Question

Show that the following numbers are irrational.
(i)1/√2 (ii) 7√5
(iii) 6 + √2 (iv) 3 -√5

Answer 1

Answer:

iii, iv

is the answer of question

Answer 2

Proved irrational

Step-by-step explanation:

(i) 1/√2

Let us assume 1/√2 is rational. Then it can be expressed as a/b where a and b are co-primes and b is not zero.

a / b = 1/√2

So b/a = √2 where b/a is rational and √2 is irrational.

rational can't be equal to irrational.

So our assumption is incorrect.

So 1/√2 is irrational. Proved.

(ii) 7√5

Let us assume 7√5 is rational. Then it can be expressed as a/b where a and b are co-primes and b is not zero.

a / b = 7√5

So a /7b = √5 where 7a/b is rational and √5 is irrational.

rational can't be equal to irrational.

So our assumption is incorrect.

So 7√5 is irrational. Proved.

(iii) 6 + √2

Let us assume 6 + √2 is rational. Then it can be expressed as a/b where a and b are co-primes and b is not zero.

a / b = 6 + √2

So a /b -6 = √2 where a/b -6 is rational and √2 is irrational.

rational can't be equal to irrational.

So our assumption is incorrect.

So 6 + √2 is irrational. Proved.

(iv) 3 -√5

Let us assume 6 + √2 is rational. Then it can be expressed as a/b where a and b are co-primes and b is not zero.

a / b = 3 -√5

So a /b -3 = -√5 where a/b -3 is rational and -√5 is irrational.

rational can't be equal to irrational.

So our assumption is incorrect.

So 3 -√5 is irrational. Proved.