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In , a square is the result of a by itself. The verb "to square" is used to denote this operation. Squaring is the same as the power 2, and is denoted by a 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 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 may also be called a or a perfect square. In , the operation of squaring is often generalized to , 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 . That is, the square function satisfies the identity . This can also be expressed by saying that the squaring function is an .


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, is the (global) of the square function. The square of a number is less than (that is ) if and only if , that is, if belongs to the . This implies that the square of an integer is never less than the original number .

Every positive 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 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 , because squares of all real numbers are . The lack of real square roots for the negative numbers can be used to expand the real number system to the , by postulating the , 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 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.


In geometry
There are several major uses of the squaring function in geometry.

The name of the squaring function shows its importance in the definition of the : it comes from the fact that the area of a with sides of length   is equal to . The area depends quadratically on the size: the area of a shape  times larger is  times greater. This holds for areas in three dimensions as well as in the plane: for instance, the surface area of a is proportional to the square of its radius, a fact that is manifested physically by the inverse-square law describing how the strength of physical forces such as gravity varies according to distance.

The squaring function is related to through the Pythagorean theorem and its generalization, the parallelogram law. Euclidean distance is not a : the three-dimensional graph of distance from a fixed point forms a , with a non-smooth point at the tip of the cone. However, the square of the distance (denoted or ), which has a as its graph, is a smooth and analytic function. The of a with itself is equal to the square of its length: . This is further generalised to in . The in is an example of a quadratic form. It demonstrates a quadratic relation of the moment of inertia to the size ().

There are infinitely many Pythagorean triples, sets of three positive integers such that the sum of the squares of the first two equals the square of the third. Each of these triples gives the integer sides of a right triangle.


In abstract algebra and number theory
The squaring function is defined in any field or ring. An element in the image of this function is called a square, and the inverse images of a square are called .

The notion of squaring is particularly important in the Z/ p Z formed by the numbers modulo an odd . A non-zero element of this field is called a quadratic residue if it is a square in Z /pZ, and otherwise, it is called a quadratic non-residue. Zero, while a square, is not considered to be a quadratic residue. Every finite field of this type has exactly quadratic residues and exactly quadratic non-residues. The quadratic residues form a group under multiplication. The properties of quadratic residues are widely used in .

More generally, in rings, the squaring function may have different properties that are sometimes used to classify rings.

Zero may be the square of some non-zero elements. A such that the square of a non zero element is never zero is called a . More generally, in a commutative ring, a is an ideal  such that x^2 \in I implies x \in I. Both notions are important in algebraic geometry, because of Hilbert's Nullstellensatz.

An element of a ring that is equal to its own square is called an . In any ring, 0 and 1 are idempotents. There are no other idempotents in fields and more generally in . However, the ring of the integers modulo  has idempotents, where is the number of distinct prime factors of . A commutative ring in which every element is equal to its square (every element is idempotent) is called a ; an example from is the ring whose elements are , with bitwise AND as the multiplication operation and bitwise XOR as the addition operation.

In a supercommutative algebra (away from 2), the square of any odd element equals to zero.

If A is a commutative , then one has

\forall x, y \isin A \quad (xy)^2 = xy xy = xx yy = x^2 y^2 .
In the language of , this equality says that the squaring function is a "form permitting composition". In fact, the squaring function is the foundation upon which other quadratic forms are constructed which also permit composition. The procedure was introduced by L. E. Dickson to produce the out of by doubling. The doubling method was formalized by A. A. Albert who started with the field ℝ and the squaring function, doubling it to obtain the field with quadratic form x2 + y2, and then doubling again to obtain quaternions. The doubling procedure is called the Cayley–Dickson process and the structures produced are composition algebras.

The squaring function can be used with ℂ as the start for another use of the Cayley–Dickson process leading to bicomplex, biquaternion, and bioctonion composition algebras.


In complex numbers and related algebras over the reals
The square function  is a twofold cover of the , such that each non-zero complex number has exactly two square roots. This map is related to parabolic coordinates.


Other uses
Squares are ubiquitous in algebra, more generally, in almost every branch of mathematics, and also in where many units are defined using squares and inverse squares: see below.

is the standard method used with overdetermined systems.

Squaring is used in and probability theory in determining the standard deviation of a set of values, or a . The deviation of each value  from the  \overline{x} of the set is defined as the difference x_i - \overline{x}. These deviations are squared, then a mean is taken of the new set of numbers (each of which is positive). This mean is the , and its square root is the standard deviation. In , the volatility of a financial instrument is the standard deviation of its values.


See also
  • Exponentiation by squaring
  • , the representation of a non-negative polynomial as the sum of squares of polynomials
  • Hilbert's seventeenth problem, for the representation of positive polynomials as a sum of squares of rational functions
  • Square-free polynomial
  • Cube (algebra)
  • Quadratic equation
  • Sums of squares (disambiguation page with various relevant links)


Related identities
Algebraic (need a )
  • Difference of two squares
  • Brahmagupta–Fibonacci identity, related to complex numbers in the sense discussed above
  • Euler's four-square identity, related to in the same way
  • Degen's eight-square identity, related to in the same way
  • Lagrange's identity
Other
  • Pythagorean trigonometric identity
  • Parseval's identity


Related physical quantities
  • , length per square time
  • cross section (physics), an area-dimensioned quantity
  • coupling constant (has square charge in the denominator, and may be expressed with square distance in the numerator)
  • (quadratic dependence on velocity)
  • , a (square velocity)-dimensioned quantity


Footnotes

Further reading
  • Marshall, Murray Positive polynomials and sums of squares. Mathematical Surveys and Monographs, 146. American Mathematical Society, Providence, RI, 2008. xii+187 pp. ,
  • (1993). 9780521426688, Cambridge University Press.

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