5⋅5, or52 (5 squared), can be shown graphically using asquare. Each block represents one unit,1⋅1, and the entire square represents5⋅5, or the area of the square.
Inmathematics, asquare is the result ofmultiplying anumber by itself. The verb "to square" is used to denote this operation. Squaring is the same asraising to the power 2, and is denoted by asuperscript 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 inprogramming languages orplain text files, the notationsx^2 (caret) orx**2 may be used in place ofx2.The adjective which corresponds to squaring isquadratic.
One of the important properties of squaring, for numbers as well as in many other mathematical systems, is that (for all numbersx), the square ofx is the same as the square of itsadditive inverse−x. That is, the square function satisfies the identityx2 = (−x)2. This can also be expressed by saying that the square function is aneven function.
The graph of the square functiony =x2 is aparabola.
The squaring operation defines areal function called thesquare function or thesquaring function. Itsdomain is the wholereal line, and itsimage 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 amonotonic function on the interval[0, +∞). On the negative numbers, numbers with greater absolute value have greater squares, so the square is a monotonically decreasing function on(−∞,0]. Hence,zero is the (global)minimum of the square function.The squarex2 of a numberx is less thanx (that isx2 <x) if and only if0 <x < 1, that is, ifx belongs to theopen interval(0,1). This implies that the square of an integer is never less than the original numberx.
Every positivereal 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 thesquare 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 ofreal numbers, because squares of all real numbers arenon-negative. The lack of real square roots for the negative numbers can be used to expand the real number system to thecomplex numbers, by postulating theimaginary uniti, 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 areal closed field, which is anordered 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 infirst-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.
There are several major uses of the square function in geometry.
The name of the square function shows its importance in the definition of thearea: it comes from the fact that the area of asquare with sides of length l is equal tol2. The area depends quadratically on the size: the area of a shapen times larger isn2 times greater. This holds for areas in three dimensions as well as in the plane: for instance, the surface area of asphere is proportional to the square of its radius, a fact that is manifested physically by theinverse-square law describing how the strength of physical forces such as gravity varies according to distance.
There are infinitely manyPythagorean 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.
The square function is defined in anyfield orring. An element in the image of this function is called asquare, and the inverse images of a square are calledsquare roots.
The notion of squaring is particularly important in thefinite fieldsZ/pZ formed by the numbers modulo an oddprime numberp. A non-zero element of this field is called aquadratic residue if it is a square inZ/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(p − 1)/2 quadratic residues and exactly(p − 1)/2 quadratic non-residues. The quadratic residues form agroup under multiplication. The properties of quadratic residues are widely used innumber theory.
More generally, in rings, the square function may have different properties that are sometimes used to classify rings.
An element of a ring that is equal to its own square is called anidempotent. In any ring, 0 and 1 are idempotents.There are no other idempotents in fields and more generally inintegral domains. However, the ring of the integersmodulon has2k idempotents, wherek is the number of distinctprime factors of n.A commutative ring in which every element is equal to its square (every element is idempotent) is called aBoolean ring; an example fromcomputer science is the ring whose elements arebinary numbers, withbitwise AND as the multiplication operation and bitwise XOR as the addition operation.
In the language ofquadratic forms, 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 byL. E. Dickson to produce theoctonions out ofquaternions by doubling. The doubling method was formalized byA. A. Albert who started with thereal numberfield and the square function, doubling it to obtain thecomplex number field with quadratic formx2 +y2, and then doubling again to obtain quaternions. The doubling procedure is called theCayley–Dickson construction, and has been generalized to form algebras of dimension 2n over a fieldF with involution.
The square functionz2 is the "norm" of thecomposition algebra, where the identity function forms a trivial involution to begin the Cayley–Dickson constructions leading to bicomplex, biquaternion, and bioctonion composition algebras.
Oncomplex numbers, the square function is a twofoldcover in the sense that each non-zero complex number has exactly two square roots.
The square of theabsolute value of a complex number is called itsabsolute square,squared modulus, orsquared magnitude.[1][better source needed] It is the product of the complex number with itscomplex conjugate, and equals the sum of the squares of the real and imaginary parts of the complex number.
The absolute square of a complex number is always a nonnegative real number, that is zero if and only if the complex number is zero. It is easier to compute than the absolute value (no square root), and is asmoothreal-valued function. Because of these two properties, the absolute square is often preferred to the absolute value for explicit computations and when methods ofmathematical analysis are involved (for exampleoptimization orintegration).
Squares are ubiquitous in algebra, more generally, in almost every branch of mathematics, and also inphysics where manyunits are defined using squares andinverse squares: seebelow.
Squaring is used instatistics andprobability theory in determining thestandard deviation of a set of values, or arandom variable. The deviation of each value xi from themean of the set is defined as the difference. These deviations are squared, then a mean is taken of the new set of numbers (each of which is positive). This mean is thevariance, and its square root is the standard deviation.
