Question

Given that $\sqrt{3}$ is irrational, prove that $5+2\sqrt{3}$ is an irrational number.

Answer 1

step 1.

• Now we all know that for proving irrationality of any number we start assuming that number as rational and prove it irrational by contradiction.
• Let us assume, to the contrary, that 5 + 2√3 is rational.
• That is, we can find coprimes a and b (b ≠ 0) such that
  5+2√3 = $\frac{p}{q}$
  √3 = $\frac{(p-5q)}{2q}$

step 2.

• Since, p and q are integers, $\frac{(p-5q)}{2q}$ is a rational number, and so √3 is rational.
• But this contradicts the fact that √3 is irrational.
• So, we conclude that 5+2√3 is irrational.