Standard (mathematics)

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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

${\displaystyle (1,0,\dots ,0),\ (0,1,0,\dots ,0),\cdots ,(0,\dots ,0,1,0),\ (0,\dots ,0,1)}$

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