Category Archives: Math
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!
Continue reading
2 Comments | 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.
Continue reading
10 Comments | tags: base 10, base 2, binary, decimal, number systems, positional notation, Rudy Rucker | posted in Basics, Math
I misspent my younger days in the warm climes of Southern California. In particular, I went to high school and college there. I moved to the Midwest about seven years after college. For many, college was the end of anything resembling much in the way of time to call their own. I have many fond memories of idle times in perfect weather!
People who know me know I have a pretty intense work ethic. They also know I have a pretty intense party ethic. (Truth is I’m just intense. Period. Work hard; play hard; relax hard.) This past week — my first week into retirement — I’ve been relaxing hard.
And the weather has been just glorious this week. So far, retirement is aces!
Continue reading
1 Comment | tags: algebra, backgammon, humor, math humor, peach daiquiri, pi, retirement, weather | posted in From My Collection, Life, Math
The maps you find in some buildings and malls have a little marker flag that says, “You are here!” The marker connects the physical reality of where you are standing at that moment with a specific point on a little flat map.
Your GPS device provides your current location in terms of longitude and latitude. Those numbers link your physical location with a specific point on any globe or map of the Earth.
But to fully represent our location, longitude and latitude are not quite enough. (We might be high overhead in a hot air balloon!) To fully represent our position, we need a little more ‘tude, but in this case that’s altitude, not attitude.
We need three (and only three) coordinates to completely represent our location in space. This post is about why.
Continue reading
18 Comments | tags: 2D, 3D, 4D, 5D, altitude, Calabi-Yau, Cartesian, coordinates, dimensions, latitude, longitude, polar coordinates, rectangular coordinates | posted in Basics, Math, Physics, Science
It seems fitting to take this opportunity to write a Sideband post the way I had originally intended when I began them. That intent goes back to the beginning; the first Sideband post was my second post here. For better or worse, the original intent didn’t last long.
In fact, it’s hard to see much difference between the Sideband posts and the other posts. That probably reflects a lack of focus on the main topics. Sidebands were intended for stuff that was off topic. But I’ve been so all over the map on topic that at least one blogger asked if I had a short attention span. [insert here one of the short attention span jokes you’ve heard before]
That’s not going to change. I’m eclectic; so is my blog. Sidebands will evolve. For now, here’s a Sideband on yesterday’s post, just like Blogger originally intended.
Continue reading
5 Comments | tags: chaos, chaos theory, fractals, Mandelbrot | posted in Math, Physics, Sideband
You probably have some idea of what infinity means. Something that is infinite goes on forever. But it might surprise you to know that there are different kinds of infinity, and some are bigger than others!
As an example, a circle has an infinite number of points. (Yet the circumference of the circle has finite length.) Compare that to a straight line with infinite length. Both have infinitely many points but does the finite length circle have fewer? [The answer is no.]
To understand all this, we have to first talk a bit about numbers.
Continue reading
32 Comments | tags: countable, counting numbers, Georg Cantor, infinity, irrational numbers, natural numbers, pi, rational numbers, uncountable | posted in Math, Science
I suppose a “golden” date could refer to a really good time out with the perfect someone. Or it could refer to a couple of hot oldsters, past their silver years, tearing up the town.
I suppose the oldsters could double the value of their gold by being with that perfect someone. It doesn’t matter; I mean neither perfect occasions nor advanced years. I speak, literally, of the date.
It’s 11-11-11, and that’s slightly fun and slightly rare. It’s a bit like your Golden Birthday, when your age matches the date (for example, when you turn 19 on the 19th of whatever month). Today we match on the date, month and year; trifecta gold! And of course, double bonus points just before lunch at 11:11:11!!
Continue reading
4 Comments | tags: 11, binary, date, golden | posted in Life, Math
I recently mentioned a parable about grains of rice and a chessboard. If you were industrious enough to try your own interweb search for [parable 64 squares grains] you might be ahead of me. Or you may have known the parable already. For the rest of you, here’s the deal.
Stripped of the narrative, it’s about taking a chessboard and placing a single grain of rice in the first square (in some versions, it’s a grain of wheat). In the second square, place two grains of rice—double the amount in the first square. In the third square use double the grains of the second square. For each square on the chessboard, use twice as many grains of rice as used for the previous square.
I’ll come back to the punchline, but I stripped the narrative. Let me fix that.
Continue reading
Leave a comment | tags: exabyte | posted in Math, Sideband
I’d planned to do this later, probably for Sideband #64, but in honor of my parents 64th wedding anniversary (2 parents, 64 years, okay!) this numerical rumination gets queue-bumped to now.
Just recently I wrote about 64-bit numbers and how 64 bits allows you to count to the (small, compared to where we’re going) number:
264 = 18,446,744,073,709,551,616
That’s 18 exabytes (or 18 giga-gigabyes). Just to put it into perspective, if we were counting seconds, it amounts to 584,942,417,355 years; more than 500 billion years! (That’s the American, short-scale billion.)
Continue reading
Leave a comment | tags: 64-bit, 64K, 8-bit, big numbers, binary, bits, exabyte, powers of two | posted in Computers, Math, Sideband
A long, long time ago on a USENET far, far away, I was part of a debate that started with the idea that, even if we had disk drives with 64-bit addressing, people would still fill them up with videos, images and whatnot.
The idea grew from some of us old-timers reminiscing about our first brick-sized 5-meg hard drive and how we thought, “Gee, I’ll never fill that up!” (And look how that turned out; I have single image files that wouldn’t fit on that drive!)
The premise was that, even with seriously gigantic hard drives, we’d still manage to fill them and need more, more, more…
Continue reading
1 Comment | tags: 64-bit, big numbers, bytes, disk drive, exabyte, hard drive, petabyte, terabyte, USENET | posted in Computers, Math, Movies, Sideband
Logos con carne 