Prove that 5√2 is an irrational number.

1.	Prove that 5√2 is an irrational number.
Answer» Let’s assume on the contrary that 5√2 is a rational number. Then, there exist positive integers a and b such that 5√2 = \(\frac{a}{b}\) where, a and b, are co-primes ⇒ √2 = \(\frac{a}{5b}\) ⇒ √2 is rational [∵ a and b are integers ∴ \(\frac{a}{5b}\) is a rational number] This contradicts the fact that √2 is irrational. So, our assumption is incorrect. Hence, 5√2 is an irrational number.

Answer»

Let’s assume on the contrary that 5√2 is a rational number.

Then, there exist positive integers a and b such that

5√2 = \(\frac{a}{b}\) where, a and b, are co-primes

⇒ √2 = \(\frac{a}{5b}\)

⇒ √2 is rational [∵ a and b are integers ∴ \(\frac{a}{5b}\) is a rational number]

This contradicts the fact that √2 is irrational. So, our assumption is incorrect.

Hence, 5√2 is an irrational number.

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