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.
We’re still motoring through numeric waters, but hang in there; the shore is just ahead. This is the last math theory post… for now. I do have one more up my sleeve, but that one is more of an overly long (and very technical) comment in reply to a post I read years ago. If I do write that one, it’ll be mainly to record the effort of trying to figure out the right answer.
This post picks up where I left off last time and talks more about the difference between numeric values and how we represent those values. Some of the groundwork for this discussion I’ve already written about in the L26 post and its followup L27 Details post. I’ll skip fairly lightly over that ground here.
Essentially, this post is about how we “spell” numbers.
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.