Square (algebra)

In some cases when superscripts are not available, as for instance in programming languages or plain text files, the notations x^2 (caret) or x**2 may be used in place of x2.

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.

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 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: v⋅v = v2.

It demonstrates a quadratic relation of the moment of inertia to the size (length).

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.

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

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 procedure is called the Cayley–Dickson construction, and has been generalized to form algebras of dimension 2n over a field F with involution.

, where the identity function forms a trivial involution to begin the Cayley–Dickson constructions leading to bicomplex, biquaternion, and bioctonion composition algebras.

is a twofold cover in the sense that each non-zero complex number has exactly two square roots.

It is easier to compute than the absolute value (no square root), and is a smooth real-valued function.

For complex vectors, the dot product can be defined involving the conjugate transpose, leading to the squared norm.

Squaring is used in statistics and probability theory in determining the standard deviation of a set of values, or a random variable.