October 19th, 2007: A bunch of people sent me examples of situations in which addition is non-commutative: nonabelian groups, rotating objects in 3D space (try it! Pick up a book and rotate it 90 degrees left and then 90 degrees away from you, and then try the rotations in the opposite order: the book ends up in different positions, so the order in which you "add" the rotations together matters), non-commutative geometry and so on. That last one is interesting because it has a real-world application too: walking on a curved surface (like on a hill, or on our planet). If you start at the equator and go 1000k north and then 100k east, you'll end up in a different position than if you go east first and then north. So there's some math that doesn't commute and has the added benefit of having real-world applications! So that's something, huh?
Thanks to everyone who emailed me explanations! I haven't written you all back but I've read them all. I'm getting to the point where I can't respond to all the email I receive, and I'm already about a month behind, but I'm hoping to catch up.
To make sure the best gifts ever in time get to you when you want them, here are the HOLIDAY SHIPPING DEADLINES. Orders placed before these dates will get to you in time for Christmas!
December 5th: All shipping methods except U.S.P.S. First-Class Package International Service™ are safe.
December 6th: International orders made not selecting U.S.P.S. Priority Mail Express International™ shipping will likely arrive in January.
December 11th: USPS FIRST CLASS and STANDARD may not make it in time and neither will USPS EXPRESS MAIL INTERNATIONAL.
December 16th: ALMOST NOTHING IS SAFE. USPS EXPRESS is the only safe USPS option.
December 19th: A CHRISTMAS MIRACLE (IF THINGS PLACED ON OR AFTER THIS DATE ARRIVE BY THE 24TH, THAT IS)
One year ago today: can i - can i be a fictional character please