Class 9 Maths Chapter 1 Number Systems NCERT Solutions | All Exercises Step by Step

Class 9 Maths Chapter 1

Number Systems – NCERT Solutions with Step-by-Step Explanation

Mathematics begins with numbers. From counting objects to solving complex equations, numbers form the foundation of the entire subject. In Class 9 Maths Chapter 1 Number Systems, students learn how different types of numbers are related and how they are represented on the number line.

This chapter introduces rational numbers, irrational numbers, real numbers and their decimal expansions. It also explains how to represent irrational numbers on the number line and how to perform operations with real numbers.

Understanding number systems is very important because many future topics like algebra, polynomials and coordinate geometry depend on these concepts.

Basic Concepts of Number Systems

The number system can be divided into different categories.

Natural Numbers
1, 2, 3, 4, 5 …

Whole Numbers
0, 1, 2, 3, 4 …

Integers
…, −3, −2, −1, 0, 1, 2, 3 …

Rational Numbers
Numbers that can be written in the form

p / q

where p and q are integers and q ≠ 0.

Examples
1/2, −5/3, 7, 0

Irrational Numbers
Numbers that cannot be written in the form p/q.

Examples
√2 , √3 , π

Real Numbers
The collection of all rational and irrational numbers is called real numbers.

Important Formulas Used

Rational number form

p / q , where q ≠ 0

Decimal expansion of rational numbers

p / q = terminating decimal or recurring decimal

Laws of exponents

a^m × a^n = a^(m+n)

a^m / a^n = a^(m−n)

(a^m)^n = a^(mn)

a^0 = 1 (a ≠ 0)

Exercise 1.1 Solutions

Question 1

Is zero a rational number? Can it be written in the form p/q?

Solution

A rational number is a number that can be written as

p/q

where q ≠ 0.

0 can be written as

0/1
0/2
0/5

Hence

0 = p/q form

Therefore zero is a rational number.

Question 2

Find six rational numbers between 3 and 4.

Solution

Write both numbers with same denominator.

3 = 21/7
4 = 28/7

Numbers between them

22/7
23/7
24/7
25/7
26/7
27/7

These are six rational numbers between 3 and 4.

Question 3

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

Solution

Convert to denominator 10

3/5 = 6/10
4/5 = 8/10

Numbers between them

6.2/10
6.4/10
6.6/10
6.8/10
7/10

Hence these are rational numbers between the given numbers.

Exercise 1.2 Solutions

Question 1

State whether the following statements are true or false.

(i) Every irrational number is a real number.

Answer
True.

Real numbers include both rational and irrational numbers.

(ii) Every point on the number line is of the form √m where m is a natural number.

Answer
False.

Many numbers on the number line are negative or fractional.

(iii) Every real number is an irrational number.

Answer
False.

Real numbers include both rational and irrational numbers.

Question 2

Are the square roots of all positive integers irrational?

Solution

No.

Example

√4 = 2

Since 2 is a rational number, not all square roots are irrational.

Exercise 1.3 Solutions

Question

Write the following in decimal form.

(i) 36/100

Solution

36/100 = 0.36

(ii) 1/11

Solution

1/11 = 0.090909…

Recurring decimal.

(iii) 329/400

Solution

329/400 = 0.8225

Terminating decimal.

Question

Express recurring decimals as p/q.

(i) 0.666…

Let

x = 0.666…

Multiply by 10

10x = 6.666…

Subtract

10x − x = 6

9x = 6

x = 6/9

x = 2/3

Exercise 1.4 Solutions

Question

State whether the following numbers are rational or irrational.

(i) 2 − √5

Since √5 is irrational and subtraction with rational number remains irrational.

Answer
Irrational.

(ii) (3 + √23) − √23

= 3 + √23 − √23
= 3

Answer
Rational.

Exercise 1.5 Solutions

Simplify

(i) 64^(1/2)

√64 = 8

(ii) 32^(1/5)

32 = 2^5

(2^5)^(1/5) = 2

(iii) 125^(1/3)

125 = 5^3

(5^3)^(1/3) = 5

Key Points to Remember

• Rational numbers can be written as p/q.

• Irrational numbers cannot be expressed as fractions.

• Real numbers include both rational and irrational numbers.

• Decimal expansions of rational numbers are terminating or recurring.

• Laws of exponents help simplify powers of numbers.

Frequently Asked Questions (FAQs)

1. What is a number system?

A number system is a way of representing numbers and classifying them into different categories like natural, rational and real numbers.

2. What are rational numbers?

Numbers that can be written in the form p/q where q ≠ 0.

3. What are irrational numbers?

Numbers that cannot be written in the form p/q.

4. Is √2 rational?

No, √2 is an irrational number.

5. Is 0 a rational number?

Yes, because it can be written as 0/1.

6. What are real numbers?

All rational and irrational numbers together form real numbers.

7. What is a terminating decimal?

A decimal that ends after a finite number of digits.

8. What is a recurring decimal?

A decimal in which digits repeat infinitely.

9. Is π a rational number?

No, π is irrational.

10. What is the value of 64^(1/2)?

It is equal to √64 = 8.

11. What is the value of 125^(1/3)?

It is equal to 5.

12. Can irrational numbers be represented on number line?

Yes, using geometric methods.

13. Are all integers rational numbers?

Yes, because every integer can be written as p/1.

14. Is √4 irrational?

No, √4 = 2 which is rational.

15. Why is the number system important?

It forms the foundation of algebra and higher mathematics.

Conclusion

Class 9 Maths Chapter 1 Number Systems builds the base for understanding mathematics at a deeper level. Once you clearly understand rational numbers, irrational numbers, decimal expansions and exponents, many other topics become easier to learn.

Practice each NCERT exercise carefully and try to solve questions on your own before checking the solution.

For more NCERT solutions, maths concepts and exam preparation guides, visit

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