Archive for the 'classics' category

My Favorite Strange Number: Ω (classic repost)

Dec 31 2008 Published by under classics, Computation, Numbers

I'm away on vacation this week, taking my kids to Disney World. Since I'm not likely to
have time to write while I'm away, I'm taking the opportunity to re-run an old classic series
of posts on numbers, which were first posted in the summer of 2006. These posts are mildly
revised.

Ω is my own personal favorite transcendental number. Ω isn't really a specific number, but rather a family of related numbers with bizarre properties. It's the one real transcendental number that I know of that comes from the theory of computation, that is important, and that expresses meaningful fundamental mathematical properties. It's also deeply non-computable; meaning that not only is it non-computable, but even computing meta-information about it is non-computable. And yet, it's almost computable. It's just all around awfully cool.

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Continued Fractions (classic repost)

Dec 30 2008 Published by under classics, Numbers

I'm away on vacation this week, taking my kids to Disney World. Since I'm not likely to have time to write while I'm away, I'm taking the opportunity to re-run an old classic series of posts on numbers, which were first posted in the summer of 2006. These posts are mildly revised.

One of the annoying things about how we write numbers is the fact that we generally write things one of two ways: as fractions, or as decimals.

You might want to ask, "Why is that annoying?" (And in fact, that's what I want you to ask, or else there's no point in my writing the rest of this!)

It's annoying because both fractions and decimals can both only describe rational numbers - that is, numbers that are a perfect ratio of two integers. The problem with that is that most numbers aren't rational. Our standard notations are incapable of representing the precise values of the overwhelming majority of numbers!

But it's even more annoying than that: if you use decimals, then there are lots of rational numbers that you can't represent exactly (i.e., 1/3); and if you use fractions, then it's hard to express the idea that the fraction isn't exact. (How do you write π as a fraction? 22/7 is a standard fractional approximation, but how do you say π, which is almost 22/7?)

So what do we do?

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15 responses so far

Idiotic Gitt: AiG and Bad Information Theory (classic repost)

Dec 29 2008 Published by under classics, Intelligent Design

I'm away on vacation this week, taking my kids to Disney World. Since I'm not likely to have time to write while I'm away, I'm taking the opportunity to re-run some old classic posts which were first posted in the summer of 2006. These posts are mildly revised.

Back when I first wrote this post, I was taking a break from some puzzling debugging.
Since I was already a bit frazzled, and I felt like I needed some comic relief, I decided to
hit one of my favorite comedy sites, Answers in Genesis. I can pretty much always find
something sufficiently stupid to amuse me on their site. On that fateful day, I came across a
gem called Information, science and biology", by the all too appropriately named
"Werner Gitt". It's yet another attempt by a creationist twit to find some way to use
information theory to prove that life must have been created by god.

This article really interested me in the bad-math way, because I'm a big fan of information theory. I don't pretend to be anything close to an expert in it, but I'm
fascinated by it. I've read several texts on it, taken one course in grad school, and had the incredible good fortune of getting to know Greg Chaitin, one of the co-inventors of algorithmic information theory. Basically, it's safe to say that I know enough about
information theory to get myself into trouble.

Unlike admission above, it looks like the Gitt hasn't actually read any real
information theory much less understood it. All that he's done is heard Dembski presenting
one of his wretched mischaracterizations, and then regurgitated and expanded upon them.
Dembski was bad enough; building on an incomplete understanding of Dembski's misrepresentations, misunderstandings, and outright and errors produces a result
that is just astonishingly ridiculous. It's actually a splendid example of my mantra on this blog: "the worst math is no math"; the entire article pretends to be doing math - but it's actual mathematical content is nil. Still, to the day of this repost, I continue
to see references to this article as "Gitt's math" or "Gitt's proof".

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e: the Unnatural Natural Number (classic repost)

Dec 27 2008 Published by under classics, Numbers

I'm away on vacation this week, taking my kids to Disney World. Since I'm not likely to
have time to write while I'm away, I'm taking the opportunity to re-run an old classic series
of posts on numbers, which were first posted in the summer of 2006. These posts are mildly
revised.

Anyway. Todays number is e, aka Euler's constant, aka the natural log base. e is a very odd number, but very fundamental. It shows up constantly, in all sorts of strange places where you wouldn't expect it.

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35 responses so far

Roman Numerals and Arithmetic

Dec 26 2008 Published by under classics, Numbers

. I'm away on vacation this week, taking my kids to Disney World. Since I'm not likely to
have time to write while I'm away, I'm taking the opportunity to re-run an old classic series
of posts on numbers, which were first posted in the summer of 2006. These posts are mildly
revised.

I've always been perplexed by roman numerals.

First of all, they're just weird. Why would anyone come up with something so strange as a
way of writing numbers?

And second, given that they're so damned weird, hard to read, hard to work with, why do
we still use them for so many things today?

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19 responses so far

i: the Imaginary Number (classic repost)

Dec 25 2008 Published by under classics, Numbers

I'm away on vacation this week, taking my kids to Disney World. Since I'm not likely to
have time to write while I'm away, I'm taking the opportunity to re-run an old classic series
of posts on numbers, which were first posted in the summer of 2006. These posts are mildly
revised.

After the amazing response to my post about ze ro, I thought I'd do one about something
that's fascinated me for a long time: the number i, the square root of -1. Where'd
this strange thing come from? Is it real (not in the sense of real numbers, but in the sense
of representing something real and meaningful)? What's it good for?

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31 responses so far

Zero (classic repost)

Dec 24 2008 Published by under classics, Numbers

This post originally came about as a result of the first time I participated in a DonorsChoose fundraiser. I offered to write articles on requested topics for anyone who donated above a certain amount. I only had one taker, who asked for an article about zero. I was initially a bit taken aback by the request - what could I write about zero? This article which resulted from it ended up turning out to be one of the all-time reader-favorites for this blog!

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Navier-Stokes: Another Historic Problem Solved!?

Oct 05 2006 Published by under classics

A few months ago, I wrote about the Poincare conjecture, and the fact that it appeared to finally have been solved by a reclusive russian mathematician named Grisha Perelman. Now there's news that *another* classic problem may have been solved. This time, it's the Navier-Stokes equation, apparently solved by [Professor Penny Smith](http://comet.lehman.cuny.edu/sormani/others/smith.html) of Lehigh University. She's published the steps leading up to her solution in top peer-reviewed journals, and a [preprint of the final paper is now available via arxiv](http://arxiv.org/abs/math/0609740). There's also a pretty good detailed description of the solution on [Christina Sormani's website](http://comet.lehman.cuny.edu/sormani/others/SmithNavierStokes.html).
The Navier-Stokes equations form a classic problem that I actually know a bit more about, although I have to admit that the proof of the solution is beyond my ability to understand. Why should you care? Aside from the fact that it's a famous problem with a million dollar reward posted by the Clay Institute for a solution, it's *useful*. Unlike the Poincare conjecture, the reason why we care about solving the Navier-Stokes equations isn't just theoretical. If Professor Smith's solution and proof do turn out to be correct, it would be a really incredible accomplishment, with direct, immediate, practical implications.

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Notices of the AMS special issue on Kurt Godel

Jun 21 2006 Published by under classics, goodmath

Harald Hanche-Olsen, in the comments on my earlier post about the Principia Mathematica, has pointed out that this months issue of the Notices of the American Mathematical Society is a special issue in honor of the 100th anniversary of Kurt Gödels birth. The entire issue is available for free online
I haven't read much of the journal yet; but Martin Davis's article The Incompleteness Theorem is a really great overview of the theorem abnd the proof, how it works, and what it means.

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Extreme math: 1 + 1 = 2

Jun 17 2006 Published by under classics

Finally, I have found online, a copy of the magnificent culmination of the 20th century's most ambitious work of mathematics. The last page of Russel and Whitehead's proof that 1+1=2. On page 378 (yes, three hundred and seventy eight!) of the Principia Mathematica.. Yes, it's there. The whole thing: the entire Principia, in all of its hideous glory, scanned and made available for all of us to utterly fail to comprehend.
oneplusone.jpg
For those who are fortunate enough not to know about this, the Principia was, basically, an attempt to create the perfect mathematics: a complete formalization of all things mathematical, in which all true statements are provably true, all false statements are provably false, and no paradoxical statements can even be written, much less proven.
Back at the beginning of the 20th century, there was a lot of concern about paradox. Set theorists had come across strange things - things like the horrifying set of all sets that don't contain themselves. (If it contains itself, then it doesn't contain itself, but if it doesn't contain itself, then it contains itself. And then really bad actors need to pretend to be robots short-circuiting while Leonard Nimoy looks on smugly.)
So some really smart guys named Bertrand Russell and Alfred North Whitehead got together, and spent years of their lives trying to figure out how to come up with a way of being able to do math without involving bad actors. 378 pages later, they'd managed to prove that 1+1=2. Almost.
Actually, they weren't there yet. After 378 pages, they were able to talk about how you could prove that 1+1=2. But they couldn't actually do it yet, because they hadn't yet managed to define addition.
And then, along came this obnoxious guy by the name of Kurt Godel, who proceeded to show that it was all a big waste of time. At which point I assume Russell and Whitehead went off and had their brains explode, pretty much the same way that the bad actors would later pretend to do.

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