Quaternary NumbersDate: 28 Jun 1995 10:05:49 -0400 From: Jonathan Rose Subject: Quaternery Numbers How do quaternary numbers work? As I understand it, it's q = r + a.i + b.j + c.k where r is real and i^2 = j^2 = k^2 = -1 but what happens when you start multiplying and dividing i, j and k? does i x j = k or -k? what is i / j ? Presumably the modulus | q | = SQR( r^2 + a^2 + b^2 + c^2) Please tell me all! I want to render 4d julia sets! Date: 28 Jun 1995 16:16:44 -0400 From: Dr. Ken Subject: Re: Quaternery Numbers Hello there! Jonathan Rose's message: How do quaternary numbers work? As I understand it its q = r + a.i + b.j + c.k where r is real and i^2 = j^2 = k^2 = -1 Yes, you're 100% right so far. but what happens when you start multiplying and dividing i, j and k? does i x j = k or -k? Well, i x j (I'll just write it as ij from now on) equals k, but ji = -k. You see, the quaternions aren't a commutative system. Perhaps you already knew that. Here's the best way I know of to remember the multiplication facts. Put the three elements in a circle like this: i k j Kind of looks like a triangle when there are only three of them, huh. Then you go around the circle to figure out the multiplication facts. For instance, you know that ij is either k or -k, and to find out which one it is, you see whether you go around clockwise or counterclockwise in the circle to get from i to j. Clockwise is positive, counterclockwise is negative. Alternatvely, you could write ijkijkijkijkijk...., and if you see your pattern in the sequence forwards, like jk, ij, or ki, then the product will be positive. If it's in the sequence backwards, then the product is negative. I guess the short answer, though, is this: ij = k ji = -k jk = i kj = -i ki = j ik = -j. what is i / j ? Well, i/j means the same thing as "i times the inverse of j." So we need to find out the inverse of j. That's just what you multiply j by to get 1, right? Since j*j = -1, j*(-j) = 1. So i/j = i*(-j) = -(i*j) = -k. The same thing holds in the similar cases. presmably the modulus | q | = SQR( r^2 + a^2 + b^2 + c^2) Yes, that's right. please tell me all! I want to render 4d julia sets! Wow, sounds fun! We're working on a 4D project here of our own. Good luck! -K |
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