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A square is a regular quadrilateral.

That is, a regular polygon with $4$ sides.

That is, a square is a plane figure with four sides all the same length and whose angles are all equal.


Abstract Algebra

Let $\struct {S, \circ}$ be an algebraic structure.

Let $f: S \to S$ be the mapping from $S$ to $S$ defined as:

$\forall x \in S: \map f x := x \circ x$

This is usually denoted $x^2$:

$x^2 := x \circ x$

Square Function on Numbers

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

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 $x$.

Square Number

Square numbers are those denumerating a collection of objects which can be arranged in the form of a square.

They can be denoted:

$S_1, S_2, S_3, \ldots$

Definition 1

An integer $n$ is classified as a square number if and only if:

$\exists m \in \Z: n = m^2$

where $m^2$ denotes the integer square function.

Euclid's Definition

In the words of Euclid:

A square number is equal multiplied by equal, or a number which is contained by two equal numbers.

(The Elements: Book $\text{VII}$: Definition $18$)

Definition 2

$S_n = \begin{cases} 0 & : n = 0 \\ S_{n-1} + 2 n - 1 & : n > 0 \end{cases}$

Definition 3

$\displaystyle S_n = \sum_{i \mathop = 1}^n \left({2 i - 1}\right) = 1 + 3 + 5 + \cdots + \left({2 n - 1}\right)$

Definition 4

$\forall n \in \N: S_n = P \left({4, n}\right) = \begin{cases} 0 & : n = 0 \\ P \left({4, n - 1}\right) + 2 \left({n - 1}\right) + 1 & : n > 0 \end{cases}$

where $P \left({k, n}\right)$ denotes the $k$-gonal numbers.