Square (algebra)

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5⋅5, or 52 (5 squared), can be shown graphically using a square. Each block represents one unit, 1⋅1, and the entire square represents 5⋅5, or the area of the square.

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 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 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 Template:Mvar), the square of Template:Mvar 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 squaring function is an even function.

In real numbers

y = x2. The graph of a quadratic function has a parabolic shape. The squares of numbers make a power law.

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 Template:Closed-open. On the negative numbers, numbers with greater absolute value have greater squares, so squaring is a monotonically decreasing function on Template:Open-closed. Hence, zero is its global minimum. The only cases where the square x2 of a number is less than Template:Mvar occur when 0 < x < 1, that is, when Template:Mvar belongs to an open interval Template:Open-open. 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 Template:Mvar, 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. The real closed fields can not 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 area: it comes from the fact that the area of a square with sides of length  Template:Mvar is equal to l2. The area depends quadratically on the size: the area of a shape Template:Mvar times larger is n2 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.


Fresnel's zone plates have rings with equally spaced squared distances to the center

The squaring function is related to distance through the Pythagorean theorem and its generalization, the parallelogram law. Euclidean distance is not a smooth function: the 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 d2 or r2), 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: vv = v2. This is further generalised to quadratic forms in linear spaces. 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).

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 square roots.

The notion of squaring is particularly important in the finite fields Z/pZ formed by the numbers modulo an odd prime number Template:Mvar. 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 (p − 1)/2 quadratic residues and exactly (p − 1)/2 quadratic non-residues. The quadratic residues form a group under multiplication. The properties of quadratic residues are widely used in number theory.

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 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 Template:Mvar such that implies . 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. {{safesubst:#invoke:anchor|main}}There are no other idempotents in fields and more generally in integral domains. However, the ring of the integers modulo Template:Mvar has 2k idempotents, where Template:Mvar is the number of distinct prime factors of Template:Mvar. 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 numbers, 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.

In complex numbers and related algebras over the reals


The composition of the tiling Image:ConformId.jpg (understood as a function on the complex plane) with the complex square function

{{#invoke:see also|seealso}} The complex square function z2 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.

{{safesubst:#invoke:anchor|main}}Another, more well known, function is the square of the absolute value | z |2 = z [[complex conjugate|Template:Overline]], which is real-valued. It is very important for quantum mechanics: see probability amplitude and Born rule. Complex numbers form one of four possible Euclidean Hurwitz algebras that are defined with a real quadratic form Template:Mvar; here q(z) = | z |2. In a Euclidean Hurwitz algebra this Template:Mvar equals to the square of the distance to 0 discussed above, and the absolute value | z | can be defined as the (arithmetical) square root of q(z). Multiplicativity of Template:Mvar in these algebras explains (or relies upon) certain algebraic identities (see below).

Other uses

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

Least squares is the standard method used with overdetermined systems.

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 Template:Mvar from the mean  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 the variance, and its square root is the standard deviation. In finance, the volatility of a financial instrument is the standard deviation of its values.

See also

Related identities

Algebraic (need a commutative ring)

Related physical quantities


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
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