Question

Prove that√2 -√3 is irrational number

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Answer 1

Let us assume that √2 - √3 is a rational number.
Then, √2 - √3 = p/q
⇒ p/q + √3 = √2
squaring both sides, we get
⇒ (p/q + √3)² = (√2)²
⇒ p²/q² + 2p/q√3 + 3 = 2
⇒ p²/q² + 3 - 2 = - 2√3p/q
⇒ p²/q² + 1 = -2√3p/q
⇒ {(p²+q²)/q² × q/-2p = √3
⇒ {(p²+q²)/-2p} = √3
⇒ √3 is a rational number [∴ pq are integers and (p²+q²)/-2p is rational]
This contradicts the fact that √3 is irrational.
So, our assumption at the starting of the answering of this question was incorrect.
Here, √2 - √3 is irrational. 
Hence proved.