View source for Square (algebra)

{{short description|Result of multiplying a number, or other expression, by itself}}
{{Redirect|²|typography of superscripts|subscript and superscript}}
{{no footnotes|date=August 2015}}
[[Image:Five Squared.svg|thumb|right|168px<!-- at 160px and 200px lines render with unequal widths 
-->|{{math|5&sdot;5}}, or {{math|5<sup>2</sup>}} (5 squared), can be shown graphically using a [[square]]. Each block represents one unit, {{math|1&sdot;1}}, and the entire square represents {{math|5&sdot;5}}, or the area of the square.]]
In [[mathematics]], a '''square''' is the result of [[multiplication|multiplying]] a [[number]] by itself. The verb "to square" is used to denote this operation. Squaring is the same as [[exponentiation|raising to]] the power&nbsp;[[2 (number)|2]], and is denoted by a [[superscript]] 2; for instance, the square of 3 may be written as 3<sup>2</sup>, which is the number 9.
In some cases when superscripts are not available, as for instance in [[programming language]]s or [[plain text]] files, the notations <kbd>''x''^2</kbd> or <kbd>''x''**2</kbd> may be used in place of <kbd>''x''<sup>2</sup></kbd>.

The adjective which corresponds to squaring is ''[[wikt:quadratic|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 [[polynomial]]s, other [[expression (mathematics)|expressions]], or values in systems of mathematical values other than the numbers. For instance, the square of the [[linear function (calculus)|linear polynomial]] {{math|''x'' + 1}} is the [[quadratic polynomial]] {{math|1=(''x''+1)<sup>2</sup> = ''x''<sup>2</sup> + 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 {{mvar|x}}), the square of {{mvar|x}} is the same as the square of its [[additive inverse]] {{math|&minus;''x''}}. That is,  the square function satisfies the identity {{math|1=''x''<sup>2</sup> = (−''x'')<sup>2</sup>}}. This can also be expressed by saying that the square function is an [[even function]].

== In real numbers ==
[[Image:Parabola2.svg|thumb|240px|The graph of the square function {{math|1=''y'' = ''x''<sup>2</sup>}} is a [[parabola]].]]
The squaring operation defines a [[real function]] called the '''{{vanchor|square function}}''' or the '''{{vanchor|squaring function}}'''. Its [[domain of a function|domain]] is the whole [[real line]], and its [[image (function)|image]] 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 {{closed-open|0, +∞}}. On the negative numbers, numbers with greater absolute value have greater squares, so the square is a monotonically decreasing function on {{open-closed|−∞,0}}. Hence, [[zero]] is the (global) [[minimum]] of the square function.
The square {{math|''x''<sup>2</sup>}} of a number {{math|''x''}} is less than {{mvar|x}} (that is {{math|''x''<sup>2</sup> < ''x''}}) if and only if {{math|0 < ''x'' < 1}}, that is, if {{mvar|x}} belongs to the [[open interval]] {{open-open|0,1}}. This implies that the square of an integer is never less than the original number {{mvar|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. 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 number]]s, 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 number]]s, by postulating the [[imaginary unit]] {{mvar|i}}, which is one of the square roots of&nbsp;&minus;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.

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

The name of the square function shows its importance in the definition of the [[area]]:  it comes from the fact that the area of a [[square]] with sides of length &nbsp;{{mvar|l}} is equal to {{math|''l''<sup>2</sup>}}. The area depends quadratically on the size: the area of a shape {{mvar|n}}&nbsp;times larger is {{math|''n''<sup>2</sup>}}&nbsp;times greater. This holds for areas in three dimensions as well as in the plane: for instance, the surface area of a [[sphere]] 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.

{{anchor|r²}}[[Image:Zonenplatte Cosinus.png|thumb|right|Fresnel's [[zone plate]]s have rings with [[arithmetic progression|equally spaced]] squared distances to the center]]
The square function is related to [[distance]] through the [[Pythagorean theorem]] and its generalization, the [[parallelogram law]]. [[Euclidean geometry|Euclidean]] distance is not a [[smooth function]]: the [[graph of a function of two variables|three-dimensional graph]] of distance from a fixed point forms a [[cone]], with a non-smooth point at the tip of the cone. However, the square of the distance (denoted {{math|''d''<sup>2</sup>}} or {{math|''r''<sup>2</sup>}}), which has a [[paraboloid]] as its graph, is a smooth and [[analytic function]]. The [[dot product]] of a [[Euclidean vector]] with itself is equal to the square of its length: {{math|1='''v'''&sdot;'''v''' = v<sup>2</sup>}}. This is further generalised to [[quadratic form]]s in [[linear space]]s. The [[inertia tensor]] in [[mechanics]] is an example of a quadratic form. It demonstrates a quadratic relation of the [[moment of inertia]] to the size ([[length]]).

There are infinitely many [[Pythagorean triple]]s, 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 square function is defined in any [[field (mathematics)|field]] or [[ring (mathematics)|ring]]. An element in the image of this function is called a ''square'', and the inverse images of a square are called ''[[square root]]s''.

The notion of squaring is particularly important in the [[finite field]]s  '''Z'''/''p'''''Z''' formed by the numbers modulo an odd [[prime number]] {{mvar|p}}. A non-zero element of this field is called a [[quadratic residue]] if it is a square in '''Z'''/''p'''''Z''', 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 {{math|(''p'' − 1)/2}} quadratic residues and exactly {{math|(''p'' − 1)/2}} quadratic non-residues. The quadratic residues form a [[group (mathematics)|group]] under multiplication. The properties of quadratic residues are widely used in [[number theory]].

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

Zero may be the square of some non-zero elements. A [[commutative ring]] such that the square of a non zero element is never zero is called a [[reduced ring]]. More generally, in a commutative ring, a [[radical ideal]] is an ideal&nbsp;{{mvar|I}} such that <math>x^2 \in I</math> implies <math>x \in I</math>. 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 [[idempotent]]. In any ring, 0 and 1 are idempotents. {{anchor|integral domains}}There are no other idempotents in fields and more generally in [[integral domain]]s. However, 
the ring of the integers [[modular arithmetic|modulo]]&nbsp;{{mvar|n}} has [[power of two|{{math|2<sup>''k''</sup>}}]] idempotents, where {{mvar|k}} is the number of distinct [[integer factorization|prime factors]] of&nbsp;{{mvar|n}}.
A commutative ring in which every element is equal to its square (every element is idempotent) is called a [[Boolean ring]]; an example from [[computer science]] is the ring whose elements are [[binary number]]s, with [[Bitwise operation|bitwise AND]] as the multiplication operation and bitwise XOR as the addition operation.

In a [[totally ordered ring]] {{math|''x''<sup>2</sup> ≥ 0}} for any {{mvar|x}}. Moreover, {{math|1=''x''<sup>2</sup> = 0 ⇔ ''x'' = 0}}.

In a [[supercommutative algebra]] where 2 is invertible, the square of any ''odd'' element equals to zero.

If ''A'' is a [[commutative semigroup]], then one has
:<math>\forall x, y \isin A \quad (xy)^2 = xy xy = xx yy = x^2 y^2  .</math>
In the language of [[quadratic form]]s, this equality says that the square function is a "form permitting composition". In fact, the square 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 [[octonion]]s out of [[quaternion]]s by doubling. The doubling method was formalized by [[A. A. Albert]] who started with the [[real number]] [[field (mathematics)|field]] ℝ and the square function, doubling it to obtain the [[complex number]] field with quadratic form {{math|''x''<sup>2</sup> + ''y''<sup>2</sup>}}, and then doubling again to obtain quaternions. The doubling procedure is called the [[Cayley–Dickson process]] and the structures produced are [[composition algebra]]s.

The square function can be used{{how|date=September 2019}} 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==
{{anchor|In complex analysis}}
{{see also|Exponentiation #Powers of complex numbers}}
The [[complex numbers|complex]] square function&nbsp;{{math|''z''<sup>2</sup>}} is a twofold cover of the [[complex plane]], such that each non-zero complex number has exactly two square roots. This map is related to [[parabolic coordinates]].<!-- unfortunately, incompatible coefficients and orientation conventions hinder a simple explanation such as σ+iτ → (σ,τ)-parabolic -->

== Other uses ==
Squares are ubiquitous in algebra, more generally, in almost every branch of mathematics, and also in [[physics]] where many [[unit of measurement|units]] are defined using squares and [[multiplicative inverse|inverse]] squares: see [[#Related physical quantities|below]].

[[Least squares]] is the standard method used with [[overdetermined system]]s.

Squaring is used in [[statistics]] and [[probability theory]] in determining the [[standard deviation]] of a set of values, or a [[random variable]]. The deviation of each value&nbsp;{{mvar|x<sub>i</sub>}} from the [[mean]]&nbsp;<math>\overline{x}</math> of the set is defined as the difference <math>x_i - \overline{x}</math>. These deviations are squared, then a mean is taken of the new set of numbers (each of which is positive). This mean is the [[variance]], and its square root is the standard deviation. In [[finance]], the [[volatility (finance)|volatility]] of a financial instrument is the standard deviation of its values.

==See also==
<!--
* [[von Neumann regular ring]], a ring such that every ideal is generated by an idempotent
 too remote from the topic? -->
* [[Exponentiation by squaring]]
* [[Polynomial SOS]], the representation of a non-negative polynomial as the sum of squares of polynomials
* [[Hilbert's seventeenth problem]], for the representation of [[positive polynomial]]s as a sum of squares of [[rational function]]s
* [[Square-free polynomial]]
* [[Cube (algebra)]]
* [[Metric tensor]]
* [[Quadratic equation]]
* [[Polynomial ring]]
* [[Sums of squares (disambiguation)|Sums of squares]] (disambiguation page with various relevant links)

=== Related identities ===
; Algebraic<span style="font-weight:400"> (need a [[commutative ring]])</span>:
* [[Difference of two squares]]
* [[Brahmagupta–Fibonacci identity]], related to complex numbers [[#.7Cz.7C²|in the sense discussed above]]
* [[Euler's four-square identity]], related to [[quaternion]]s in the same way
* [[Degen's eight-square identity]], related to [[octonion]]s in the same way
* [[Lagrange's identity]]
; Other
* [[Pythagorean trigonometric identity]]
* [[Parseval's identity]]

=== Related physical quantities ===
* [[acceleration]], 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)
* [[kinetic energy]] (quadratic dependence on velocity)
* [[specific energy]], a (square velocity)-dimensioned quantity

== Footnotes ==
{{reflist}}

==Further reading==
* Marshall, Murray Positive polynomials and sums of squares. Mathematical Surveys and Monographs, 146. American Mathematical Society, Providence, RI, 2008. xii+187 pp. {{isbn|978-0-8218-4402-1}}, {{isbn|0-8218-4402-4}}
* {{cite book | title=Squares | volume=171 | series=London Mathematical Society Lecture Note Series | first=A. R. | last=Rajwade | publisher=[[Cambridge University Press]] | year=1993 | isbn=0-521-42668-5 | zbl=0785.11022 }}

[[Category:Algebra]]
[[Category:Elementary arithmetic]]
[[Category:Exponentials|2]]
[[Category:Squares in number theory]]
[[Category:Unary operations]]