Question

Prove that 2+root3 is irrational

Answer 1

Let 2+√3 be a rational number, also, we know that 2 is a rational number and √3 is an irrational number.

Now, if

Rational - Rational = Rational

=> 2+√3-2 = Rational

=> √3 = Rational

Clearly, √3 is an irrational number. Hence, contradiction of our supposition, and so, 2+√3 is an irrational number.

Answer 2

Let's assume to reach the contradiction that  2+√3  is rational.

Let  x = 2+√3,  where x is a rational number which values the RHS.

$x=2+\sqrt{3} \\ \\ \\ x^2=(2+\sqrt{3})^2 \\ \\ \\ x^2=2^2+(\sqrt{3})^2+2 \times 2 \times \sqrt{3} \\ \\ \\ x^2=4+3+4\sqrt{3} \\ \\ \\ x^2=7+4\sqrt{3} \\ \\ \\ x^2-7=4\sqrt{3} \\ \\ \\ \displaystyle \frac{x^2-7}{4}=\sqrt{3}$

At the final step, it seems that √3 can be written in p/q form. This creates a contradiction.

So  2+√3  is not rational.

Hence proved!!!