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**Exercise 1.3Ques 1: Prove that âˆš5 is irrational.**

âˆ´ We have to find two integers a and b (where, b â‰ 0 and a and b are coprime) such that

a/b = âˆš5

â‡’ a = âˆš5.b

Squaring both sides, we have

a

âˆ´ 5 divides a

â‡’ 5 divides a ...(2)

[âˆµ a prime number â€˜pâ€™ divides a

âˆ´ a = 5c, where c is an integer.

âˆ´ Putting a = 5c in (1), we have

5c = âˆš5.

or (5c)

â‡’ 25c

â‡’ 5c

â‡’ 5 divides b

â‡’ 5 divides b ...(3)

From (2) and (3)

a and b have at least 5 as a common factor.

i.e., a and b are not coprime.

âˆ´ Our supposition that âˆš5 is rational is wrong.

Hence, âˆš5 is irrational.

âˆ´ We can find two coprime integers â€˜aâ€™ and â€˜bâ€™ such that

[3+2âˆš5]

âˆ´

â‡’

â‡’

âˆµ a and b are integers,

âˆ´

âˆ´

â‡’ (1) is a rational

â‡’ âˆš5 is a rational

But this contradicts the fact that âˆš5 is irrational.

âˆ´ Our supposition is wrong.

3+2âˆš5 is an irrational.

(ii) 7âˆš5

(iii) 6+âˆš2

Let 1/âˆš2 be rational, and

such that â€˜aâ€™ and â€˜bâ€™ are coprime integers and b â‰ 0.

...(1)

since, the division of two integers is rational.

âˆ´ 2a/b is a rational.

From (1), âˆš2 is a rational number which contradicts the fact that âˆš2 is irrational.

âˆ´ Our assumption is wrong.

Thus, 1âˆš2 is irrational.

(ii) Let us suppose that 7âˆš5 is rational.

Let there be two coprime integers â€˜aâ€™ and â€˜bâ€™.

such that 7âˆš5 = a/b , where b â‰ 0

Now, = 7âˆš5 = a/b

â‡’

â‡’ âˆš5 is a rational

This contradicts the fact that âˆš5 is irrational.

âˆ´ We conclude that 7âˆš5 is irrational.

(iii) Let us suppose that 6+âˆš2 is rational.

âˆ´ We can find two coprime integers â€˜aâ€™ and â€˜bâ€™ (b â‰ 0), such that

6+âˆš2 = a/b

âˆ´

or ...(1)

âˆµ a and b are integers,

âˆ´

[âˆµ subtraction of integers is also an integer]

[âˆµ Division of two integers is a rational number]

â‡’ is a rational number.

From (1), âˆš2 is a rational number, which contradicts the fact that âˆš2 is an irrational number.

âˆ´ Our supposition is wrong.

â‡’ 6+ âˆš2 is an irrational number.**Revisiting Rational Numbers and Their Decimal Expansion****Remember:**

I. The decimal expansion of every rational number is either terminating or non terminating and repeating.

II. For any rational number p/q having terminating decimal representation, the prime factorisation of â€˜qâ€™ is of the form 2^{n} 5^{m}, where n and m are non-negative integers.

III. For any rational number p/q, if the prime factorisation of q is of the form 2^{n }5^{m}, where n and m are non-negative integers, then its decimal representation is terminating.

IV. For any rational number p/q, if the prime factorisation of q is not of the form 2^{n} 5^{m}, where m and n are non-negative integers, then its decimal representation is non terminating and repeating.

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