Standard (mathematics)

This article is about the standard in mathematics. For other uses of the term standard, please see standard (disambiguation).

In mathematics, the word standard is often used in its non-technical sense to indicate the "normally" or "most frequently" used of several options. Words with a similar, but usually more formal and more technical meaning are: normal, natural, canonical. Thus there may be, for instance,

• a "standard" symbol that is usually (but not necessarily always) used, or
• a "standard" way to write an equation that need not be considered as a normal form.

Some very common examples for the use of the word "standard" are:

• The standard basis in d-dimensional real or complex vector spaces (or, more generally, in any d-dimensional vector space K^d over a field K)

is the basis formed by the d d-tuples

$(1,0,\dots ,0),\ (0,1,0,\dots ,0),\cdots ,(0,\dots ,0,1,0),\ (0,\dots ,0,1)$

• In set theory, standard model of the natural numbers usually refers to the the set $\mathbb {N}$ constructed inductively from the empty set.

• The term standard (natural, real, complex, etc.) numbers is used to distinguish the usually used numbers from their nonstandard counterparts.

• In statistics the standard deviation is the most commonly used measure of variation.

• In mathematical physics there is a well-known standard model of particle physics.