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 ≠ 0)
Such that $\sqrt 5 = \frac{a}{b}$
$\Rightarrow $ b $\sqrt 5 $=a
Squaring both sides, we get
$\Rightarrow $ $5b ^2 =a^2 ... (1)$
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) ,
$5b^2 = 25c^2$
$\Rightarrow b^2 =5c^2$
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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