Natural Square Roots

Why

We want to solve equations with squares.

Definition

Let $m$ be a square number. We want to find $n$ to satisfy

\[ n^2 = m. \]

This result motivates definining a function the square root function which square numbers to their roots.

Notation

Let $S \subset \N $ denote the set of square numbers. Let $f: S \to N$ be the function such that $f(s)$ is the square root of $s$. We denote the result of $f$ on $n$ by $\sqrt{n}$.