# Square (algebra)

*https://en.wikipedia.org/wiki/Square_(algebra)*

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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 3^{2}, 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

`may be used in place of`

*x***2`.`

*x*^{2}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} = *x*^{2} + 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 x), the square of x is the same as the square of its
additive inverse −*x*. That is, the square function satisfies the identity *x*^{2} = (−*x*)^{2}. This can also be expressed by saying that the square function is an
even function.

## In real numbers

The squaring operation defines a
real function called the **square function** or the **squaring function**. Its
domain is the whole
real line, and its
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 [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 square *x*^{2} of a number *x* is less than x (that is *x*^{2} < *x*) if and only if 0 < *x* < 1, that is, if x belongs to the
open interval (0,1). 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. 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 i, 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 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 l is equal to *l*^{2}. The area depends quadratically on the size: the area of a shape n times larger is *n*^{2} 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.

The square 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 *d*^{2} or *r*^{2}), 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: **v**⋅**v** = v^{2}. This is further generalised to
quadratic forms in
linear spaces via the
inner product. 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 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 square 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**/*p***Z** formed by the numbers modulo an odd
prime number 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 (*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 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 I 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. There are no other idempotents in fields and more generally in
integral domains. However,
the ring of the integers
modulo n has
2^{k} idempotents, where k is the number of distinct
prime factors of 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 numbers, with
bitwise AND as the multiplication operation and bitwise XOR as the addition operation.

In a
totally ordered ring, *x*^{2} ≥ 0 for any x. Moreover, *x*^{2} = 0 if and only if *x* = 0.

In a
supercommutative algebra where 2 is invertible, the square of any *odd* element equals zero.

If *A* is a
commutative semigroup, then one has

In the language of
quadratic 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 by
L. E. Dickson to produce the
octonions out of
quaternions by doubling. The doubling method was formalized by
A. A. Albert who started with the
real number
field ℝ and the square function, doubling it to obtain the
complex number field with quadratic form *x*^{2} + *y*^{2}, and then doubling again to obtain quaternions. The doubling procedure is called the
Cayley–Dickson construction, and has been generalized to form algebras of dimension 2^{n} over a field *F* with involution.

The square function *z*^{2} is the "norm" of the
composition algebra ℂ, where the identity function forms a trivial involution to begin the Cayley–Dickson constructions leading to bicomplex, biquaternion, and bioctonion composition algebras.

The ordinary
complex square function *z*^{2} 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.

The **absolute square** of a complex number is the product *z*^{*} *z* involving its
complex conjugate.^{
[1]}^{
[2]}^{
[3]}^{
[4]}^{
[5]}^{
[6]}^{
[7]}^{
[8]}
It is also known as **modulus squared** or **magnitude squared**, after the
real-value square of the
complex-number modulus (
magnitude or absolute value), |*z*|^{2}.
It equals the sum of real-valued squares of the complex number's
real and imaginary parts, .
It can be generalized to vectors as the *
complex dot product*.

The squared modulus is applied in signal processing, to relate the Fourier transform and the power spectrum, and also in quantum mechanics, relating probability amplitudes and probability densities.

## 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 x_{i} 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

- 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 polynomials as a sum of squares of rational functions
- Square-free polynomial
- Cube (algebra)
- Metric tensor
- Quadratic equation
- Polynomial ring
- Sums of squares (disambiguation page with various relevant links)

### Related identities

- Algebraic (need a commutative ring)

- Difference of two squares
- Brahmagupta–Fibonacci identity, related to complex numbers in the sense discussed above
- Euler's four-square identity, related to quaternions in the same way
- Degen's eight-square identity, related to octonions in the same way
- Lagrange's identity

- Other

### 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

**^**Weisstein, Eric W. "Absolute Square".*mathworld.wolfram.com*.**^**Moore, Thomas (January 9, 2003).*Six Ideas That Shaped Physics: Unit Q - Particles Behaves Like Waves*. McGraw-Hill Education. ISBN 9780072397130 – via Google Books.**^**Blanpied, William A. (September 4, 1969).*Physics: Its Structure and Evolution*. Blaisdell Publishing Company. ISBN 9780471000341 – via Google Books.**^**Greiner, Walter (December 6, 2012).*Quantum Mechanics: An Introduction*. Springer Science & Business Media. ISBN 9783642579745 – via Google Books.**^**Burkhardt, Charles E.; Leventhal, Jacob J. (December 15, 2008).*Foundations of Quantum Physics*. Springer Science & Business Media. ISBN 9780387776521 – via Google Books.**^**Senese, Fred (August 24, 2018).*Symbolic Mathematics for Chemists: A Guide for Maxima Users*. John Wiley & Sons. ISBN 9781119273233 – via Google Books.**^**Steiner, Mark (June 30, 2009).*The Applicability of Mathematics as a Philosophical Problem*. Harvard University Press. ISBN 9780674043985 – via Google Books.**^**Maudlin, Tim (March 19, 2019).*Philosophy of Physics: Quantum Theory*. Princeton University Press. ISBN 9780691183527 – via Google Books.

## 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
- Rajwade, A. R. (1993).
*Squares*. London Mathematical Society Lecture Note Series.**171**. Cambridge University Press. ISBN 0-521-42668-5. Zbl 0785.11022.