Prove that√3is an irrational number.

1.	Prove that√3is an irrational number.
Answer» RONG>To PROVE:- $\bf\implies √3 \: is \: irrational$ SOLUTION:- $\bf\implies By \: Contradiction\: Method$ $\bf\implies Assuming \: √3 is \: rational$ $\bf\implies √3=\frac{p}{q} \: [p \: and \: q \: are \: co-prime$ $\bf\implies p=√3q \: ..(1)$ $\bf\implies p²=3q²$ $\bf\implies \frac{p²}{3}=q²$ $\bf\implies p² \: divides \: 3$ $\bf\implies p \: also \: divides \: 3$ $\bf\implies p=3m \: [m \: is \: any \: integer]$ $\bf\implies √3q=3m$ $\bf\implies q=\frac{3m}{√3}$ $\bf\implies q=√3m$ $\bf\implies q²=3m²$ $\bf\implies \frac{q²}{3}=m²$ $\bf\implies q² \: divides\: 3$ $\bf\implies q \: divides \: 3$ $\bf\implies p \: divides \: 3$ $\bf\implies q \: also \: divides \: 3$ $\bf\implies But \: p \: and \: q \: are \: co-primes$ $\bf\implies This \: is \: because \: of \: our \: wrong \: assumption$ $\bf\implies √3 \: is \: irrational$