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प्रश्न
Prove that the following is irrational:
`1/sqrt2`
उत्तर
`1/sqrt2`
`1/sqrt2 xx sqrt2/sqrt2 = sqrt2/2`
Let a = `(1/2)sqrt2` be a rational number.
∴ `1/2 (sqrt2)` is rational.
Let `1/2 (sqrt2) = a/b`, such that a and b are co-prime integer and b ≠ 0.
∴ `sqrt2 = (2a)/b` ...(1)
Since the division of two integers is rational.
∴ `(2a)/b` is rational.
From (1), `sqrt2` is rational, which contradicts the fact that `sqrt2` is irrational.
∴ Our assumption is wrong.
Thus, `1/sqrt2` is irrational.
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