Number Systems Class 9 MCQ and Most Important Questions

Number Systems Class 9 NCERT Notes with Questions & HOTS

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Number Systems Class 9 (NCERT Based)

Number Systems is the first chapter of Class 9 Mathematics and forms the foundation of higher mathematics. Here students learn about (Q), Irrational Numbers, Real Numbers, Decimal Expansions, and Laws of Exponents. Understanding these concepts is essential for future chapters and competitive examinations.

हिंदी में:
कक्षा 9 गणित का पहला अध्याय Number Systems है। यह अध्याय गणित की मजबूत नींव तैयार करता है। इसमें (Q), Irrational Numbers, Real Numbers, Decimal Expansion तथा Exponents के नियमों का अध्ययन किया जाता है।

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Table of Contents

1. Introduction to Number Systems
2. Types of Numbers
3. Quotient of two integers
4. (I) Irrational Numbers
5. Real Numbers
6. Decimal Expansion
7. Operations on Real Numbers
8. Laws of Exponents
9. Important Formulas
10. Key Points
11. NCERT Important Questions
12. HOTS Questions
13. Frequently Asked Questions
14. Chapter Summary

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Abbreviations

Abbreviation	Full Form
N	Natural Numbers
W	Whole Numbers
Z	Integers
Q	Rational Numbers
I	Irrational Numbers
R	Real Numbers
HCF	Highest Common Factor
LCM	Least Common Multiple

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1. Introduction to Number Systems

A Number System is a way of representing numbers. Various sets of numbers together form the Number System.

Hindi:
संख्याओं को दर्शाने की विभिन्न पद्धतियों को Number System कहा जाता है।

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2. Types of Numbers

A. Natural Numbers (N)

Numbers used for counting:

1, 2, 3, 4, 5, …

Example

• 8
• 25
• 100

Hindi:
गिनती में उपयोग होने वाली संख्याएँ Natural Numbers कहलाती हैं।

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B. Whole Numbers (W)

Natural Numbers + Zero

0, 1, 2, 3, 4, 5…

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C. Integers (Z)

Positive Numbers + Negative Numbers + Zero

…, -3, -2, -1, 0, 1, 2, 3 …

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D. Rational Numbers (Q)

A number which can be written as:

$\frac{p}{q},\ q\neq0$

where p and q are integers. Rational numbers have terminating or recurring decimal expansions.

Examples

• 3/4
• -7/8
• 5
• 0

Hindi:
जो संख्या p/q के रूप में लिखी जा सके, जहाँ q ≠ 0 हो, वह Rational Number कहलाती है।

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E. Irrational Numbers (I)

Numbers that cannot be expressed in p/q form.

Examples:

• √2
• √3
• π
• √5

Irrational numbers have non-terminating and non-recurring decimal expansions.

Hindi:
जो संख्या p/q के रूप में व्यक्त नहीं की जा सकती, वह Irrational Number कहलाती है।

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F. Real Numbers (R)

Real Numbers = Rational Numbers + Irrational Numbers. Every point on the number line represents a real number.

Formula

$R=Q\cup I$

Hindi:
सभी Rational और Irrational Numbers मिलकर Real Numbers बनाते हैं।

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Number System Chart ( Number Systems Class 9 )

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3. Decimal Expansion ( Number Systems Class 9 )

Rational Numbers

Terminating Decimal

Example:

1/2 = 0.5

5/8 = 0.625

Non-Terminating Recurring Decimal

Example:

1/3 = 0.3333…

7/11 = 0.636363…

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Irrational Numbers

Non-Terminating Non-Recurring

Example:

√2 = 1.41421356…

π = 3.14159265…

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4. Operations on Real Numbers

Operation	Result
Rational + Rational	Rational
Rational + Irrational	Irrational
Irrational + Irrational	May be Rational/Irrational
Rational × Irrational	Irrational

Examples:

• 2 + √3 = Irrational
• √2 + √2 = 2√2 (Irrational)
• √2 × √2 = 2 (Rational)

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5. Laws of Exponents

For a > 0

$a^m\times a^n=a^{m+n}$

$\frac{a^m}{a^n}=a^{m-n}$

$(a^m)^n=a^{mn}$

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Important Formulas

Square Root Identity

$(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b})=a-b$

Rationalisation Formula

$\frac{1}{\sqrt{a}+b}\times\frac{\sqrt{a}-b}{\sqrt{a}-b}$

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Key Points for Revision

Every Integer is a Rational Number.

Decimal expansion of Rational Numbers is terminating or recurring

Every Rational Number is a Real Number.

Decimal expansion of Irrational Numbers is non-terminating and non-recurring.

Every Whole Number is an Integer.

√2, √3, √5, π are Irrational Numbers.

Every Irrational Number is a Real Number.

Rational + Irrational = Irrational.

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NCERT Important Questions with Answers

Q1. Is zero a rational number?

Answer: Yes.

0 = 0/1

Hence Rational Number.

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Q2. Is √49 rational?

Answer:

√49 = 7

7 = 7/1

Therefore Rational Number.

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Q3. Is √7 rational or irrational?

Answer:

√7 cannot be expressed as p/q.

Hence Irrational.

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Q4. State True or False

Every irrational number is a real number.

Answer: True.

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Q5. Classify the numbers

(i) 0.625

(ii) √11

(iii) 4/9

Answer

(i) Rational

(ii) Irrational

(iii) Rational

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HOTS Questions (Higher Order Thinking Skills)

HOTS 1

Can the sum of two irrational numbers be rational?

Answer

Yes.

√2 + (-√2) = 0

0 is Rational.

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HOTS 2

Can the product of two irrational numbers be rational?

Answer

Yes.

√2 × √2 = 2

2 is Rational.

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HOTS 3

Find three irrational numbers between 2 and 3.

Answer

• √5
• √6
• √7

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HOTS 4

Is 0.101001000100001… rational?

Answer

No.

It is non-terminating and non-recurring.

Hence Irrational.

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Frequently Asked Questions (FAQ)

What is a Number System?

A system used to represent numbers.

What is a Rational Number?

A number expressible in p/q form.

What is an Irrational Number?

A number not expressible in p/q form.

Is π Rational?

No, π is Irrational.

Is every irrational number real?

Yes.

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One-Shot Chapter Summary

• Natural Numbers → Counting Numbers.
• Whole Numbers → Natural Numbers + 0.
• Integers → Positive, Negative and Zero.
• Rational Numbers → p/q form.
• Irrational Numbers → Not in p/q form.
• Real Numbers → (Q)+(I)
• Rational decimals are terminating or recurring.
• Irrational decimals are non-terminating and non-recurring.
• Laws of Exponents help simplify powers.
• Number Systems is the foundation chapter for all higher mathematics.

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