RD Sharma Class 9 Solutions Chapter 1 Exercise 1.4 2021| Download Free PDF

RD Sharma Class 9 Solutions Chapter 1 Exercise 1.4

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RD Sharma Class 9 Solutions Chapter 1 Exercise 1.4

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RD Sharma Solutions Class 9 Chapter 1 Number System Ex 1.4

Question 1: Define an irrational number.

Solution:

A number that cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0. It is a non-terminating or non-repeating decimal.

Question 2: Explain how irrational numbers differ from rational numbers.

Solution:

An irrational number is a real number that can be written as a decimal but not as a fraction i.e. it cannot be expressed as a ratio of integers.

It cannot be expressed as a terminating or repeating decimal.

For example, √2 is an irrational number

A rational number is a real number that can be written as a fraction and as a decimal, i.e. it can be expressed as a ratio of integers.

It can be expressed as a terminating or repeating decimal.

For example, 0.10 and 5/3 are rational numbers

Question 3: Examine whether the following numbers are rational or irrational:

Solution:

(i) √7

Not a perfect square root, so it is an irrational number.

(ii) √4

A perfect square root of 2.

We can express 2 in the form of 2/1, so it is a rational number.

(iii) 2 + √3

Here, 2 is a rational number, but √3 is an irrational number

Therefore, the sum of a rational and irrational number is an irrational number.

(iv) √3 + √2

√3 is not a perfect square, thus an irrational number.

√2 is not a perfect square, thus an irrational number.

Therefore, the sum of √2 and √3 gives an irrational number.

(v) √3 + √5

√3 is not a perfect square, and hence, it is an irrational number

Similarly, √5 is not a perfect square, and it is an irrational number.

Since the sum of two irrational numbers is an irrational number, √3 + √5 is an irrational number.

(vi) (√2 – 2)²

(√2 – 2)² = 2 + 4 – 4 √2

= 6 – 4 √2

Here, 6 is a rational number, but 4√2 is an irrational number.

Since the sum of a rational and an irrational number is an irrational number, (√2 – 2)2 is an irrational number.

(vii) (2 – √2)(2 + √2)

We can write the given expression as;

(2 – √2)(2 + √2) = ((2)² − (√2)²)

[Since, (a + b)(a – b) = a² – b²]

= 4 – 2 = 2 or 2/1

Since 2 is a rational number, (2 – √2)(2 + √2) is a rational number.

(viii) (√3 + √2)²

We can write the given expression as;

(√3 + √2)² = (√3)² + (√2)² + 2√3 x √2

= 3 + 2 + 2√6

= 5 + 2√6

[using identity, (a+b)² = a² + 2ab + b²]

Since the sum of a rational number and an irrational number is an irrational number, (√3 + √2)² is an irrational number.

(ix) √5 – 2

√5 is an irrational number, whereas 2 is a rational number.

The difference between an irrational number and a rational number is an irrational number.

Therefore, √5 – 2 is an irrational number.

(x) √23

Since, √23 = 4.795831352331…

As the decimal expansion of this number is non-terminating and non-recurring, it is an irrational number.

(xi) √225

√225 = 15 or 15/1

√225 is a rational number as it can be represented in the form of p/q, and q is not equal to zero.

(xii) 0.3796

As the decimal expansion of the given number is terminating, it is a rational number.

(xiii) 7.478478……

As the decimal expansion of this number is a non-terminating recurring decimal, it is a rational number.

(xiv) 1.101001000100001……

As the decimal expansion of the given number is non-terminating and non-recurring, it is an irrational number

Question 4: Identify the following as rational or irrational numbers. Give the decimal representation of rational numbers:

Solution:

(i) √4

√4 = 2, which can be written in the form of a/b. Therefore, it is a rational number.

Its decimal representation is 2.0.

(ii) 3√18

3√18 = 9√2

Since the product of a rational and an irrational number is an irrational number, 3√18 is an irrational number.

Or 3 × √18 is an irrational number.

(iii) √1.44

√1.44 = 1.2

Since every terminating decimal is a rational number, √1.44 is a rational number.

And its decimal representation is 1.2.

(iv) √9/27

√9/27 = 1/√3

Since the quotient of a rational and an irrational number is irrational numbers, √9/27 is an irrational number.

(v) – √64

– √64 = – 8 or – 8/1

Therefore, – √64 is a rational number.

Its decimal representation is –8.0.

(vi) √100

√100 = 10

Since 10 can be expressed in the form of a/b, such as 10/1, √100 is a rational number.

And its decimal representation is 10.0.

Question 5: In the following equation, find which variables x, y, z etc. represent rational or irrational numbers:

Solution:

(i) x² = 5

Taking square root on both sides,

x = √5

√5 is not a perfect square root, so it is an irrational number.

(ii) y² = 9

y² = 9

or y = 3

3 can be expressed in the form of a/b, such as 3/1, so it is a rational number.

(iii) z² = 0.04

z² = 0.04