MP Board · Class 10 · Mathematics · Real NumbersProve that $\sqrt{2}$ is an irrational number by explaining the core concept.

To prove that $\sqrt{2}$ is irrational, we begin by assuming the contrary that it is a rational number. This means it can be expressed in the simplest form $p/q$ where $p$ and $q$ are coprime integers and $q \neq 0$. Squaring both sides yields $p^2 = 2q^2$, implying that 2 divides $p^2$ and consequently divides $p$. Continuing this contradiction leads to the conclusion that our initial assumption was false.