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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 ''x''2. 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 is the quadratic polynomial . One of the important properties of squaring, for numbers as well as in many other mathematical systems, is that (for all numbers ), the square of is the same as the square of its additive inverse . That is, the square function satisfies the identity . This can also be expressed by saying that the squaring function is an even function. == In real numbers == 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 . On the negative numbers, numbers with greater absolute value have greater squares, so squaring is a monotonically decreasing function on . Hence, zero is its global minimum. The only cases where the square of a number is less than occur when , that is, when belongs to an open interval . This implies that the square of an integer is never less than the original number. 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. Zero is the square of only one number, itself. For this reason, it is possible to define the square root function, which associates with a non-negative real number the non-negative number whose square is the original number. No square root can be taken of a negative number within the system of real numbers, because squares of all real numbers are non-negative. The lack of real square roots for the negative numbers can be used to expand the real number system to the complex numbers, by postulating the imaginary unit , which is one of the square roots of −1. The property "every non negative real number is a square" has been generalized to the notion of a real closed field, which is an ordered field such that every non negative element is a square and every polynomial of odd degree has a root. The real closed fields cannot be distinguished from the field of real numbers by their algebraic properties: every property of the real numbers, which may be expressed in first-order logic (that is expressed by a formula in which the variables that are quantified by ∀ or ∃ represent elements, not sets), is true for every real closed field, and conversely every property of the first-order logic, which is true for a specific real closed field is also true for the real numbers. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Square (algebra)」の詳細全文を読む スポンサード リンク
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