I
Days Without:
- Alcohol: 31
- Ice Cream: 31
- Finishing a book: 3
- Finishing a comic: 0
- Watching a movie: 2
- Watching TV: 0
When my adviser emailed me earlier in the week asking for a progress report I had to remind myself how much my life sucks. With his second email said adviser proved himself to be an asshole, but this much was known. I've been sitting on this thing for years with little progress since from the beginning I've hardly cared about it. For all that I've read or written I've barely scratched the surface and it seems madness to think my work will be an "original contribution" to the literature. I hate the pressure to produce, pressure that will just continue were I to get my dream job. As for my current job, it's dead-end enough and won't take me anywhere. It's stop-gap at best. I've grown to hate the current database project, for no matter what I give them the main boss will misunderstand it, it will be lacking some functionality I want it to have, those who use it after I leave will f*ck it up, and eventually it will be shown to be a wasted effort when they move to some other software. It's nearly every single person's favorite month, the month and holiday when we're reminded how how lonely and empty our lives are at the inter-personal level. I'm in a city that has plenty of dating opportunities for young 20-somethings but which offers little for unattractive 30-somethings; that's the curse of a college town. When my father was my age, well, I came into the world, whereas I'm single with no family, no kids, no prospects. In short, I'm not going anywhere.
II
It's a simple question: what is the most irrational number?
We all know what irrational numbers are: real numbers that cannot be expressed as the ratio of two integers. Which is to say, for example, that 1, 2, 3, 1/2, 45/13, and so are rational (the set of numbers Q), but the square root of 2, for example, is not. √2—that's the best I can do.
√—√
But I digress.
So which number is the most irrational?
One argument might go like this: it should be furthest away from the rational numbers. Digression the first: a field, which in simplified language is a set of objects with addition, subtraction, multiplication and division. If you take any two elements of the field and perform one of those operations, you will get another element of the field, which is a way of saying a (or the) field is "closed" (under addition and multiplication). You'll note that √2 is not in Q; you can't add or multiply (or subtract or divide) any rational numbers to get √2. So let's just add √2 to Q, and call that E=Q(√2), or the rationals extended by the square root of 2. E is also a field and every element of E can be written as a + b√2, where a and b are rational numbers. And let's say we find another number, call it c, such that c^{2} is in E but c isn't ... then let's create F=E(√c) ... and so on and so on ...
We're expanding the rational numbers (Q) with quadratic field extensions. Look at everything in E that's not already in Q—it's only one extension away from being rational. All the things in F that aren't already in E? Two extensions away ...
We now clearly have a metric for determining how "irrational" a number is: we just count how many "steps" away from "rational" it is ... well, we're leaving out a bunch of steps, such as figuring out: what's the smallest number of extensions necessary to get to a certain number? I mean, you won't find √3 in E, but you will find it in E(√3) ... two extensions. But you'd also find it in Q(√3) ... just one extension. But let's assume we can work out these details ... if so, then we have a metric, and it tells us ...
... that transcendental numbers must be pretty irrational. Things like e and π. And that would be because all the numbers you can get to by extending and extending Q with more and more roots, those turn out to be the same as the numbers that are solutions ("zeros") to polynomial equations with rational (or integer, if you wish) coefficients ... things like x^{2}+2x+3, or more generally a_{0}x^{n} + a_{1}x^{n-1} + a_{2}x^{n-2} + ... a_{n-1}x + a_{n}. And since π & Co. are not such solutions, they're not in such field extensions, which gives them a level of "irrationality" that one might term "infinite."
Or at least unbridgeable.
Quite a discrete solution we have here. In a way.
But there might be other ways to measure irrationality, or you might have another reason for asking the question. If you have any ol' number given to you, odds are, it's irrational. Odds are, it's transcendental. You can count the rationals; you can't count the reals. But there are still infinitely many rational numbers, they're easy to work with, so perhaps you want to approximate that irrational number you've been given with something that looks like a fraction.
This is, conveniently enough, called rational approximation.
If I have an irrational number, call it n, I can approximate it with a/b (a, b being integers), and then I can find another rational number, say c/d such that a/b < c/d < n. c/d approximates n better than a/b does; it's "closer." For the sake of argument all my rational numbers will be in "reduced form" (no 2/4 and 3/9 nonsense ... 1/2 and 1/3 it is, for example).
Hurwitz's Irrational Number Theorem gives us a "best possible" method of approximation. Our interest is in reducing the "error." We use rational approximations all the time, such as 22/7 being a "decent" approximation of π, with the error being about 0.00126, and 355/113 giving an error of only 0.000000266. We would judge how "good" such an approximation is—given that there are infinitely many such approximations—by providing, for example, a limit on the size of the denominator, such as the best approximations where the denominator is less than 10, less than 200, and so on.
Another and related way to approximate a number—and we do it all the time in trig, etc.—is through a sequence of partial sums of an infinite series. One of the most well-known would probably be the sum of the reciprocal of squares, which gives us π/6. We can look at the partial sums of the decimal expansion of a number, which leads us to continued fractions. The more quickly a series converges, the better the approximation. The result of this line of thinking is that—thinking again back to Hurwitz and that √5 in the denominator—the golden mean/ratio, which is difficult to approximate accurately, in fact the most difficult to approximate accurately.
Transcendental numbers like π are in some regards notoriously easy to approximate, as if they were just a little be off from being rational. The thing about the golden mean/ratio, though, is that it's the solution to a very simple quadratic equation, for its definition comes from finding a and b such that (a + b)/a = a/b, which is to say, the solution to x^{2} - x - 1 = 0. Which is to say, x = (1 +/- √5)/2.
And going back to our first metric for how irrational a number is, it's worth noting that (1 +/- √5)/2 would be in Q(√5), which is to say, just "one step" from rational.
I always find this amusing. In a sense because I'm easily amused. It's not really "meaningful" in any "meaningful" way, but what I enjoy is not just that two reasonable definitions of "degree of irrationality" give different answers, but they give opposite answers. Furthermore, it's worth noting that the method of quadratic field extensions is a nice, discursive method that shows us how to "reach" (get to, achieve) a certain category of numbers; it provides a path, yet the harder a number is to reach, the easier it is to approximate, within limits, blah blah blah (my boilerplate disclaimer was eaten by my long-dead dog).
Up next: Kant's Third Critique: How the Critique of Judgment is both the least and most important element of Kant's critical project.
Aside: My two favorite books, covering quadratic field extensions (and Galois Theory, etc.) on the one hand and basic number theory on the other are—
- Hadlock, Charles, Field Theory and its Classical Problems. Carus Mathematical Monographs 19—I have an "old" hardcover of this, but I think I left my copy in my parents' spare bedroom. I love this book. It's clear, it's for people who love geometric constructions, and it's a bottom-up, historically-oriented approach to algebra (from concrete problems, to field theory, then to the abstraction of rings and the abstraction of groups).
- Niven, Ivan, Hebert S. Zuckerman and Hugh L. Montgovery, An Introduction to the Theory of Numbers. 5th Edition. New York: John Wiley & Sons, 1991—this is my favorite "Budapest" book (except, perhaps, Lovász's Combinatorial Problems and Exercises, almost the only combinatorics book you'll ever need).
I'll leave out today's segment on numbers, synesthesia, and visualizing the calendar.
III
Today:
- Lost—picked up where last season left off, and last season's cliffhanger was, well, a whopper. But so far, while I'm not entirely convinced they can pull off something interesting in the limited season available to them, I'm intrigued. They've kept my attention for another week.
- Captain America, Vol. 4 No. 34: Last issue we were told what was obvious to most readers from the beginning—that Bucky would be the new Cap. This issue he was "unveiled" and got some breaking-in action. The writing is still tight enough. The AV Club's comics panel covers this issue of Captain America as well as BKV's Y: The Last Man.
- We had an ice quake this afternoon—the ice in a nearby lake shifted, and it shook buildings hundreds of feet away. They article in ISTHMUS mentions that it barely registered on the Richter scale (0.2), but when you're that close to the so-called "epicenter" and on the top floor of a building, an old building, and the floor wobbles like JELLO, well, it reminded me for a moment of the Northridge quake, which woke me from my sleep just after 4:30 in the morning back in 1994. H and I were the only people in our building, on our floor at least, this afternoon, and it actually felt as if something had impacted the building, but it stopped after a few seconds. H called someone she knew on the ground floor, and they'd felt it, too. We looked around, out windows, to see whether something, a large truck or bus, perhaps, had crashed. Students were just walking as normal, as if nothing was out of the ordinary, but since we'd both felt it we knew it was "real." Later in the afternoon I was at Ye Olde Coffee Shoppe, sipping my afternoon brew, when I heard a newcomer tell the guy she was with a table away from me what she'd experienced in the afternoon, and that's when I first heard the ice hypothesis floated, but I only listened in, for I didn't want to interrupt their chat. And since she'd been in a different, taller building, I knew it wasn't entirely localized, at least not to my office area.
- The Year My Parents Went On Vacation: I'm not sure I need another "touching" movie about the 70s or such, but there are a number of potentially interesting aspects that make it appear to be more than mere melodrama. Even though the trailer is not explicit on this matter, it seems obvious that the "vacation" the parents are on is of a political nature. Furthermore, about halfway through the trailer one of my favorite songs, "Chiribim Chiribom," is heard. Alas, the renditions you find online tend toward the puerile and insipid. The same song can be heard part of the way through Fight Club—pay attention when the guy washing the sidewalk sprays the pedestrian with the hose—and my favorite version, written as "Dschiribim Dschiribam," is performed by Arik Brauer.
- I began watching the beginning of (so ... imprecise, unwilling to commit, that formulation ...) Eli Stone after Lost, and while it was written well enough, and it featured Victor Garber, and there was George Michael and other music and fantasy, etc., the premise offended me enough that I couldn't keep watching it. If someone else did and found it worthwhile, not the utter and sentimental trash I fear, please let me know.
When my adviser emailed me I went through a night of stress. It took me hours to fall asleep. Since then I've mainly ignored him and the past few days I've stopped by the library in the afternoon, and been reminded how much my life rocks. I experienced a resurgent interest in my work. I've not exhausted my research topic and I'm always finding new aspects and perspectives on it. Sometimes it's a detail, a tidbit, an aspect of depth I'd missed before, and it seems impossible to trace all the threads, to run out. Other times I turn the corner and encounter something new and unexpected; I add it to my work as if I've added a new axiom or something else orthogonal, and my vector space explodes. Although I need to bring this path of work and research to an end at some point, while I continue with it I at least have my editorial work, which is of interest to me of itself, but it also puts me in contact, as did my last editorial position, with scholars and colleagues in a myriad fields and subfields, and the topics touched upon seem limitless. I felt comfortable taking on the database project because it touched upon my own database wants and needs and so the professional dovetailed with the private. Along the way I picked up more python knowledge and decided to forgo my own personal framework for Django, which is a pleasure enough for a hobbyist to work with. I'm not tied down by a relationship; I have no noticeable biological clock and so do not force myself into bad relationships just because a "body" or "other" is necessary to make me feel "whole." I look at too many of my friends, often in other, larger cities, and they jump job to job because of the stress, and the 70-hour weeks leave them no personal time, but they feel the need to meet Mr. or Ms. Right before it's "too late" or in order to please the parents, whereas I have the occasional company of my friends when I need such company and time to myself when I don't feel like being social. This stage of life is a holding pattern, sure, but I could be worse off, I'm not in debt, I'm not tied down, and there are plenty of years ahead of me for that whole "settling down" thing.
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