Question

proove that √2 is irrational​

Answer 1

Answer:

Proof that the square root of 2 is irrational

Assume is rational, i.e. it can be expressed as a rational fraction of the form , where and are two relatively prime integers. ... However, two even numbers cannot be relatively prime, so cannot be expressed as a rational fraction; hence is irrational

Answer 2

Let us assume that Square root 2 is rational No.

Suppose a and b have a common factor other than 1, then we can divide by the common factor , and assume that a and b are coprime

So, b Square root 2=a

Squaring on both side , and rearranging we get 2b square =a square .

Therefore, a2 is divisible by 2 , it follows that a is also divisible by 2 .

So, we can write a=2c for some integer c.

Substituting for a , we get 2b2 = 4 c2 , that is , b2 = 2c2.

This mean that b2 is divisible by 2, and so b is also divisible by 2 ..

Therefore, a and b have at least 2 common factor .

But this contradicts the fact that a and b are coprime .

This contradicts has arisen bcuz of our incorrect assumption that Square root 2 is rational , So we conclude that square root 2 is irrational...

Too long Ufff..