Editing Hypercomplex number (section)

== Two-dimensional real algebras ==
'''Theorem:'''<ref name=KS78/>{{rp|14,15}}<ref>{{citation |author-link=Isaak Yaglom |first=Isaak |last=Yaglom |year=1968 |title=Complex Numbers in Geometry |pages=10–14}}</ref><ref>{{citation |editor-first=John H. |editor-last=Ewing |year=1991 |title=Numbers |page=237 |publisher=Springer |isbn=3-540-97497-0}}</ref> Up to isomorphism, there are exactly three 2-dimensional unital algebras over the reals: the ordinary [[complex number]]s, the [[split-complex number]]s, and the [[dual number]]s. In particular, every 2-dimensional unital algebra over the reals is associative and commutative.

Proof: Since the algebra is 2-dimensional, we can pick a basis {{nowrap|{{mset|1, ''u''}}}}.  Since the algebra is [[closure (mathematics)|closed]] under squaring, the non-real basis element ''u'' squares to a linear combination of 1 and ''u'':
: <math>u^2 = a_0 + a_1 u</math>
for some real numbers ''a''<sub>0</sub> and ''a''<sub>1</sub>.

Using the common method of [[completing the square]] by subtracting ''a''<sub>1</sub>''u'' and adding the quadratic complement ''a''{{su|b=1|p=2}}{{nnbsp}}/{{nnbsp}}4 to both sides yields
: <math>u^2 - a_1 u + \frac{1}{4}a_1^2 = a_0 + \frac{1}{4}a_1^2.</math>

Thus <math display="inline">\left(u - \frac{1}{2}a_1\right)^2 = \tilde{u}^2</math> where <math display="inline">\tilde{u}^2~ = a_0 + \frac{1}{4}a_1^2.</math>
The three cases depend on this real value:
* If {{nowrap|1=4''a<sub>0</sub>'' = −''a''<sub>1</sub><sup>2</sup>}}, the above formula yields {{nowrap|1=''ũ''<sup>2</sup> = 0}}. Hence, ''ũ'' can directly be identified with the [[nilpotent]] element <math>\varepsilon</math> of the basis <math>\{ 1, ~\varepsilon \}</math> of the dual numbers.
* If {{nowrap|4''a<sub>0</sub>'' > −''a''<sub>1</sub><sup>2</sup>}}, the above formula yields {{nowrap|''ũ''<sup>2</sup> > 0}}. This leads to the split-complex numbers which have normalized basis <math>\{ 1 , ~j \}</math> with <math>j^2 = +1</math>. To obtain ''j'' from ''ũ'', the latter must be divided by the positive real number <math display="inline">a \mathrel{:=} \sqrt{a_0 + \frac{1}{4}a_1^2}</math> which has the same square as ''ũ'' has.
* If {{nowrap|4''a<sub>0</sub>'' < −''a''<sub>1</sub><sup>2</sup>}}, the above formula yields {{nowrap|''ũ''<sup>2</sup> < 0}}. This leads to the complex numbers which have normalized basis <math>\{ 1 , ~i \}</math> with <math>i^2 = -1</math>. To yield ''i'' from ''ũ'', the latter has to be divided by a positive real number <math display="inline">a \mathrel{:=} \sqrt{\frac{1}{4}a_1^2 - a_0}</math> which squares to the negative of ''ũ''<sup>2</sup>.

The complex numbers are the only 2-dimensional hypercomplex algebra that is a [[Field (mathematics)|field]].
[[Split algebra]]s such as the split-complex numbers that include non-real roots of 1 also contain [[idempotent element|idempotent]]s <math display="inline">\frac{1}{2}(1 \pm j)</math> and [[zero divisor]]s <math>(1 + j)(1 - j) = 0</math>, so such algebras cannot be [[division algebra]]s. However, these properties can turn out to be very meaningful, for instance in representing a [[light cone]] with a [[null cone]].

In a 2004 edition of ''[[Mathematics Magazine]]'' the 2-dimensional real algebras have been styled the "generalized complex numbers".<ref>{{citation |first1=Anthony A. |last1=Harkin |first2=Joseph B. |last2=Harkin |title=Geometry of Generalized Complex Numbers |journal=[[Mathematics Magazine]] |volume=77 |issue=2 |pages=118–129 |year=2004 |doi=10.1080/0025570X.2004.11953236 |s2cid=7837108 |url=http://people.rit.edu/harkin/research/articles/generalized_complex_numbers.pdf}}</ref> The idea of [[cross-ratio]] of four complex numbers can be extended to the 2-dimensional real algebras.<ref>{{citation |first=Sky |last=Brewer |title=Projective Cross-ratio on Hypercomplex Numbers |journal=[[Advances in Applied Clifford Algebras]] |volume=23 |issue=1 |pages=1–14 |year=2013 |doi=10.1007/s00006-012-0335-7 |arxiv=1203.2554|s2cid=119623082 }}</ref>