The last Sideband discussed two algorithms for producing digit strings in any number base (or radix) for integer and fractional numeric values. There are some minor points I didn’t have room to explore in that post, hence this followup post. I’ll warn you now: I am going to get down in the mathematical weeds a bit.
If you had any interest in expressing numbers in different bases, or wondered how other bases do fractions, the first post covered that. This post discusses some details I want to document.
The big one concerns numeric precision and accuracy.
Fractional base basis.
I suspect very few people care about expressing fractional digits in any base other than good old base ten. Truthfully, it’s likely not that many people care about expressing factional digits in good old base ten. But if you’re in the tiny handful of those with an interest in such things — and don’t already know all about it — read on.
Recently I needed to figure out how to express binary fractions of decimal numbers. For example, 3.14159 in binary. And I needed the real thing — true binary fractions — not a fake that uses integers and a virtual decimal point.
The funny thing is: I think I’ve done this before.
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.
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.
Sidebands are 10; a full hand; a (very small) odometer moment.
The accident of genetics and evolution that gives us ten fingers (and ten toes) causes us to count in tens and celebrate things that occur on tens boundaries.
Turning 30, 40 or 50 years of age is viewed as cause to bring out the black balloons and mocking birthday cards. Yet celebrating 30, 40, 50 or 60 years of marriage is increasingly cause to celebrate (especially these divorce-prone days).