How to prove ✓ 2 ✓ 5 is irrational?

Proof that the sum of the square root of two and the square root of five is irrational

To prove that $\sqrt{2} + \sqrt{5}$ is irrational, we use proof by contradiction. We begin by assuming the opposite: that the sum is a rational number. By definition, any rational number can be expressed as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers, $q \neq 0$, and the fraction is in its simplest form.

Therefore, we can write: $\sqrt{2} + \sqrt{5} = \frac{p}{q}$.

Next, we square both sides of the equation to eliminate the radicals. Squaring the left side gives: $(\sqrt{2} + \sqrt{5})^2 = 2 + 5 + 2\sqrt{10} = 7 + 2\sqrt{10}$. Equating this to the square of the right side gives $7 + 2\sqrt{10} = \frac{p^2}{q^2}$.

By rearranging this equation to isolate the radical term, we find that $\sqrt{10}$ can be expressed entirely in terms of rational numbers (integers $p$ and $q$). However, $10$ is not a perfect square, which means its square root is inherently an irrational number with a non-terminating, non-repeating decimal expansion ($\approx 3.1622776\dots$).

This creates a clear mathematical contradiction: a purely rational value cannot possibly equal an irrational value. Because our initial assumption that $\sqrt{2} + \sqrt{5}$ is rational leads to an impossible conclusion, that assumption must be false. Hence, $\sqrt{2} + \sqrt{5}$ is irrational.