. Prove that the following are irrational


(i)irrational

(ii)irrational

(iii)irrational

Best Answer

Explanation: (i) irrational

Let us assume, that irrational is rational.

Then, there exist positive co-primes a and b ≠ 0 such that irratilonal= a/b

Rearranging, we get

irrational= b/a

Since   and  are integers, b/a is rational, and so √2 is rational.

 

But this contradicts the fact that √2 is irrational.

 

So, irrational is irrational.

 

(ii) irrational

Let us assume, that irrational is rational.

Then, there exist positive co-primes a and b  ≠ 0 such that irrational= a/b

Rearranging, we get

irrational=a/7b

Since   and  are integers,b/7a  is rational, and sois rational√5.

But this contradicts the fact that √5 is irrational.

So,irrational  is irrational.

 

(iii) irrational

Let us assume, that irrational is rational.

Then, there exist positive co-primes a and b ≠ 0 such that irrational= a/b

Rearranging, we get

√2=a/b-6

Since   and  are integers, a/b-6 is rational, and so √2 is rational.

But this contradicts the fact that √2 is irrational.

So,  is irrational.

6+√2

Final Answer: (i) irrational is irrational.

(ii) irrational is irrational.

 

(ii) irrational is irrational.

 

Talk to Our counsellor