Definition:Square

Algebra
Let $$x$$ be a number.

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

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

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

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.