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 $\GF$ is the mapping $f: \GF \to \GF$ defined as:

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

where $\times$ denotes multiplication.

Definition 2

The square (function) on $\GF$ is the mapping $f: \GF \to \GF$ 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.

$\blacksquare$