---
title: 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
β 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 `iβ`), as well as every
sum and product of those numbers (i.e. it is closed under `+` and `Γ`).
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 (Γ B i))`, where
`A` and `B` are `rational` (and potentially `zero`). A `complex` number will
never require *more* parts, like `(Γ C i i)`, `(Γ D i i i)`, etc. since we know
that `(= (Γ i i) -1)` (from the above definition of imaginary units), so all
higher powers of `i` will reduce down to the `(+ A (Γ 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 `iβ`, 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 (`iβ`, `iβ` and `iβ`) gives another useful
system called the quaternions; which is especially useful for describing 3D
rotations. The numerical tower in [Kawa
Scheme](https://www.gnu.org/software/kawa/Quaternions.html) 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](https://en.wikipedia.org/wiki/Octonion) 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 `dβ`, we get the system
of [dual numbers](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/Dual_quaternion), 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 `hβ`, to get
[the hyperbolic numbers](https://en.wikipedia.org/wiki/Split-complex_number).