# Complex And Hypercomplex Numbers

You created this dream out of bits and pieces

Filed away in your mind

You’re caught inside a fantasy

But we’ll find the truth inside— Star One,

Cassandra Complex

Scheme’s existing tower defines a level called `complex`

,
which contains the `rational`

numbers and a single imaginary
unit `i`

(AKA `i₀`

), as well as every sum and
product of those numbers (i.e. it is closed under `+`

and
`×`

).

If you’ve never encountered `complex`

numbers before, they
have two important properties which will be relevant to our more-general
framework of GA. Firstly every `complex`

number, no matter
how much we mix and nest sums and products, will always reduce down to a
*single* sum, of the form `(+ A (× B i))`

, where
`A`

and `B`

are `rational`

(and
potentially `zero`

). A `complex`

number will never
require *more* parts, like `(× C i i)`

,
`(× D i i i)`

, etc. since we know that
`(= (× i i) -1)`

(from the above definition of imaginary
units), so all higher powers of `i`

will reduce down to the
`(+ A (× B i))`

form. Secondly there is no meaningful way to
further reduce this sum, so a `complex`

number is always made
of two “parts”; despite being a *single* number!

Racket’s notation for `complex`

numbers is hence
`A+Bi`

(with no spaces); or using `-`

instead of
`+`

when `B`

is negative.

The problem with a dedicated `complex`

level is that it
gives preferential treatment to imaginary units relative to dual and
hyperbolic units. Some might find this desirable, but I’ve decided to
extend my `geometric`

level to encompass
`complex`

, which makes the tower simpler and more
consistent:

### Aside: The
Many Structures Found Inside `geometric`

Whilst `complex`

is certainly a useful type of number, the
reason I don’t want it as a level above `rational`

is there
are other numbers above `rational`

, which are neither above
or below `complex`

.

The and vice “sibling” other non-`rational`

units form
perfectly there are two numbers As mentioned above, these
non-`rational`

units have appeared in various theories over
the course of several centuries. You don’t need to know or care about
these different algebras, since they crop up naturally as patterns in
GA, but since a numerical tower is all about representing such nested
structures it seems prudent to define them for those who care!

#### Complex Numbers

If we extend the `rational`

numbers with a single
imaginary unit, say `i₀`

, we get a self-contained numerical
system called the complex numbers. This has found uses which is useful
in 2D geometry, wave mechanics, electrical engineering, etc. Indeed,
this already exists in the standard Scheme tower, as the
`complex`

level!

#### Quaternions And Hyperimaginary Numbers

Extending `complex`

with another imaginary unit doesn’t
give a useful theory, but having *three* imaginary units
(`i₀`

, `i₁`

and `i₂`

) gives another
useful system called the quaternions; which is especially useful for
describing 3D rotations. The numerical tower in Kawa
Scheme has a `quaternion`

level above
`complex`

, so we’ll do the same!

There is actually an infinite family of such “hypercomplex” theories,
each with twice as many units as the last (when counting all the
imaginary units *and* the unique `rational`

unit
`1`

); I’ll call these *hyperimaginary*, to distinguish
them from the other flavours. The octonions have seven
imaginary units, the sedeneons have fifteen, and so on. However, the
further we go, the less useful those theories become, as they follow
fewer (and weaker) algebraic rules. In particular, everything past the
quaternions violates associativity, which I don’t consider “numeric”
enough to live in our tower!

#### Dual Numbers

If we extend `rational`

with a single *dual* unit,
say `d₀`

, we get the system of dual numbers,
which is useful for e.g. automatic differentiation. Dual numbers don’t
include an imaginary unit, and `complex`

numbers don’t
include a dual unit, so neither is a sub-set of the other. Hence they’ll
need to live side-by-side in our tower!

#### Dual Quaternions

The combination of `quaternion`

and `dual`

forms a useful theory called the dual
quaternions, which are used to describe rotation and translation in
3D space. Thankfully there’s a perfect spot for it in our
unfortunately-wonky tower:

#### Hyperbolic Numbers

We’ve got one flavour of unit left, the hyperbolics, and you may have
guessed that we can *also* extend `rational`

with one
of those, say `h₀`

, to get the hyperbolic
numbers.