# Equivalence of Definitions of Square Function

## Theorem

Let $\F$ denote one of the standard classes of numbers: $\N$, $\Z$, $\Q$, $\R$, $\C$.

The following definitions of the concept of Square Function are equivalent:

### Definition 1

The square (function) on $\F$ is the mapping $f: \F \to \F$ defined as:

$\forall x \in \F: \map f x = x \times x$

where $\times$ denotes multiplication.

### Definition 2

The square (function) on $\F$ is the mapping $f: \F \to \F$ defined as:

$\forall x \in \F: \map f x = x^2$

where $x^2$ denotes the $2$nd power of $x$.

## Proof

By definition of $n$th power (for positive $n$):

$x^n = \begin{cases} 1 & : n = 0 \\ x \times x^{n - 1} & : n > 0 \end{cases}$

Thus:

 $\ds x^2$ $=$ $\ds x \times x^1$ $\ds$ $=$ $\ds x \times x \times x^0$ $\ds$ $=$ $\ds x \times x \times 1$ $\ds$ $=$ $\ds x \times x$

Hence the result.

$\blacksquare$