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प्रश्न
Prove that `2/sqrt(7)` is irrational, given that `sqrt(7)` is irrational.
सिद्धांत
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उत्तर
Given: `sqrt(7)` is irrational.
To Prove: `2/sqrt(7)` is irrational.
Proof [Step-wise]:
1. Write `2/sqrt(7)` in a form that factors out `sqrt(7)`:
`2/sqrt(7) = 2/sqrt(7) xx sqrt(7)/sqrt(7)`
= `2/7 xx sqrt(7)`
2. Note that `2/7` is a nonzero rational number.
3. Suppose, for contradiction, that `2/sqrt(7)` is rational.
Then `2/7 xx sqrt(7)` is rational.
4. If a nonzero rational `2/7` times a number is rational, then that number must be rational.
Multiply the rational result by the rational `7/2` to recover `sqrt(7)`.
Hence, `sqrt(7)` would be rational.
5. This contradicts the given that `sqrt(7)` is irrational.
Therefore, the assumption in step 3 is false.
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