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Question

Prove that 3-427 is an irrational number, given that 2 is an irrational number.

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Solution

Let us assume that 3-427 is a rational number.

Thus, 3-427 can be represented in the form of pq, where p and q are integers, q ≠ 0, p and q are co-prime numbers.

3-427=pq3-42=7pq42=3-7pq42=3q-7pq2=3q-7p4qSince, 3q-7p4q is rational2 is rationalBut, it is given that 2 is an irrational number.Therefore, our assumption is wrong.Hence, 3-427 is an irrational number.

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