Summary
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 (caret) or x**2 may be used in place of x2. The adjective which corresponds to squaring is quadratic. 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 + 1)2 = x2 + 2x + 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 x2 = (−x)2. This can also be expressed by saying that the square function is an even function. The squaring operation defines a real function called the or the . Its domain is the whole real line, and its is the set of nonnegative real numbers. The square function preserves the order of positive numbers: larger numbers have larger squares. In other words, the square is a monotonic function on the interval . On the negative numbers, numbers with greater absolute value have greater squares, so the square is a monotonically decreasing function on . Hence, zero is the (global) minimum of the square function. The square x2 of a number x is less than x (that is x2 < x) if and only if 0 < x < 1, that is, if x belongs to the open interval . This implies that the square of an integer is never less than the original number x. Every positive real number is the square of exactly two numbers, one of which is strictly positive and the other of which is strictly negative.
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