1.

Prove that each of the following numbers is irrational:5 + 3√2

Answer»

Let, 5 + 3√2 be rational. 

Hence, 5 and 5 + 3√2 are rational. 

∴ (5 + 3√2 – 5) = 3√2 = rational [∵Difference of two rational is rational] 

∴ 1 3 × 3√2 = √2 = rational [∵Product of two rational is rational] 

This contradicts the fact that √2 is irrational. 

The contradiction arises by assuming 5 + 3√2 is rational. 

Hence, 5 + 3√2 is irrational.



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