Question

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

Answer 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.

• Step 1: Let us assume, to the contrary, that (5+ 3√2) is rational.

• That is, we can find coprime a and b (b ≠ 0) such that (5+3√2) = p/q.

write: To the contrary, let us assume (5+3√2) = p/q

• Step 2:Rearranging the terms we will get:

write:

3√2 = (p / q) - 5

√2 = (p - 5q) / 3q

Step 3: Since, p and q are integers, (p - 5q) / 3q is a rational number, and so √2 is rational but this contradicts the fact that √2 is irrational. So, we conclude that 5+3√2 is irrational