(0.1010010001\ldots\) and classifies it as:
    (A) Rational (B) Irrational (C) Terminating (D) Repeating
    Answer: (B)
23. She had to identify an irrational number from a group:
    (A) \(-\dfrac{3}{4}\) (B) \(2.75\) (C) \(\sqrt{3}\) (D) \(5\)
    Answer: (C)

Answer Key

Section A:

Section B:

9. Definitions with examples
10. \(\dfrac{7}{9}\)
11. Non-terminating repeating
12. Represented conceptually with number line
13. -1.5, -1, -0.5
14. Rational, Irrational, Rational

Section C:

15. Proved
16. Proved
17. Example: \(\dfrac{67}{100}, \dfrac{68}{100}, \dfrac{69}{100}, \dfrac{7}{10}, \dfrac{71}{100}\)
18. Represented geometrically using compass method

Section D: