Question 3

Prove that the following are irrationals:

Answer

(i) Let us assume 1/√2 is a rational number.

1/√2 = p/q , where q ≠ 0 and p and q are co primes.

On reciprocal,

√2 = qp ................(1)

Since, q and p are integers and q/p is also a rational number
As we know √2 is an irrational number.
From (1)

√2 ≠ q/p

Thus our assumption is wrong 1/√2 is not a rational number.
Hence, proved

(ii) Let us suppose 7√5 is a rational number.
7√5 = p/q, where p and q are co primes and q ≠ 0
On solving , √5 = (p/q)7 .....................(1)

Since p, q and 7 integers and (p/q)7 is also a rational number.
And we know √5 is an irrational number.
From (1)

√5 ≠ (p/q) / 7

So our supposition Is wrong 7√5 is not a rational number.
Hence, proved.

(iii) Let us suppose 6 + √2 is a rational number.

6 + √2 = a/b, where a, b are co primes and b ≠ 0.
On solving,

√2 = a/b - 6 .....................(1)

Since a, b and 6 are integers and a/b - 6 is also a rational number.
And we know that √2 is an irrational number.
From (1)

√2 ≠ a/b - 6

Thus our Superposition is wrong 6√2 is not a rational number.
Hence, proved.

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