Question

Prove that 3 + 2√3 is an irrational number. ​

Answer 1

Answer:

Let 3+2√3 is a rational number.

A rational number can be written in the form of p/q.

p,q are integers then (p-6q)/2q is a rational number. ...

Therefore,3+2√3 is an irrational number.

Answer 2

Answer:

Let us assume, to the contrary, that 3 + 2√3 is a rational number.

That is, we can find coprime a and b (b not equal to 0) where a and b are integers.

Therefore, 3 + 2√3 = a/b

Rearranging the equation, we get 2√3 = a-3b/b

I.e √3 = a-3b/2b

Since a and b are integers, therefore a-3b/2b is a rational and therefore √3 is rational.

But this contradicts the fact that √3 is irrational.

This contradiction has arisen due to our incorrect assumption that 3 + 2√3 is rational.

Therefore, 3 + 2√3 is an irrational number.