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:
number
┌─┐ ┌──┐ ┌─┐
│ └─┘ └─┘ │
│geometric |
├──────────┤
│ rational │
├──────────┤
│ integer │
────┼──────────┼────
│ natural │
├──────────┤
│ zero │
└──────────┘complex numbers.
Aside: The
Many Structures Found Inside geometric
geometric
═════════════════════
║dual-quaternion║ hb │
├────╥──────────╢ yo │
│ ║quaternion║ pl │
│dual╟──────────╢ ei │
│ ║ complex ║ rc │
├────╨──────────╨────┤
╲ rational ┌╯
├──────────┬──────╯
│ integer │
────┼──────────┼────────────
│ natural │
├──────────┤
│ zero │
└──────────┘hyperbolic numbers contain rational, and
are contained in geometric; but are distinct from
dual, complex and their extensions
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!
number
┌─┐ ┌──┐ ┌─┐
│ └─┘ └─┘ │
│geometric |
├──────────┤
│quaternion│
├──────────┤
│ complex │
├──────────┤
│ rational │
├──────────┤
│ integer │
────┼──────────┼────
│ natural │
├──────────┤
│ zero │
└──────────┘quaternion level contains complex, and is
contained by geometric
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!
number
┌─┐ ┌──┐ ┌─┐
│ └─┘ └─┘ │
│geometric |
┌─┴──╥──╳───╳┴──┐
│ ║quaternion│
│dual╟──────────┤
│ ║ complex │
└┬┬╳─╨───────┬┬─┘
\│ rational ├╯
├──────────┤
│ integer │
────┼──────────┼────
│ natural │
├──────────┤
│ zero │
└──────────┘dual numbers requires bodging our tower, so multiple
“levels” can occur at the same height!
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:
number
┌─┐ ┌──┐ ┌─┐ ╔╦╕
│ └─┘ └─┘ │ ╢╟├╯
│geometric ╞╧╣╚╡
┌┴──────────┴─╨─┤
│dual-quaternion│
├────╥──────────┤
│ ║quaternion│
│dual╟──────────┤
│ ║ complex │
└┬┬╳─╨───────┬┬─┘
\│ rational ├╯
├──────────┤
│ integer │
────┼──────────┼────
│ natural │
├──────────┤
│ zero │
└──────────┘dual-quaternion level combines values from both
dual and quaternion
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.