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प्रश्न
Prove that `3 - 2sqrt(5)` is an irrational number, given that `sqrt(5)` is an irrational number.
उत्तर
To Prove: `3 - 2sqrt(5)` is an irrational number
Given `sqrt(5)` is an irrational number
Let `3 - 2sqrt(5)` is a rational number
∴ `3 - 2sqrt(5) = p/q` ...(Where q ≠ 0)
`3 - 2sqrt(5)q` = p
3q – p = `2sqrt(5)q`
`(3q - p)/(2q) = sqrt(5)`
p and q of are integers
∴ `(3p - q)/(2q)` is a rational number but `sqrt(5)` is an irrational number
Hence rational number ≠ irrational number
So our assumption is wrong by contradiction fact
∴ `3 - 2sqrt(5)` is an irrational number.
Hence Proved.
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