Square (arithmetic)

In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a superscript 2; for instance, the square of 3 may be written as 3², which is the number 9. In some cases when superscripts are not available, as for instance in programming languages or plain text files, the notations x^2 or x**2 may be used in place of x².

The adjective which corresponds to squaring is .

The square of an integer may also be called a square number or a perfect square. In algebra, the operation of squaring is often generalized to polynomials, other expressions, or values in systems of mathematical values other than the numbers. For instance, the square of the linear polynomial $x + 1$ is the quadratic polynomial $x 2 + 2 x + 1$ .

One of the important properties of squaring, for numbers as well as in many other mathematical systems, is that (for all numbers $x$ ), the square of $x$ is the same as the square of its additive inverse $- x$ . That is, the square function satisfies the identity $x 2 = (- x) 2$ . This can also be expressed by saying that the squaring function is an even function.

The squaring function preserves the order of positive numbers: larger numbers have larger squares. In other words, squaring is a monotonic function on the interval $[0, +\infty)$ . On the negative numbers, numbers with greater absolute value have greater squares, so squaring is a monotonically decreasing function on $(-\infty,0]$ . Hence, zero is its global minimum. The only cases where the square $x 2$ of a number is less than $x$ occur when $0 < x < 1$ , that is, when $x$ belongs to an open interval $(0,1)$ . This implies that the square of an integer is never less than the original number.

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