The norm of a mathematical object is a quantity that in some (possibly abstract) sense describes the length, size, or extent of the object. Norms exist for complex
numbers (the complex modulus, sometimes also
called the complex norm or simply "the norm"), Gaussian
integers (the same as the complex modulus,
but sometimes unfortunately instead defined to be the absolute
square), quaternions (quaternion
norm), vectors (vector
norms), and matrices (matrix
norms). A generalization of the absolute value known as the *p*-adic
norm is also defined.

Norms are variously denoted ,
, , or .
In this work, single bars are used to denote the complex modulus, quaternion
norm, *p*-adic norms, and vector
norms, while the double bar is reserved for matrix norms.

The term "norm" is often used without additional qualification to refer to a particular type of norm (such as a matrix norm
or vector norm). Most commonly, the unqualified term
"norm" refers to the flavor of vector norm
technically known as the L2-norm. This norm is variously
denoted , , or ,
and gives the length of an *n*-vector . It can be computed as

The norm of a complex number, 2-norm of a vector, or 2-norm of a (numeric) matrix is returned by `Norm`[*expr*].
Furthermore, the generalized -norm
of a vector or (numeric) matrix is returned by `Norm`[*expr*,
*p*].

The norm (length) of a vector should not be confused with a normal vector (a vector perpendicular to a surface).