Folded into the mixed baklava of my 2018, was a special mathematical bit of honey. With the help of some excellent YouTube videos, the light bulb finally went on for me, and I could see quaternions. Judging by online comments I’ve read, I wasn’t alone in the dark.
There does seem a conceptual stumbling block (I tripped, anyway), but once that’s cleared up, quaternions turn out to be pretty easy to use. Which is cool, because they are very useful if you want to rotate some points in 3D space (a need I’m sure many of have experienced over the years).
The stumbling block has to do with quaternions having not one, not two, but three distinct “imaginary” numbers.
17 Comments | tags: complex numbers, math theory, mathematics, natural numbers, number systems, numbers, octonion, quaternion, real numbers | posted in Math, Sideband
Over the last two days I’ve written about a way of viewing words, sentences, even entire books, as single (very large) numbers. We do that by treating the characters in the string as “digits” in a number system we define. Technically speaking, we interpret the string as a number written in some large radix.
This is actually what we do every time we look at a written number. For example, we interpret the four-character text string “2013” as representing the numeric value two-thousand-and-thirteen. We do this easily, because we’ve grown up with the base 10 number system, decimal. The systems I’ve written about simply extend the concept.
Today, as a Sideband, I thought I’d get into some of the more technical details.
Leave a comment | tags: ASCII, CP-1252, decimal, hexadecimal, Mayans, number systems, Python, radix, Sumerians, Unicode, UTF-8 | posted in Computers, Math, Sideband
Yesterday I introduced you to the idea of words as numbers. There are many ways to create a map between words and numbers. For example, we could assign them the number that represents their position in the dictionary. That would make words that start with “A” have smaller numbers while words that start with “Z” would have the largest numbers.
There are also ways to treat the words themselves as numbers. We can interpret the letters the same way we do digits. Each letter has an assigned numeric value, and then a string of letters—just like string of digits—forms a number. The scheme I showed you yesterday allows us to treat (only!) single words as numbers.
Now let’s extend this so that entire sentences—or even entire books—become numbers!
1 Comment | tags: big numbers, number systems, numbers, positional notation, Rudy Rucker, words, zero | posted in Basics, Math
Today I’d like to introduce you to a concept I picked up from mathematician Rudy Rucker in his 1987 book, Mind Tools (The Five Levels of Mathematical Reality). I’ll warn you now that there is some math ahead (but no math homework—unless you want to). It won’t get any more complicated than multiplication and addition, but we will be dealing with some extremely large numbers (so large they are more ideas than numbers).
The end result is that we’re going to tie together the written word with numbers. I’m going to show you how every word, every sentence, every book, magazine and blog article can be reduced to a single (very large) number. That we can do this provides a foundation we can use to discover some amazing things about mathematical reality.
It may sound dry or intimidating, but stick with it! You just might find it worthwhile.
9 Comments | tags: base 10, base 2, binary, decimal, number systems, positional notation, Rudy Rucker | posted in Basics, Math