Question

show that √2 is not a rational number

Answer 1

Let √2 be a rational no. say a/b such that a and b are integers b not equal to 0 and a and b are co primes.
Therefore, √2= a/b
SBS, 2 = a^2/b^2
2b^2=a^2
Therefore,
2 is a factor of 2b^2
2 is a factor of a^2
2 is a factor of a. ........(I)
Let a=2x
2b^2=(2x)^2
2b^2=4x^2
b^2=2x^2
Therefore,
2 is a factor of 2x^2.
2 is a factor of b^2.
2 is a factor of b. ..........(ii)
From (I) and (ii),
2 is a common factor of a and b.
But this is not possible.
Since a and b are co primes.
Our supposition was wrong.
√2 is irrational.