Question

prove that √2 is an irrational no.?​

Answer 1

Answer:

Step-by-step explanation:

Let it be assumed that √2 is rational.

Then √2= p/q (where p and q are integers and q not equal to zero. Also assume that p and q are coprimes.)

√2 = p/q

=>(√2)² = (p/q)²

[ Squaring both the sides]

=>2 = p²/q²

=>p² = 2q²

=> 2 divides p²

=> 2 divides p .........1

Further assume that p = 2r

=>(2r)² = 2q²

=>4r² = 2q²

=>q² = 2r²

=>2 divides q²

=>2 divides q ..........2

From eq 1 and 2 it is clear that 2 is a factor of p and q which contradicts our assumption that p and q are coprimes. Hence , √2 is irrational.[proved]