Editing Interval (mathematics) (section)

==Topological algebra==
{{more citations needed|section|date=September 2023}}

Intervals can be associated with points of the plane, and hence regions of intervals can be associated with [[region (mathematical analysis)|region]]s of the plane. Generally, an interval in mathematics corresponds to an ordered pair {{math|(''x'', ''y'')}} taken from the [[direct product]] <math>\R \times \R</math> of real numbers with itself, where it is often assumed that {{math|''y'' > ''x''}}. For purposes of [[mathematical structure]], this restriction is discarded,<ref>Kaj Madsen (1979) [https://www.ams.org/mathscinet/pdf/586220.pdf Review of "Interval analysis in the extended interval space" by Edgar Kaucher]{{dead link|date=November 2017 |bot=InternetArchiveBot |fix-attempted=yes }} from [[Mathematical Reviews]]</ref> and "reversed intervals" where {{math|''y'' &minus; ''x'' < 0}} are allowed. Then, the collection of all intervals {{math|[''x'', ''y'']}} can be identified with the [[topological ring]] formed by the [[direct sum of modules#Direct sum of algebras|direct sum]] of <math>\R</math> with itself, where addition and multiplication are defined component-wise.

The direct sum algebra <math>( \R \oplus \R, +, \times)</math> has two [[ideal (ring theory)|ideal]]s, { [''x'',0] : ''x'' ∈ R } and { [0,''y''] : ''y'' ∈ R }. The [[identity element]] of this algebra is the condensed interval {{math|[1, 1]}}. If interval {{math|[''x'', ''y'']}} is not in one of the ideals, then it has [[multiplicative inverse]] {{math|[1/''x'', 1/''y'']}}. Endowed with the usual [[topology]], the algebra of intervals forms a [[topological ring]]. The [[group of units]] of this ring consists of four [[quadrant (plane geometry)|quadrant]]s determined by the axes, or ideals in this case. The [[identity component]] of this group is quadrant I.

Every interval can be considered a symmetric interval around its [[midpoint]].  In a reconfiguration published in 1956 by M Warmus, the axis of "balanced intervals" {{math|[''x'', &minus;''x'']}} is used along with the axis of intervals {{math|[''x'', ''x'']}} that reduce to a point. Instead of the direct sum <math>R \oplus R,</math> the ring of intervals has been identified<ref>[[D. H. Lehmer]] (1956) [https://www.ams.org/mathscinet/pdf/81372.pdf Review of "Calculus of Approximations"]{{dead link|date=November 2017 |bot=InternetArchiveBot |fix-attempted=yes }} from Mathematical Reviews</ref> with the [[hyperbolic number]]s by M. Warmus and [[D. H. Lehmer]] through the identification
:<math>z = \tfrac12(x + y) + \tfrac12(x - y)j,</math>
where <math>j^2 = 1.</math>

This linear mapping of the plane, which amounts of a [[ring isomorphism]], provides the plane with a multiplicative structure having some analogies to ordinary complex arithmetic, such as [[polar decomposition#Alternative planar decompositions|polar decomposition]].