Square (algebra)
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In
The square of an
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
In real numbers
The squaring operation defines a
The square function preserves the order of positive numbers: larger numbers have larger squares. In other words, the square is a
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
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 l2. The area depends quadratically on the size: the area of a shape n 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.
The square function is related to
The
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/pZ 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/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 square function may have different properties that are sometimes used to classify rings.
Zero may be the square of some non-zero elements. A
An element of a ring that is equal to its own square is called an
In a
In a supercommutative algebra where 2 is invertible, the square of any odd element equals zero.
If A is a
In the language of
The square function z2 is the "norm" of the composition algebra , where the identity function forms a trivial involution to begin the Cayley–Dickson constructions leading to bicomplex, biquaternion, and bioctonion composition algebras.
In complex numbers
On complex numbers, the square function is a twofold cover in the sense that each non-zero complex number has exactly two square roots.
The square of the
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 a
For
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 xi 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.
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.
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 0-8218-4402-4
- Rajwade, A. R. (1993). Squares. London Mathematical Society Lecture Note Series. Vol. 171. Zbl 0785.11022.