Why does the decimal expansion of √2 never end or repeat?

The decimal expansion of $\sqrt{2}$ never ends or repeats because it is an irrational number. By definition, an irrational number cannot be expressed as the quotient of two integers, or $p/q$.12

If $\sqrt{2}$ were a rational number, it would necessarily have a repeating decimal expansion, as all rational numbers do. However, mathematical proof—famously explored by ancient Greek mathematicians—demonstrates that assuming $\sqrt{2}=p/q$ leads to a logical contradiction. Specifically, if you assume the ratio is in its simplest form, the algebra forces both $p$ and $q$ to be even, which contradicts the premise that they share no common factors.23

Because $\sqrt{2}$ cannot be represented as a fraction of integers, its decimal representation must be infinite and non-repeating. This property identifies it as an algebraic irrational number, serving as a solution to the equation $x^2-2=0$.412