Number Systems Redux

Just What Exactly Is A Number?

Lots of feedback about those hypercomplex posts. Well, some. An email flooded in. Actually to tell the truth, the interest had less to do with hypercomplex numbers as such, more with the programmable field length string formatting trick used in printing the multiplication tables for Cayley-Dickson algebras. So, okay, I'll write that up as a separate mini-subject; but only after this recapitulation.

What I want to do is revisit and spell out the basics of the earlier progression, from one number system to the next, via the sequence of equations which force the introduction of new concepts into the realm of what we are willing to call a number. This is not the only such path, but it is an interesting one. Remember that at every stage, the "current" number system contains everything in every preceding system. Click on each title / picture for the related Wikipedia article.

The Natural Numbers N

We start like Romans, with the everyday system of object counting numbers, N = {1, 2, 3, 4, ...}; or more accurately I guess {I, II, III, IV, ...}; and ask if there are any problems it can't solve. It doesn't take long to turn up one:

x + 1 = 1

None of our counting numbers has this property, that if you add one to it, the result is one. So like the somewhat brighter-than-Roman Arabs, we introduce a new concept: the number zero. Adding this into N, we find the solution we were after, namely x = 0, in...

The Whole Numbers W

Or "nonnegative integers", for a less ambiguous term. Still, our work has only now begun. Immediately, we are confronted by another unsolved problem, another simple equation without a solution:

x + 1 = 0

What kind of thing becomes nothing, when you add something to it? Answer: a negative number. Adding all of these into W, we find our particular solution, x = -1, in...

The Integers Z

Still, we haven't yet solved even the full set of the very simplest equations. What about this one:

x + x = 1

That's going to require that we bring in fractions, too. Let's add in every possible number of the form p/q, where p and q are integers (and q is nonzero). Including x = 1/2, the solution to our latest problem. Then we have...

The Rationals Q

Surely that's everything? Well no, it's actually very easy to show that even with rational numbers as finely grained as we wish, there's still no solving this:

x² = 2

For that we need the so-called radicals. And while we're there, we might as well add in every solution, positive or negative, that we can find to any polynomial equation (in x, with rational coefficients). Then we will have...

The Algebraics

These include such radicals as x = √2, which was just what we needed right then. And while it might seem we've come a long way, still in truth we don't actually have any more numbers than we started out with in N. The algebraics are still just countably infinite. But all that's about to change, as soon as we solve the next candidate equation,

x^x = 2

No algebraic number satisfies this equation (proof here). To solve it, we have to extend our number system to include the transcendentals. Once we do so, we obtain our first uncountably infinite set of numbers, namely...

The Reals R

But even this power of the continuum leaves half the world's polynomials unsolved. Here's one such:

x² = -1

To solve this, we have to add yet another new kind of number, an imaginary number x = i, which squares to yield -1. Between them, the real and imaginary numbers in combination give us numbers of the form x + iy. Like the rationals Q before them, these are essentially nothing more than ordered pairs of preexisting numbers. We call these particular pairs...

The Complex Numbers C

And like the reals, they are a powerful lot. One ditty, called The Fundamental Theorem of Algebra, tells us we don't have to look any further for a full solution to any polynomial equation in x. Yet still we persevere; what if we want to mess with the basic rules of algebra itself? What if we want to solve something like maybe,

(xy - yx)² = -4

Inspecting this curiosity, plainly if we want xy to differ from yx, then by definition we need a new kind of number that does not commute. Historically, the first non-commutative algebra discovered was...

The Quaternions H

These are obtained from C simply by adding another new square root of -1. Call it j. Once done, we can use the identities i² = j² = (ij)² = ij(ij) = -1, easily to prove that x = i, y = j solves our non-commutative equation above. And naturally, we can go still further. What about

(x(yz) - (xy)z)² = -4

This time we need x(yz) and (xy)z to differ. Historically again, it happens that the first non-associative algebra discovered was...

The Octonions O

And that once again, these were constructed by adjoining another completely independent and brand new square root of -1, to H. Casually defying more than a century and a half of mathematical history, we'll call it k. Once we sort out its multiplication table, we can verify that x = i, y = j, z = k satisfies our previous, non-associative equation.

Now, let's drop back down from three variables to two. Consider:

((xy)y - x(y²))² = -4

This can no longer be solved in O, because now we need (xy)y to be different from x(y²), whereas these quantities are always equal in O. We express this by saying that while O is not associative, it is alternative. Historically, the first non-alternative algebra discovered was...

The Sedenions S

These are formed in the "usual" way, by adding yet another unique √-1 which we'll call l. Again with a suitable multiplication table, we can readily show that x = i, y = ij + kl satisfies that non-alternative equation.

There, once again, we stop. Not of course because we have run out of algebras. The Cayley-Dicksons alone go on forever! And not because these algebras have become any less interesting, or too "pathological" with the loss of the alternative property. But simply because we have to stop somewhere. Applications exist of algebras beyond S, including certain topological investigations and many physical curiosities.

Previously:

A Few Hypercomplex Numbers
A Few Hypercomplex Onions