**Hyperbolic Numbers**

Define
the

*dual*, or*hyperbolic*numbers, as number pairs with pairwise operations:
(a;b)*(c;d)
= (a*c;
b*d)

1
/(c;d) = (1/c; 1/d)

(a;b)+(c;d)
= (a+c;
b+d)

-(a;b) = (-a: -b)

0 = (0; 0)

1 = (1; 1)

r = (r; r) for
any real number r

e = (1;-1) this
is the

*dual unit*.
This
system is a ring; * and + are commutative, associative, have identities 0 and
1, inverses –x, and sometimes 1/x. Reciprocal fails for the non-zero numbers
(a;0) and (0;b).

The
dual numbers can be written in terms of 1 and e:

(a;b) = (a+b)/2 + e
(a-b)/2

Written
with e, dual numbers have these laws:

(x +
ey) + (z + ew) = (x+z)
+ e(y+w)

e

^{2}= 1
(x +
ey) * (z + ew) = (xz+yw)
+ e(xw+yz)

1 / (x
+ ey) = (x/(x

^{2}-y^{2})) - e (y/(x^{2}-y^{2})) if x^{2}-y^{2}0
These
rules resemble the rules for complex multiplication, with some signs reversed.
The analog to cis in complex numbers is hyperbolic cis, or ‘chesh’:

chesh(t) = cosh(t) + e
sinh(t)

This
rule applies:

chesh(a)
* chesh(b) = chesh(a+b)

And
just as cis(a)*z is a rotation of z in the complex plane, so too is chesh(a)*z
a Lorentz transformation of z in the hyperbolic plane.

Each of
the three arithmetics on triple ratios is isomorphic to the dual numbers. The
e’s correspond to negatives in the other two arithmetics. For instance, in
arithmetic 3,

x +

_{3}(-1_{3})y is subtraction: x -_{3}y;
but x +

_{3}(-1_{1})y is a dual number;
as is its conjugate x +

_{3}(-1_{2})y = x -_{3}(-1_{1})y
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