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Question

Prove that 2+53 is an irrational number, given that 3 is an irrational number.

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Solution

Let us assume that 2+53 is a rational number.

Thus, 2+53 can be represented in the form of pq, where p and q both are integers, q ≠ 0, p and q are co-prime numbers.

2+53=pq53=pq-253=p-2qq3=p-2q5q

since, p-2q5q is rational ⇒ 3 is rational.

But, it is given that 3 is an irrational number.

Therefore, our assumption is wrong.

Hence, 2+53 is an irrational number.

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