WEEKLY WHINE

The future of munbers

As you may have noticed, work at CITMTL has ground to a halt. The California Institute of Technology Munber Theory Laboratory, once one of the most powerful forces on campus, has plunged into a lengthy period of recession. No work has been published in this long interval that covers this entire term. I think that this is a great mistake on my part, so I would like to take a break here and submit some of my excuses, as well as a look at some of the progress [although not very much] that's been made in that downtime.

First, a look at munber theory as a whole. We know what munbers are, how to make them equal and identical, how to add and multiply them, how to draw them, and how to find degrees. We can generalise this to determine how to perform any operations, unary or binary, how to determine zero and unit munbers, and how to compute various properties of their members. But so much more remains, and it's hard to tell exactly where we go from here.

I have been shorting you on munbers for several reasons. One is that my overall Web work has included several other projects, beginning with the upcoming refurbishment of GoobNet. This will go with the refurbishment of 45 Ricketts and will be amongst many things to be refurbished. Another is the Caltech SEDS site, refurbished back in DEC 1998.

It was also modified in a different way in that Caltech SEDS Space News was added to it. Regardless of whether you're at Caltech, you can go there to find out all you could care to know about space exploration, astronomic matters, and so forth. It's unusual as far as my recent Web work, since it requires regular updates rather than merely "suggesting" them, as it were. I have to check the Internet daily to find reports for Space News, and since I write the copy, that adds quite a bit of time.

I have been overloading this past term, but that's no excuse. The truth is that I often think of things to do with munbers, but I never actually sit down and type another chapter of Simple Introduction to Munber Theory. That's a problem, since Chapter 5 is due to be the most important one. In fact, I could probably combine the first four chapters into a single überchapter, but that's not an immediate concern at this moment.

Chapter 5 will be about what I call "supermunbers". That's really a misnomer, since I think that these objects are the true future of munbers. The "munbers" that we've been working with so far aren't true munbers in my mind; they're more of "infermunbers". The next generation of munbers is not going to be restricted to member numbers that appear only once. So far, you've seen only f(1) munbers, so named because the frequency of each member number is fixed at 1. [Of course, for numbers that are unequal to that munber, their frequency is 0, but that is assumed to be a possibility.] I'd like to progress to f(R) munbers, meaning that the member numbers can have any real frequency. Within this set is the set of f(Z⁺) munbers, those munbers in which the member numbers all have positive integer frequencies.

You may be wondering exactly what this does for us. I think that this is several orders of magnitude beyond existing work because with variable frequencies, we can ensure that adding or multiplying two munbers yields a munber with a degree equal to the product of the operands' degrees. None of this at most junk will be seen. I guarantee that these munbers will be the real deal. That is why I believe that the original form of munbers was meritless.

I should do something to those original four chapters, since they're now obsolete. I may condense them or just ditch them entirely and start from scratch. It's important to establish the mechanisms, but we should be able to do that as we explain the frequencies anyway. In any case, that will be determined in the first edition of my book. For now, the Zeroth Edition is going to resemble a zeroth edition, since there's no other choice for me. After all, I carry around a hard copy of the first four chapters.