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प्रश्न
Prove that `1/(sqrt(3) + 1` is an irrational number.
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उत्तर
Given: To prove that the number `1/(sqrt(3) + 1` is an irrational number.
To Prove: `1/(sqrt(3) + 1` is irrational.
Proof:
1. Start with the expression: `1/(sqrt(3) + 1`.
2. Rationalise the denominator by multiplying the numerator and denominator by the conjugate `sqrt(3) - 1`:
`1/(sqrt(3) + 1) xx (sqrt(3) - 1)/(sqrt(3) - 1)`
= `(sqrt(3) - 1)/((sqrt(3) + 1)(sqrt(3) - 1))`
3. Expand the denominator using the difference of squares formula:
`(sqrt(3))^2 - (1)^2`
= 3 – 1
= 2
4. Thus, `1/(sqrt(3) + 1) = (sqrt(3) - 1)/2`.
5. Note that `sqrt(3)` is an irrational number.
6. Assume, for the sake of contradiction, that `(1/(sqrt(3) + 1))` is rational.
7. Then, `((sqrt(3) - 1)/2)` would be rational since they are equal.
8. Since 2 is rational, for `((sqrt(3) - 1)/2)` to be rational, `sqrt(3) - 1` must be rational.
9. If `sqrt(3) - 1` is rational, then `sqrt(3) = (sqrt(3) - 1) + 1` would be rational because a sum of two rational numbers is rational.
10. But `sqrt(3)` is irrational.
11. This is a contradiction.
12. Hence, the assumption that `(1/(sqrt(3) + 1))` is rational is false.
