# Associative Composition Algebra/Binarions

There are three binarion AC algebras. The division binarions C is the field of complex numbers:

$z=x+yi,\ \ i^{2}=-1,\ \ x,y\in R,\quad z^{*}=x-yi.$ Several academic journals and many university texts are dedicated to the function theory of C, for example the Wikibook Complex Analysis.

As mentioned in the Introduction, C can be taken as unarion AC algebra with * just the identity map. The binarions from this unarion field are bibinarions, also known as bicomplex numbers. For example,

$b=z+hw,\ h^{2}=-1,\quad b^{*}=z-hw.$ Then $N(b)=bb^{*}=z^{2}+w^{2}\in C.$ Sir James Cockle wrote about tessarines in 1848, describing features of this AC algebra to be denoted T. For instance, there is the element $hi\in T\ {\text{satisfying}}\ (hi)^{2}=h^{2}i^{2}=+1.$ The third binarion AC algebra is $D=\{x+yhi:x,y\in R\}\subset T.$ This real plane in T is called the split-complex numbers, or split binarions, in the AC algebra context, but has found many applications and other labels.

Usually hi is written j so the generic split complex number is $z=x+yj,\ \ j^{2}=+1,\ \ z^{*}=x-yj.$ Then in D, $N(z)=zz^{*}=x^{2}-y^{2},$ so that when N(z) is constant z is on a hyperbola. Hence the necessity of the Transcendental paradigm in the description of this AC algebra.

## Division binarions

With $z=x+iy,\ \ zz^{*}=x^{2}+y^{2}=N(z)$  is the square of the Euclidean distance from 0 to z. Furthermore, the vectors from 0 to w and z in C are perpendicular, z ⊥ w, when $zw^{*}+wz^{*}=0.$  These features make C an ideal vehicle for geometric display.

As an example, consider the perpendicularity of the diagonals of a rhombus $\{0,z,w,z+w\},\ \ zz^{*}=ww^{*}.$  One diagonal is z + w and the other is parallel to z – w. They are perpendicular because

$(z+w)(z-w)^{*}+(z-w)(z+w)^{*}=0.$ .

### Möbius transformations

Möbius transformations act on the projective line over division binarions. The points on this line use projective coordinates: (a,b)~(c,d) if there is a non-zero u such that ua=c and ub=d. This binary relation on pairs of division binarions is an equivalence relation, where an equivalence class is written [a:b] for any pair (a,b) in the class.

As projective linear transformations, a Möbius transformation may be written

$[z,w]{\begin{pmatrix}a&c\\b&d\end{pmatrix}}=[az+bw,\ cz+dw].$

The point [z,0] corresponds to a point at infinity with respect to the rest of C represented by [z,w]=[zw−1, 1].

Exercise: Show that the Möbius transformation takes infinity to a/c.

For other points, let w = 1 so

$[z,1]{\begin{pmatrix}a&c\\b&d\end{pmatrix}}=[az+b,\ cz+d]=[{\frac {az+b}{cz+d}},1]$  when z ≠ −d/c.

To avoid singular transformations, adbc is taken to be non-zero. Particular transformations coming under the Möbius umbrella include:

• Case 1: b=c=0, d=1. a>1 magnification, 0<a<1 contraction, a=−1 reflection in 0, aa*=1 rotation
• Case 2: c=0, a=d=1, b=t translation of division binarion plane by t in C
• Case 3: a=d=0, b=c=1 multiplicative inversion of C extended to 0 and infinity.

Note that the rotation z to uz leaves [0,1] and [1,0] fixed. Furthermore, any two distinct points p and q can be placed to these polar opposites by

$[z,1]{\begin{pmatrix}1&1\\-q&-p\end{pmatrix}}=[z-q,\ z-p].$

The image of a third point r cannot be [1,1] since pq, but r can be moved there: Let $w={\frac {r-p}{r-q}}.$  Then

$[r,1]{\begin{pmatrix}w&1\\-wq&-p\end{pmatrix}}=[rw-wq,\ r-p]=[1,1].$

Thus the constructed transformation maps p,q,r to infinity, 0, 1 respectively. Applied to a fourth division binarion z, the image is the cross ratio [z,p,q,r].

Exercises:

1. Construct the transformation taking −i to [1,0], 0 to [i,1] and i to [1,1]. What is the image of the unit disk zz* < 1 under this transformation?
2. Two points in the projective line are fixed with a rotation. Show that a Möbius transformation that leaves three points fixed must be the identity mapping.
3. If μ is a circle or line and g is a Möbius transformation, then μg is a circle or line.
4. If the image of z under the cross ratio transformation of p, q, and r is a real number, then the four points z, p, q, and r lie on a circle or line.

## Bibinarions

The procedure for bibinarions replaces R with C in the construction of real binarions, except that C is deprived of complex conjugation, the conjugation being replaced by the identity mapping. The result of the construction is an algebra of two dimensions over division binarions, (z,w), with conjugation (z,w)* = (z, − w), and $N(z,w)\ =\ z^{2}+w^{2}.$  The algebra has null vectors and is a split AC algebra. Note in particular that (0,1)(0,1) = (−1,0) so that (0,1) is an imaginary unit.

The idea of an algebra with two imaginary units that commute was considered in mid-19th century Britain. Hamilton used a commuting h with his biquaternions. James Cockle saw that the square of the product of these imaginary units was plus one, thus creating "a new imaginary in algebra" as he wrote in Philosophical Magazine in 1848. His use of the letter j,   j2 = +1, has been widely adopted. Although Hamilton provided a vocabulary of vector operations (including the del operator), these explorations preceded set theory, group theory, and the unfolding of mathematical notation. With 1 on the real axis, the two imaginary units, and their product, Cockle's commutative algebra T (tessarines) has a real basis of four elements. By the end of the 19th century tessarines and quaternions were referred to as hypercomplex numbers.

In 1892 Corrado Segre introduced bicomplex number in Mathematische Annalen (v 40: 455 to 67). In 1919 Leonard Dickson explained the development of composition algebras. He was working on the step from division quaternions to octonions (beyond the scope of this text). In 1942 A. A. Albert regularized the science of composition algebras over R and C. Though bibinarions are naturally an algebra over division binarions, the temptation to view it as a real algebra reveals the split binarion plane. Nevertheless, the division binarion basis of the algebra is essential to the construction of biquaternions in the next chapter.