Question

show that 1/2+root 3 is irrational

Answer 1

Hello buddy..
We can rationalize the denominator of the above expression, & then we can proceed with our proof…

After rationalization

1 / (2+√3) * (2-√3) / (2-√3)

= (2-√3) / (4–3)

= 2-√3

Now, we can prove that 2-√3 is an irrational..

We assume 2-√3 is a rational ……(1)

=> 2-√3 = p/q ( where, p, q belong to the set of integers, q is not equal to 0)

=> 2- p/q = √3

Here, LHS is a rational number, as we know that the difference of 2 rationals is always a rational.

But √3 is an irrational ( It's a theorem)

=> in the above LHS should not be equal to RHS..

This contradiction arises because of our wrong assumption.

So, 2-√3 should be an irrational number…. (corrected our assumption (1) )

=> 1/(2+√3) is an irrational number

[ Hence Proved]

Hope it helps you buddy...