Definition: In [mathematics], a [canonical form] of a mathematical [entity] is a [well-defined] [one-to-one] [mapping] on the [set] of entities to a [unique] [representative] of each [equivalence class] of an [equivalence relation].

The [definition] notably contains a plethora of [indefinite article|indefinite articles]. There is no [unique] [canonical form], although there is typically one which meets the subjective definition of "simplest". Specification of a particular canonical form is required before the [idea] can become a [thing].

For example, the set of [rational number|rational numbers] p/q for integers p and non-zero q has the canonical form (relative to the usual equivalence relation, [equality]), given by the reduction of the rational number to lowest terms, by removing the [highest common factor] of p and q. Note that this alone is not sufficient to ensure the canonical form is well-defined. Applying this [algorithm] to 2/4 and (-2)/(-4) does not give identical results, although the two are [equal]. It is [sufficient] to ensure q is [positive] to complete the definition of this canonical form. Furthermore, note that a canonical form is not necessarily an [invariant]. The sum of two rational numbers a/b and c/d, (ad+bc)/cd is not necessarily in the canonical form, even if the source [operand|operands] were, such as 1/3+1/6 = 9/18 = 1/2.

Once a canonical form is defined, their usefulness stems from the observation that any two entities are [equivalent] if their canonical forms are [identical]. Typically, if objects are reduced to a particular canonical form, operations on and comparisons of those objects are far simpler, a principle that extends outside of mathematics. It is much easier to work with [measurement|measurements] that are in a [consistent] set of [units], and failure to do so may result in [Mars Climate Orbiter|disaster]. [Case sensitivity in computing] is an area where canonicalization can ensure names, such as [URI|uniform resource identifiers], refer to the same entity even though the [binary] [representation] of the two names may differ. Technologies such as [Unicode] can make canonicalization highly [nontrivial], but it is essential before operations such as [comparison] can effectively be used. Incorrect canonicalization can lead to [Nimda|undesirable results].

Canonical form (idea)