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Concept
Square (algebra)
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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Related concepts (24)
Quadratic function
In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before the 20th century, the distinction was unclear between a polynomial and its associated polynomial function; so "quadratic polynomial" and "quadratic function" were almost synonymous. This is still the case in many elementary courses, where both terms are often abbreviated as "quadratic".
Square number
In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals 32 and can be written as 3 × 3. The usual notation for the square of a number n is not the product n × n, but the equivalent exponentiation n2, usually pronounced as "n squared". The name square number comes from the name of the shape. The unit of area is defined as the area of a unit square (1 × 1).
Euler's four-square identity
In mathematics, Euler's four-square identity says that the product of two numbers, each of which is a sum of four squares, is itself a sum of four squares. For any pair of quadruples from a commutative ring, the following expressions are equal: Euler wrote about this identity in a letter dated May 4, 1748 to Goldbach (but he used a different sign convention from the above). It can be verified with elementary algebra. The identity was used by Lagrange to prove his four square theorem.
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