Category Archives: Math

Dimensional Coordinates

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.

Sideband #43: Chaos!

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.

Infinity is Funny

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 a simple example, a small circle is infinite in the sense that you can loop around and around the circle forever. At the same time, your entire path along the circle is bounded in the small area of the circle. Compare that to the straight line that extends to infinity. If you  travel that line, you follow a path that goes forever in some direction.

What if we draw a larger circle outside the small circle. If there are an infinite number of points on the small circle and an infinite number of points on the large circle, does the larger circle have the same number of points as the small one? [The answer is yes.]

To understand all this, we have to first talk about bit about numbers.

A Golden Date

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. And 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!!

[By the way, (BTW: remind me about "by the way"), speaking of "oldsters," number me among those who find such phrases as "67 years young" … at best too silly to utter, if not in fact insultingly stupid. I passed the half-century mark in the last decade, so I am at least beginning to have some say in these matters.]

Anyway, today’s date is slightly fun in a numerical sense. We only get twelve triple-matches every 100 years, and we’re nearing the end of our twelve for the next 100.  This is, as its numbers imply, the eleventh this century. We’re just about done with golden dates for the 2000′s—no more until the 2100′s. Next year will be the last one for until nearly 100 years.

These magical dates only blossom, one per year, in the first part of any given century. Back, 100  years ago, in the early 1900s, the equivalent of today was 11-11-1911. In 100 years, someone will be noting that it’s 11-11-2111 (which has an extra “1″!).

We began this era’s progression ten years ago, in the true and proper first year of the millennium, 2001,  on the first month of the first (true and proper) year, and, in fact, on the very first day of the first month of the first (true and proper) year. Specifically: on 01-01-01. After that they came a month later each year until this year the golden date is in November. Our last one this century takes place on December 12th next  year (12 minutes past noon).

Today’s date is also fun, because it’s binary! So was a golden date from last year: 10-10-10. These two are the only others except for the one already mentioned above, the first one of any century: 1-1-1. And that’s it on binary dates!

I while back I mentioned some of my favorite CS jokes, probably my most favorite of which is the one about there being 10 kinds of people: those who can count in binary and those who can’t. (A joke that’s funny only to the people the joke is actually about.)

The most excellent xkcd featured a new spin on that joke recently:

xkcd 953

As always, the real punch, or a really funny secondary punchline, is in the image’s hover text. In this case it reads, “If you get an 11/100 on a CS test, but you claim it should be counted as a ‘C’, they’ll probably decide you deserve the upgrade.

And I confess, I didn’t fully get it until I was explaining it to someone else. I was thinking the  test score, 11/100, was intended to be a binary score. It wasn’t… the student actually got eleven out of 100. And that’s certainly not a ‘C’ grade!

Sideband #40: Chessboard of Rice

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 redress that matter. I was going to tailor my own from the whole cloth, but I decided to look around to refresh my mind about the pattern and instead found an off-the-rack number that looks just great. It’s recently minted, and it’s called The Power of Compounding. I couldn’t put it better myself, so I didn’t.

If you would like to dig deeper on your own, you can start at the Wikipedia article that discusses it: Wheat and chessboard problem.

Here’s the punchline: if you could pour grains of rice for all 64 squares on the chessboard, you end up pouring a total of 18,446,744,073,709,551,615 grains of rice.

If that number sounds familiar, you may have been reading this blog. That number is 18 exabytes (18 giga-giga-bytes), which I first mentioned writing about loading a lot of movies on a very large disc drive. Specifically, that is the number of unique addresses you get with 64 bits: 18 exabytes.

Actually that is one less than the number I mentioned. That number ended with 616; the number above ends with 615. The difference is that here we’re not counting the empty chessboard with no grains of rice. In previous talk about 64 bits, all 64 bits being off (all chessboard squares empty) is the number zero, which I’ve also mentioned before. The parable skips the empty chessboard and starts with one square, one grain of rice.

We can think of the chessboard as a binary number. Each chessboard square is one bit. The amount of rice that is poured for a square (twice the amount used for the previous square) is the value of that bit. Each added bit is worth twice as much rice as the bit before it.

As the parable mentions, at first the doubling doesn’t seem to amount to much: 1, 2, 4, 8, 16, 32, 64, 128… At eight squares, even after doubling seven times, we’ve just broken 100. We can pause here to note that the total amount of rice poured is 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 = 255. That is the sum of the bit values.

Let me draw the connection to binary numbers more clearly. If a square has rice poured on it, that bit is “on” (is set to “1″). If there is no rice for a square, that bit is “off” (is set to “0″). In the parable the squares are set progressively. No square is ever reset to off once filled. From a binary number point of view, it progresses like this: 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111. The last one represents eight squares filled with grain, each one holding twice as much as the one before.

If we imagined sweeping away the rice from the middle four squares and leaving the rice in the two end squares on each side, we’d have 1 + 2 + 64 + 128 = 195 grains of rice. The binary number represented by the squares now is 11000011. (I haven’t said anything about which way is which on the bit strings. For now it doesn’t matter.)

As the parable also mentions, after a while the doubling becomes serious. 16 thousand, 32 thousand, 64 thousand, 128 thousand, 256 thousand. Wait, that’s a quarter million, and we’re only 18 squares in! By 25 squares we hit 33 million; at 30 squares we cross the billion mark. At 40 squares we reach trillion (million million), and at 50 squares we’re over 1000 times beyond that. By 60 squares, we’ve got over 1 billion billion grains of rice, and we reach the final count four squares later: 18,446,744,073,709,551,615 grains of rice.

The interweb returns varying numbers for “how many grains of rice in a pound.” The number goes as low as 16,000 to as high as 29,000. One source measured it at 22,680.  Short grain rice would have a higher number of grains per pound; maybe as much as three or four times.

Let’s assume 30,000 grains of rice in a pound and see what we have on the chessboard.

Well, that’s 307,445,734,561 tons of rice. The Empire State Building weighs in around 370,000 tons, so there’s almost 1,000,000 times as much rice.

One million Empire State Buildings worth of rice.

Sideband #28: 2 ^ 64, ‘K!

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.)

Now the number, 64, is a significant one to computer programmers. For one thing, it’s 8×8, a full chessboard. Among programmers, the numbers 2, 8, 64, 256  and 1024 may be the most significant, although all powers of two have some significance.

The number 2, of course, is central to computer programming. A bit–the fundamental information unit–has two values: zero and one. (Hence my second-favorite bumper sticker regarding computer science: Calm down! It’s just ones and zeros!) Everything is based on this, which is why powers of two have the significance they do.

The number 8 is also central to computing in that it’s (by far) the most common grouping of bits. The well known computer word, “byte,” by default means eight bits. (And it does get its name from being a “bite” of  data. The half-sized package of four bits, a half-bite of data, is known as a nybble.) In fact, the history of computing contains six-bit bytes and nine-bit bytes, so the term “octet” is used to remove ambiguity. But these days, byte pretty much means eight.

Eight bits allows you to count up to 256, which is why it’s a significant number in computing. But these days 256 isn’t as useful as it used to be. In the early (American) days, 256 was more than enough codes to represent all the letters (both upper- and lower-case), all ten digits, all the punctuation symbols and plenty left over for control codes and drawing symbols. There was even room for some non-American characters, such as ñ and ö and Ω.

28 = 256; 82 = 64

Two to the power of eight is 256; eight to the power of two is 64. And 64 is the number of squares on a chessboard. It is also the number of bits used as a single package of data in many computers. (A 64-bit package is 8 8-bit bytes.)

And as mentioned above, a 64-bit number counts a good deal higher than an 8-bit number.  If you’re interested in the progression, it goes like this:

one byte (eight bits): 28 = 256
two bytes (16 bits): 216 = 65,536  (“64 K”)
four bytes (32 bits): 232 = 4,294,967,296
eight bytes (64 bits): 264 = 18,446,744,073,709,551,616

(Programmers tend to double things, which is why the powers go from 8 to 16 to 32 to 64.)

Which at least brings us to the point of this post! Two to the power of 64 counts to a rather big number, but it’s a number that has a name: 18+ exabytes. What if you had 64 K (65,536) bits?  How high would that count? First, here’s a  handful of ways to notate it:

2 ^ 64K
= 2 ^ (2 ^ 16)
= 2 ^ (2 ^ (2 ^ 4))
= 2 ^ (2 ^ (2 ^ (2 ^ 2)))

And here it is written out (ready?):

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87895905719156736

Now that’s a big number!

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. And a couple of my computers with many hundreds of spinning gigabytes are showing much less free space than not. On the other hand, hard drives currently are in the terabyte range, which seems very spacious (I haven’t come close to filling mine). Plus there is cloud computing, which removes the burden of storing your own data, and there’s the idea of streaming video and music rather than owning it. This all appears to, perhaps, put an end to the tendency to fill the available space. Or even need much of it.

So it may be moot now, but a dozen years ago it was a debatable point (this was USENET; everything was a debatable point) that even a drive with 64-bit addressing might end up being not enough. We’ll leave off the detail that disk space is never actually addressed at the byte level, but at the sector or cluster level, so the available bytes are actually quite a bit more than the address space suggests. It turns out to be a detail of no significance.

Given the see-saw history of storage filling and capacity growth, it did seem possible. Until one really looks at the numbers. (In fact, I wrote my own arbitrary-precision calculator decades ago for the very purpose of exploring such topics.)

With 64 bits, you can count to 18,446,744,073,709,551,616. The size of that number alone should give you pause. It’s beyond terabyte, beyond petabyte, it’s 18+ exabytes! And while large data storage organizations might need that kind of storage, it seems rather a great amount for any individual.

Still, five megabytes seemed pretty big once, and each jump in storage seemed to offer an all but unfillable wealth of space, but each time we did find ourselves wanting more, more, more. So it’s possible that 18 exabytes might somehow be too much for an individual user with a passion for owning movies.  Let’s run some numbers.

I started by considering a very long movie: Gone with the Wind. It runs 238 minutes; nearly four hours. (This was before Lord of the Rings or other long epics came out.) I bumped that up to just over 4.5 hours to give me a nice round 16K seconds per movie. I assumed a frame was 8K pixels by 4K pixels (which at the time seemed ultra-high fidelity). I also assumed 32-bit color, 60 frames per second and eight-channel, 24-bit audio with a 96 kHz sample frequency.

This gave me 131,979,144,069,120 (uncompressed) bytes per very long movie. That’s nearly 132 terabytes for a single movie. You might think you could fill up an 18-exabyte disk pretty easily at that rate.

You’d be wrong.

It would take 139,770 movies that size to fill your disk. And they would take you over 600,000 hours (72.6+ years!) to view. And that’s assuming you could watch 24 hours per day, every day of those 70-some years.

I suppose you could fill such a disk (at least in theory), but what would be the point? For that matter, can you even imagine over 100,000 movies you’d want to see? (Forget the 100,000… can you imagine the 39,770??)

And there is the matter of exactly how you would fill it up. Downloading the data would be a bit of a problem. Assuming a 1-gigabit internet connection (125 megabytes) running full-out, each (uncompressed) 132-terabyte movie takes over 12 (24-hour) days. (If that seems improbable, keep in mind the movie we’re considering streams at a rate of 8 gigabytes per second. Stuff that down your Netflix streaming connection!)

Bottom line: filling your disk with the 139,770 movies would take 4,671 years!

Which all tells you something about the bit rate of real world information as well as the degree of data compression involved in our media.

Sideband #13: The Number 42

Nearly all science fiction fans share a meme about the number 42. This meme comes from the Douglas Adams book, The Hitchhiker’s Guide to the Galaxy, one of the great “modern classics” (an apparent oxymoron, but it is just shorthand for ‘a recent work that is so good that someday it will be counted among the classics’). The book is the first in the “increasingly misnamed” trilogy that shares its name.

The trilogy is “increasingly misnamed” in that it now has five books. The joke is that, in science fiction, trilogies are as common as aliens, spaceships and time travel. In fact, depending on the context, there are a two trilogies that have earned the sobriquet, “The Trilogy.” (Issac Asimov‘s Foundation series in the context of pure SF; and, of course, J.R.R. Tolkien‘s Lord of the Rings books in the context of SF + fantasy.)

In any event, the number, 42, is the answer to the question.

Sideband #10: A Full Hand

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).

Despite the (admittedly very pedantic) fact that the new millennium actually began in 2001—the first year of the new epoch—most people celebrated the odometer change from 1999 to 2000. (In our IT department we had to deal with the Y-to-K issue. We spent a huge amount of time going through all corporate documentation and changing all those “Y”s to “K”s.  It was never clear why that was so important, but we got it done and just in time for the party.)

The baseball world was all agog this past week, because New York Yankee Derek Jeter got his 3000th hit (and he did it with a home run, a feat only ever equaled by Wade Boggs back in 1999—the last year of the previous millennium). Seems like getting 2999 (or 3001) hits is a pretty big deal, but 3000 is a record book entry.

In particular, we revere the major odometer numbers—the ones with all zeros (except for the “1″ on the very left): 10; 100; 1000; 10,000; 100,000; 1,000,000; etc. There is, perhaps, an instinctive reason for this. These numbers represent the digit positions themselves and are the basis of how we naturally represent numbers.

And they progress upon themselves:

• 100 = (10 x 10)
• 1000 = (10 x 10 x 10)
• 10,000 = (10 x 10 x 10 x 10)
• (and so forth)

As we’ll explore some other time, they can also be represented like this:

• 10 = 101
• 100 = 102
• 1000 = 103
• 10,000 = 104
• (and so forth)

But for now, Sidebands are 10.  Happy 1oth!