Prove that 5 is irrational. - Vidyadip

Question

Prove that $\sqrt{5}$ is irrational.

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Answer

Let us prove $\sqrt{5}$ irrational by contradiction.
Let us suppose that $\sqrt{5}$ is rational. It means that we have co-prime integers a and b $(b \neq 0)$
Such that $\sqrt{5}=\frac{a}{b}$
$\Rightarrow b \sqrt{5}=a$
Squaring both sides, we get
$\Rightarrow 5 b^2=a^2$ ... (1)
It means that 5 is factor of $a^2$
Hence, 5 is also factor of a by Theorem. ... (2)
If, 5 is factor of a , it means that we can write a = 5c for some integer c .
Substituting value of a in (1) ,
$5 b^2=25 c^2$
$\Rightarrow b^2=5 c^2$
It means that 5 is factor of $b ^2$
Hence, 5 is also factor of b by Theorem. ... (3)
From (2) and (3) , we can say that 5 is factor of both a and b .
But, a and b are co-prime .
Therefore, our assumption was wrong. $\sqrt{5}$ cannot be rational. Hence, it is irrational.

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