Question

how to prove that root 2 is an irrational number​

Answer 1

First of all decimal expansion of root 2 is not accurate and it is non-terminating non-repeating number so it is a irrational number

Answer 2

Answer:

Let us assume that root 2 is a rational number then there exists two positive integers a and b such that

root 2=a/b,where a and b are co-primes then HCF is 1

(root2)^2=(a/b)^2

2=a^2/b^2

2b^2=a^2

b^2=a^2/2

2/a^2 (by the theorem p/a^2,p/a)

2/a -(1)

now,

a=2c,where c is some positive integers

a^2=4c^2

2b^2=4c^2 (a^2=2b^2)

b^2=2c^2

c^2=b^2/2

2/b^2 (by the theorem p/a^2,p/a)

2/b -(2)

from equation (1) and (2) a and b have atleast 2 as a common factor.But this is contradicts a and b have no common factor other then 1.

So,our suppositions is wrong.

therefore,root 2 is a irrational number.