Question

show that rute2 is irrational​

Answer 1

Answer:

Step-by-step explanation:

if possible √2 be rational and simplest form is p/q

then ,a and b are integer it have no common factor other than 1 and bis not equal to 0.

then ,

√2=a/b =)2 =a^2/b^2( squaring on both side)

2b^2=a^2

2 divides a^2

and so ,(2 divides 2b^2 )

=) 2 divides a ,but 2 is prime and divides by b^2 =) 2 divides b .

let a=2c for some integer c .

putting a =2c in (1) we get .

2b^2=4c^2 =)b^2=2c^2

=)2 divides b^2 ( so,2 divides 2c^2 )

=)2 divides b

so, 2 is prime and 2 divides b^2 =)2 divides b

thus two is common factor of a and b .

but this is contradiction fact that a and b have no common. common factor other than 1 .

the contradiction arises by assuming that √2 is rational .

hence √2 is irrational..