Prove that  $\sqrt{2}$ is an irrational number.

We need to prove that is an irrational number. Let us assume that is a rational number.
Therefore, $\sqrt{2}$= 𝑝 , where 𝑝 and 𝑞 are co-primes.
On squaring both sides, we get

2 = $\frac{p^2}{q^2}$

𝑝² = 2𝑞²

Therefore, 𝑝² is divisible by 2 and hence 𝑝 is divisible by 2.... (𝑖)

Now, let 𝑝 = 2𝑟 for some integer 𝑟.Therefore, on squaring we get

⇒ 𝑝² = 4𝑟²      

⇒ 2𝑞²  = 4𝑟²

⇒ 𝑞² = 2𝑟²

Therefore, 𝑞² is divisible by 2 and hence 𝑞 is divisible by 2.      ... (𝑖𝑖)

From the above two equations, it can be said that 𝑝 and 𝑞 are divisible by 2, which is in contradiction with the fact that 𝑝 and 𝑞 are co-primes.

Therefore, our assumption is false.

Hence, $\sqrt{2}$    is an irrational number that is proved.