Question

prove that underroot 5 is not a rational no

Answer 1

Question:- Prove that ✓5 is irrational.
Solution:- Let us assume to the contrary that ✓5 is rational. Then there exist two co-primes(the no.'s which have nothing in common between them) "a" and "b".
Then, ✓5=a/b (a and b are positive odd integers.)
b✓5=a
Now squaring both sides we get,
5b^2=a^2 ............ (1)
Therefore, 5 divides a^2, and according to theorem if p divides a, then p also divides a^2.

Therefore 5 also divides a, so we can write,
a=5c , for some integer c
Squaring both sides we get,
(a)^2= (5c)^2
a^2= 25c^2
Now we will replace the value of a^2 by 5b^2.
5b^2 = 25c^2 (By equation 1)
Now 5 divides 25,
b^2/5 = c

Again by the theorem, If b^2 divides 5, then b also divides 5.
But this contradicts the fact that a and b are co-primes, and they have 5 in common.
Our assumption is wrong.
Hence ✓5 is irrational.