A Few Hypercomplex Numbers

Adam's Apples

Adam learned to count sitting at home. Various number systems occurred to him, each one seeming to answer still more new questions...

Initially the natural numbers N (1, 2, 3 and their friends) arrived, letting him answer stuff like "How many apples should I bring home, to avoid an argument?" Later, zero was added, giving him the whole numbers, and answering "How many of a surplus does that leave for unexpected guests?" Then negative numbers extended this arrangement to the integers Z; now he could keep track of apple debts as well as stocks.

Naturally, all of this led to apple shortages, as others began to stockpile; soon he was looking at puzzles like "There's just one apple, and two of us; how many apples each can we have?" Hence the concept of a new type of number, expressed as a sequence of two or more other, pre-existing numbers: the ratios, or rational numbers Q, combining for example 1 and 2 into the single new number ½. Yet still he couldn't answer posers like "What length's the diagonal of my mile-square orchard?" Adding radicals like √2 to obtain the algebraic numbers, still he would find other measurements, like the circumference of a two- or three-inch diameter apple, impossible to pin down.

R - The Reals

For once he got lucky, and defined a (real) transcendental number as anything that wasn't already on the line! Finally, he had the continuum of the real numbers R stretching from -∞ on the left, past his nose, and onward to +∞ on the right. He even learned enough about projective geometry to include ∞ itself, when context called for such extremes.

Everything after N and up to this point was an attempt to solve an ever growing set of equations; to answer an ever expanding realm of questions. Suppose that x = 1. Then some of these questions were,

• What's x - x? - Add zero; get the whole numbers.
• What's x - 2? - Add negatives; get the integers Z.
• What's x / (x+1)? - Add fractions; get the rationals Q.
• What's y, if y² = x + 1? - Add radicals; get the algebraics.
• What's an apple's circumference, if x is its diameter? - Add everything else on the line; get the reals R.

But damn, he thought, damn and stop me if you've heard this one before, but damn if it doesn't all go serpent shaped again, just as soon as I leave the familiar domain of apple accountancy! Here's one example still without an answer:

• What's x, if x² + 1 = 0?

But all of that was quite some time ago. The answer lay nearby, but in another dimension; in this next domain...

C - The Complex Numbers

Just give this elusive result a whole new name, and call it a number. The new name is i; one example of a square root of minus one. Another such example is -i. Now, since i is a number, it can be multiplied it by any real number y; so doing produces an imaginary number, iy. Also, any real number x can then be added to this new thing, making a complex number, x + iy.

This new number i has effectively generated a second copy of the real number line. The new line is conventionally shown at right angles to the original, intersecting at zero in an Argand Diagram (see figure, from Wikipedia). The two lines define a complex plane. So whereas real numbers were represented by points on a line, these newfangled complex numbers are instead points on a plane.

Complex numbers have proved to be insanely useful in various mathematical, scientific and engineering disciplines. At their basics, complex addition and multiplication are used to represent and to compose together planar transformations. For example, multiplication by i results in a 90° anticlockwise rotation. So, how about adding more dimensions? Can we solve still more equations, by adding in still more crazy so-called numbers?

H - The Quaternions

What about this situation: find p and q, where

• (pq - qp)² = -4

Obviously this equation has no solution in the real domain (where no square is negative), nor even the complex one. Multiplication of complex numbers is still commutative; for any real or complex p and q, we have (pq - qp) = 0, and that's never going to square to anything other than more zero. Yet believe it or not, if we now perform exactly the same trick as before, namely introducing a brand new square root of -1 into our system, then we can immediately find multiple solutions to this very odd equation.

The new guy is a completely independent square root of -1, distinct from both i and -i. Let's call him j. Now just as the real and imaginary lines generate the complex plane C, so too do i and j generate a plane. But multiplication in this plane is a little different from multiplication in C. To multiply i and j, you have to imagine the vector i "rotating" towards j, and by the rules of vector multiplication (the so-called cross product), generating a third vector ij orthogonal to both of these, perhaps in the direction that a right handed screw would be driven by such a rotation. But remember, the real line is already orthogonal to both i and j. This new vector ij can't be real; for then j himself would be merely imaginary to begin with, and not an entirely new kind of number after all.

No, i and j between them have to generate an entirely new direction ij, simultaneously orthogonal to both of these and to the real line. In fact this ij himself turns out to be yet another distinct square root of -1. As William Hamilton famously discovered in 1843: during the attempt to break out of two dimensions and into three, we're instead summarily deposited in the land of four! And by the preceding analogy with vector spaces, rotating in the opposite direction, notice this anticommutativity:

ji = -ij.

Numbers of the form a + bi + cj + dij (where a, b, c and d are real) are termed the quaternions, H. In general, if p and q are quaternions, then pq ≠ qp; they're not commutative.

Incidentally, don't take away the impression that these unit vectors, their products, and their negatives, are the only square roots of -1 in the new system. That may have been true in the complex domain, where only two numbers, the discrete ±i, held that distinction. But in the 3D vector subspace of the quaternions, i.e. the space remaining upon removal of the real axis, there's an uncountably infinite number of √-1. Every point on the surface of the unit sphere, a = 0, b² + c² + d² = 1, qualifies.

Returning to the equation at the head of this section, let's now set

p = i, q = j

then

(pq - qp)² = (ij - ji)² = (ij - (-ij))² = (2ij)² = 4(ij)² = 4(-1) = -4

So p = i, q = j is one solution to the very odd equation; and there are others.

Applications of the non-commutative quaternions were once much more numerous, but they have fallen out of favour, now displaced almost everywhere by matrix and vector algebras. Their one current party trick is modelling the camera in 3D computer games and other simulations, where they are less prone to the problems of interpolation and gimbal lock suffered by the alternatives. So we can at least claim them as a successful extension of the transformational complex numbers, from the plane into 3D and 4D spatial rotations (which are also non-commutative).

O - The Octonions

Now, what about this situation: find p, q and r, where

• ((pq)r - p(qr))² = -4
  

Say what you will about the non-commutative quaternions, at least those guys associated, just like the complex and the real: for any p, q and r, we had ((pq)r - p(qr)) = 0. Please struggle to contain your amazement as I reveal to you now, that by the simple expedient of adding still another independent square root of -1 to our system, we can immediately find multiple solutions to this new and extremely odd equation.

The drill is the same as before. Our new root k is orthogonal to all of i, j, ij, and the real line. So too are his uninvited pals, namely ik, jk, and ijk. That's a total of eight independent dimensions (the number doubles with every brand new √-1 added). Seven of these are imaginary, so our basis includes seven square roots of -1. The seven points and directed lines of the Fano Plane (below) offer a mnemonic to help us multiply these. Example: the central vertical line tells us that

• (jk)(ik) = ij, and (ik)(jk) = -ij;
  
• (ik)(ij) = jk, and (ij)(ik) = -jk;
  
• (ij)(jk) = ik, and (jk)(ij) = -ik; and so on.
  

The Fano Plane mnemonic for octonion multiplication

adapted from http://en.wikipedia.org/wiki/File:FanoMnemonic.PNG

The bottom line in particular makes this statement of antiassociativity:

i(jk) = -(ij)k.

Numbers of the form a + bi + cj + dij + ek + fik + gjk + hijk (where a, b, c, d, e, f, g and h are real) are termed the octonions, O. In general, if p, q and r are octonions, then (pq)r ≠ p(qr); they're not associative. Returning to the equation at the head of this section, let's now set

p = i, q = j, r = k

then

((pq)r - p(qr))² = ((ij)k - i(jk))² =(ijk + ijk)² = (2ijk)² = 4(ijk)² = 4(-1) = -4

So p = i, q = j, r = k is one solution to the extremely odd equation; and there are others.

Applications of the non-associative octonions are currently quite rare. They are useful in investigating eight-dimensional transformations in general, and 8D rotations in particular. Some mathematicians (like John Baez and Ian Stewart) have recently begun to speculate that there might be a central role for the octonions in certain advanced physical theories.

S - The Sedenions

Now, what about this situation: find p and q, where

• pq = 0, p ≠ 0, q ≠ 0.

In all of the number systems we've seen so far, if any two numbers p and q were nonzero, we could be sure that their product pq would also be nonzero; conversely, if a product was zero, then so too was at least one of its factors. But if we now add just one more square root of -1 to our evolving number system, we can immediately find multiple solutions to this new and pathologically odd equation.

Call the new root l. As usual he brings along an entourage of his pals, this time a further seven: il, jl, ijl, kl, ikl, jkl, and ijkl. So first, we need to go through the chore of working out the extension of the multiplication table from 8 to 16 dimensions. Here's the result (click for big):

The new 16-dimensional numbers are termed the sedenions, S. Notice that the top left quarter of the basis multiplication diagram contains the rules for octonions (previously shown encoded into a Fano Plane above); similarly in turn, the top left quarter of that contains the quaternions; the top left quarter of that, the complex numbers; and finally the top left quarter of that, the single cell "1", is of course the basis for the reals.

Returning to the equation at the head of this section, let's now set

p = ij + kl, q = ik + jl

then

pq = (ij + kl)(ik + jl) = ij(ik) + ij(jl) + kl(ik) + kl(jl) = -jk + il - il + jk = 0

So p = ij + kl, q = ik + jl is one solution to the pathologically odd equation; and there are others.

I'll not go into the 16-dimensional sedenions S any further - they are just a little too crazy! Technically, S and all subsequent extensions are no longer normed division algebras. There are only four of these, and we have already seen them all: they were R, C, H and O.

Incidentally, don't think of zero divisors as pathological in all number systems; it's just that they have no place in normed division algebras. For a particularly tame illustration, consider the integers modulo 12, as modelled by the numerals 0 to 11 on an analog clock face (if yours has 12 in place of 0, obviously you're not a C programmer). Define addition as clockwise progression, and multiplication as repeated addition. Then speaking modulo 12, we have

3 * 4 = 0.

Still Higher Dimensions

Just for fun, I can't resist mentioning the names of the next few algebras after S. These are:

• the 32-dimensional trigintaduonions;
• the 64-dimensional sexagintaquattuornions;
• the 128-dimensional centumduodetrigintanions; and
• the 256-dimensional ducentiquinquagintasexions.
  

In the unlikely event that any of these should darken your door again, let's hope they do so under the mercifully un-Latin alternative names proposed by Robert P.C. de Marrais and Tony Smith:

• P - the pathions, from the 32 Paths of Kabbalah;
• X - the chingons, from the 64 Hexagrams of the I Ching;
  
• U - the routons, from Route 128 of the "Massachusetts Miracle"; and
  
• V - the voudons, from the 256 deities of Voodoo's Ifa pantheon.

The number of crazy-looking equations soluble in each system also continues to rise. For example, there are nonzero routons whose squares are zero. There are nonzero voudons, having nonzero squares and cubes, yet whose fourth powers are zero. And so on.

Finally, I'd like to go on record here to propose the name austons for the next algebra in this series. Named of course for the capital city of the Lone Star State: area code 512. Sadly there's never likely to be any demand for these, as a certain Periodicity Theorem tells us that nothing new happens beyond the voudons.

Confessions

My aim in this article was to provide a contrast to the traditional presentation of these particular algebras, R, C, H, O and S, as a progression wherein "algebra gets a little bit worse with each step":

• going from R to C, the property of ordering is lost;
• going from C to H, commutativity is lost;
• going from H to O, associativity is lost;
• going from O to S, unique divisors, and our marbles, are lost.

Instead I've stressed what's gained with each step. Seeking out more and more equations to solve, building successively more powerful algebras, until eventually, there is nothing we can't do - and nothing we can!

During the exposition, I've taken various liberties with notation.

• There is in fact no consensus as to whether the set of natural or whole numbers, both, or neither, contains zero. I've arbitrarily gone with N = {1, 2, 3, ...}.
• Yes, my account of the "real line" misses completely the point of the continuum hypothesis. The reals weren't my principal target here.
  
• k has traditionally been used, since the day and hour of William Hamilton's discovery of the quaternions, to represent ij, rather than the "next" new √-1 (the generator of the octonions). I play fast and loose with these bases, coming from a background in electrical engineering. Over there we live and breathe, eat and drink complex numbers; but we defiantly (and consistently) use j in place of i.
  

There are many more exotic number systems to be found. In fact, the schemes covered in this article, the first few Cayley-Dickson algebras, these are just some of the more popular (or better known) main line stations on a network also contaning biquaternions (like quaternions, but with complex instead of real coefficients - can, open! worms, everywhere!), exterior algebras, various split and conic hypercomplex numbers, and connecting even further out to some very weird places indeed.