Definition:Square

Algebra

Let $x$ be a number.

Then the square of $x$ is $x \times x$ and can be written $x^2$.

Square Number

An integer $n$ is defined as square iff $\exists m \in \Z: n = m^2$.

It is also (in the context of polygonal numbers) called a square number.

For emphasis, such a number is sometimes referred to as a perfect square, but this could cause confusion with the concept of perfect number, so its use is discouraged.

The sequence of square numbers, for $n \in \Z: n \ge 0$ begins:

$0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, \ldots$

This sequence is A000290 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Also see the Odd Number Theorem for a well-known recurrence relation defining the square numbers.

Euclid's Definition

As Euclid defined it:

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

(The Elements: Book VII: Definition $18$)

Abstract Algebra

Let $\left({S, \circ}\right)$ be an algebraic structure.

Let $x \in S$.

Then the square of $x$ is $x \circ x$ and can be written $x^2$.

Squaring

The action of multiplying a number by itself is called squaring, and $x^2$ therefore is usually read $x$ squared.

Geometry

In geometry, a square is a four-sided regular polygon.

The Internal Angles of a Square are right angles.

The Area of a Square is $L^2$ where $L$ is the length of a side of the square.