Numbers:
I've been getting mail all day asking me to explain something that appeared in today's XKCD comic. Yes, I've been reduced to explaining geek comics to my readers. I suppose that there are worse fates. I just can't think...
Posted on April 14, 2008 9:13 PM • 25 Comments •
Even though this post seems to be shifting back to axiomatic set theory, don't go thinking that we're done with type theory yet. Type theory will make its triumphant return before too long. But before that, I want to...
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Posted on November 19, 2007 2:53 PM • 34 Comments •
Don't you dare use the number 271277229129081016424883074559900780951 under any circumstances. It's mine, mine I tell you, and if you use it, or copy it, I can have you arrested and sent to do hard time in prison. And it...
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Posted on May 9, 2007 11:44 AM • 45 Comments •
The Surreal Reals I was reading Conway's Book, book on the train this morning, and found something I'd heard people talk about, but that I'd never had time to read or consider in detail. You can use a constrained subset...
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Posted on April 5, 2007 10:09 PM • 13 Comments •
Today we're going to take our first baby-step into the land of surreal games. A surreal number is a pair of sets {L|R} where every value in L is less than every value in R. If we follow the...
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Posted on April 1, 2007 7:24 PM • 19 Comments •
In my last post on the surreals, I introduced how the surreal numbers are constructed. It's really fascinating to look back on it - to see the structure of numbers from 0 to infinity and beyond, and realize that...
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Posted on March 31, 2007 6:01 PM • 2 Comments •
What are the real numbers? Before I go into detail, I need to say up front that I hate the term real number. It implies that other kinds of numbers are not real, which is silly, annoying, and frustrating....
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Posted on February 5, 2007 8:30 AM • 40 Comments •
One of the interestingly odd things about how people understand math is numbers. It's astonishing to see how many people don't really understand what numbers are, or what different kinds of numbers there are. It's particularly amazing to listen...
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Posted on January 23, 2007 9:50 AM • 47 Comments •
Tons of folks have been writing to me this morning about the BBC story about an idiot math teacher who claims to have solved the problem of dividing by zero. This is an absolutely infuriating story, which does an excellent...
Posted on December 7, 2006 9:49 AM • 132 Comments •
While I was researching yesterdays post on Archimedes integration, one of the things I read reminded me of one of the stranger things about Greek and earlier math. They had a notion that the only valid fractions were unit fractions;...
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Posted on November 15, 2006 3:31 PM • 20 Comments •
After my binary fingermath stuff, a few people wrote to me to ask about just how binary really works. For someone who does the kinds of crazy stuff that I do, the idea of different number bases is so fundamental...
Posted on October 16, 2006 9:34 PM • 39 Comments •
Since we talked about the surreals, I thought it would be interesting to take a very brief look at an alternative system that also provides a way of looking at infinites and infinitessimals: the hyperreal numbers. The hyperreal numbers are...
Posted on August 21, 2006 9:17 PM • 17 Comments •
Last thursday, I introduced the construction of John Conway's beautiful surreal numbers. Today I'm going to show you how to do arithmetic using surreals. It's really quite amazing how it all works! If you haven't read the original post introducing...
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Posted on August 21, 2006 8:30 AM • 11 Comments •
Surreal numbers are a beautiful, simple, set-based construction that allows you to create and represent all real numbers, so that they behave properly; and in addition, it allows you to create infinitely large and infinitely small values, and have them...
Posted on August 17, 2006 5:06 PM • 7 Comments •
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...
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Posted on August 16, 2006 2:45 PM • 47 Comments •
How can you talk about interesting numbers without bringing up π? History The oldest value we know for π comes from the Babylonians. (Man, but those guys were impressive mathematicians; almost any time you look at the history of fundamental...
Posted on August 14, 2006 8:30 AM • 21 Comments •
In the comments onmy post about φ, Polymath, (whose blog is well worth checking out) provided a really spectacular link involving φ. It's an excerpt from a book called "Mathematical Gems 2", showing a problem that came from John Conway,...
Posted on August 12, 2006 10:10 AM • 1 Comments •
In email, someone pointed me at an automated proof system called Metamath. Metamath generates proofs of mathematical statements using ZF set theory. The proofs are actually relatively easy to follow, which is quite unusual for an automated theorem prover. I'll...
Posted on August 10, 2006 10:53 AM • 3 Comments •
Sorry for the delay in the category theory articles. I've been busy with work, and haven't had time to do the research to be able to properly write up the last major topic that I plan to cover in cat...
Posted on August 9, 2006 11:51 AM • 9 Comments •
Lots of folks have been asking me to write about φ, the golden ratio. I'm finally giving up and doing it. I'm not a big fan of φ. It's a number which has been adopted by all sorts of flakes...
Posted on August 8, 2006 12:25 PM • 45 Comments •
Quaternions Last week, after I wrote about complex numbers, a bunch of folks wrote and said "Do quaternions next!" My basic reaction was "Huh?" I somehow managed to get by without ever being exposed to quaternions before. They're quite interesting...
Posted on August 7, 2006 7:53 PM • 34 Comments •
Ω is my own personal favorite transcendental number. Ω isn't really a specific number, but rather a family of related numbers with bizzare properties. It's the one real transcendental number that I know of that comes from the theory of...
Posted on August 7, 2006 9:00 AM • 54 Comments •
If you look at the history of math, there've been a lot of disappointments for mathematicians. They always start off with an idea of math as a clean, beautiful, elegant thing. And they seem to often wind up disappointed. Which...
Posted on August 4, 2006 11:00 AM • 18 Comments •
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...
Posted on August 3, 2006 12:00 PM • 43 Comments •
Looks like I've accidentally created a series of articles here about fundamental numbers. I didn't intend to; originally, I was just trying to write the zero article I'd promised back during the donorschoose drive. Anyway. Todays number is e, aka...
Posted on August 2, 2006 11:17 AM • 29 Comments •
