BOOOO! BOOOOOOOOOO!
BOOOOOO!!!
>> WHY is pi so darned mathematically significant…?
Because it’s a round number? 
MathGeek:
Also, there’s not much mystery about the number 9 (or -9).
9 is mysterious in the sense that it is the only number you can turn upside down to make it a 6. Then if you put them together you get 69 which is a very positive number (if you know what I mean). Try and do that with pi!
Now, now, sailor, pi is one very important part of what makes 69 so significant.
I admit, pi is not the best source in the world for random numbers. But for low-security applications, it does just fine.
Arguing about the ‘significance’ and the ‘utility’ of pi is like arguing about the utility of any part of advanced (and currently abstract) mathematics. There are still things left to discover about mathematics. Like, for instance, why this works: The Miraculous Bailey-Borwein-Plouffe Pi Algorithm. It is a little bit more than trivial that digit-extraction methods work for pi. It raises a few questions about how random pi really is. After all, if you can pull any digit of pi out of your hat like that, it gets the intelligent among us thinking.
So, sailor, maybe pi isn’t all that trivial.
That thumping sound you can hear in the background is me smacking myself in the forehead. Normally I can do arithmetic rather well, really…
Superdude: I can’t answer your question properly, as that’s not my field of mathematics (I’m a topologist, myself), but part of the fascination with pi in particular (as opposed to, say, the square root of…ahem…82) is that pi is just so darned ubiquitous. It shows up in geometry and topology, calculus, and statistics (the probability that a number chosen at random from the natural numbers has no repeated prime divisors is 6/pi^2), as well as in more esoteric areas like the study of elliptic integrals. And despite that ubiquitousness, it’s oddly recalcitrant as well: it’s not just irrational but transcendental, its digits are not only nonperiodic but appear to be randomly distributed, and so forth.
You also asked how it was discovered that pi was irrational. That’s a much nastier question, but one proof of pi’s irrationality can be found in the book “Pi: A Source Book” by Berggren, Borwein, and Borwein. Other places to look would be in some undergraduate calculus texts; I know that “Spivak: On Calculus” contains a proof.
As Math Geek mentioned, one of the unanswered questions about pi is whether it is a normal number (in base 10) or not (normal meaning, basically, all single digits occur equally often, all strings of two digits occur equally often, all strings of three as well, strings of four, and so on).
This is obviously not the same as being irrational–the number .1010010001000010000010000001… is obviously irrational, and obviously not normal (in base ten, or base two, or any base). (Normal is also not the same as being transcendental, but that’s not quite as obvious).
The plot element of Contact about information being encoded in the digits of pi is actually a flaw in the book. It’s been proven that almost all of the real numbers are normal (“almost all” meaning that the set of numbers that are not normal has Lebesgue measure zero). Pi is thought to be normal, but that has yet to be proven. One thing about normal numbers, however, is that any normal number has all information (information that can be expressed in a finite manner, anyway) encoded in it, regardless of what encoding system you use, simply because all possible finite length strings of digits occur in it. Almost all of the real numbers have all possible information encoded in their digits–if you go out far enough, you’ll find the entire works of Shakespeare encoded in the digits, signed and dated in chronological order; or the entire history of the human race, for that matter. So it would really be not at all surprising to find information “encoded” in the digits of pi; it is to be expected.
Anyway, it also seems somewhat closed minded to me to disregard all possibilities of normal numbers vs. non-normal numbers (plus whatever other knowledge can be gained from studying the digits of pi) having practical applications in the future. Ideas that originate as abstract mathematical concepts quite often are found to have practical applications later on, so don’t be too quick to judge.
>> I admit, pi is not the best source in the world for random numbers. But for low-security applications, it does just fine.
Nope. I am not going to let you save face on this one (feel free to do the same to me). There is an absolute contradiction between digits being random (as in I have no idea what’s coming next) and being part of a string which is predefined and publicly well known and downloadable. It is not a matter of degree but of concept. The randomness of pi is zero and cannot be used for any high or low security encription. Or do you have a case to show? In fact, the very notoriety of pi makes it the very worst key you can think of. If you were going to use a number which is not random, you may at least choose an obscure one.
Low security > Toy > How long does it take a person to figure out how to decrypt it?
Hey, nerds have fun, too.
The point in Contact wasn’t that messages showed up in pi… As Sagan correctly pointed out, you’d expect to find messages… Eventually. The thing is, though, if you’re looking for a fairly long string, you wouldn’t expect it to show up until after an astronomical number of decimal places. The size of the message “received” at the end of the book isn’t specified, but we know that it’s a square of a prime number, so let’s say 169. That was in base-11, and the message was composed such that that string contained zeros and ones exclusively. The probability of a random 169 digit base-11 string containing only zeros and ones is 1.9*10[sup]-15[/sup], and the implication in the book was that they didn’t need to go to anywhere near 10[sup]15[/sup] decimal places to find the pattern. Of course, this doesn’t even take into account the fact that this long string of zeros and ones happened, in the book, to match up to a raster for a perfect circle.
In other words, the “message” received in the book was definitely statistically significant.
Thanks for the clarification, Chronos. Admittedly, I haven’t read the book myself, I was merely going by what I’d read in this post and what I’d heard elsewhere. I still find it very interesting that almost all of the real numbers have this property of containing all possible finitely encrypted information, but I do see the distinction you/Sagan make. I guess I oughta read the book.
AHunter3 writes:
> I seem to recall the governor of a Bible-belt state once
> got unduly mystical about trinities and sevens and such
> and issued an edict that set the state’s official value
> of * to 3. Of course, it was substantially ignored by
> anyone with any reason to utilize *; meanwhile, the
> population of people who shared the governor’s regard for
> threes and sevens–the natural constituency in support of
> his pronouncement–by and large had no idea what he was
> talking about and also ignored the matter, so it does not
> appear to have won him any praise or brought any
> trinitarian influence to bear on godless mathematics.
> Which just goes to show that you cannot have your pi and
> edict too.
No, that’s just a distorted version of a story about the Indiana legislature that was pretty distorted in the first place. Here’s one of many explanations of this story you can find on the web:
http://www.cs.unb.ca/~alopez-o/math-faq/mathtext/node18.html
Here’s Cecil’s take on this:
http://www.straightdope.com/classics/a3_341.html
You can find several other discussions of this by putting “pi Indiana legislature” in the Google search engine. Nobody’s motivation in trying to push this bill through the legislature was remotely religious or mystical. The “discoverer” of this theorem was just one of those nuts who thinks he knows mathematics but can’t do a mathematical proof to save his life. (As the references mention, he claimed to have shown that pi was equal to several different values.) The legislator who introduced the bill did so mostly because he was trying to do a favor for a constituent. The legislature let the bill go through without much opposition (until finally someone pointed out just how ridiculous it was) because they considered it one of those resolutions with no practical effect that get passed because the sponsor of the bill tells them that he needs the resolution passed to impress his constituents.
Pi rules. I used to have it memorized to 60 digits, well on my way to 100 for no good reason at all, when I suddenly graduated college, got a job, and promptly forgot about it. Lesse how much I can remember…
3.14159265358979323846264338…
Hmm, that’s about it, and I don’t want to check and see if I’m right. At any rate, pi is certainly a mysterious number as others here have pointed out.
Is that so?
pi*x(1 + x)=n
Convert words into the ASCII equivalent. This is a number. Convert it to base ten. Apply it as “X” in the formula to get n. Round “n” off to a specific number of decimal places (using truncation error formulas we can retain our information without an infinite number of decimal places). Now, convert n to ASCII again as [ascii1].[ascii2] where the “.” represents the decimal place and ascii 1 and 2 are the numbers on either side of it. Our string now looks like
as.gH$@!µ.uuU
and so on. so what we need is a way to seperate words themselves from other words. Again, a “.” would do just fine.
How long would my string have to be for you, having the string and knowing full well that I used “pi” for my encryption, to be able to break it and read the message? Can anyone tell me this, or have I violated using pi as a key in some way?
I don’t think I’ve magically thought of the best encryption method ever, but can anyone figure out how hard a code like this would be to break?
I’m not sure I understand your algorithm. How exactly is n applied? Could you do a sample encryption of, say, my name? Joe_Cool= 74 111 101 95 67 111 111 108
Also, how do you DEcrypt? Pi isn’t really the key here (at least from what I can tell. I may be wrong), more like it’s part of the algorithm. Is the key the number of decimal places where you round? What piece of information is there, that knowing this would make you able to retrieve the original message? In other words, given a specific ciphertext and key, is there exactly one possible plaintext (original message)? If not, then the system can’t be broken, but also can’t be used because it’s flawed.
(whips out his ASCII Converter…wait, he doesn’t have one)
Ok. All Caps, though…
As Well, due the number of decimal places on my calculator I am going to make the formula simply pi*X. (as well, you’ll see what I mean by "converting it to base 10 which was sadly off the mark)
J 74
O 79
E 69
C 67
O 79
O 79
L 76
Since, due to the underscore, this is one word I seperated it into two for a better demonstration.
Joe = 747969 so far and Cool = 67797976
OK.
n[sub]JOE[/sub]= pi747969 = 2349813.9155129 (truncated)
n[sub]COOL[/sub] = pi67797976 = 212993623.3298571 (truncated)
23 49 81 03.91 55 12 09.21 29 93 62 03.32 98 57 01
Note that they come in pairs of two here, of course, even if it is a single digit. I cannot convert these smaller numbers back into characters on the screen, however, rest assured there are characters there.
But, there is an erl encrypted name!
There are many other ways of coming up with a simple formula to convert entire phrases or any number of characters at once like this. For example, the formula
.5((a + b)[sup]2[/sup] + 3a + b)
will give you a single unique whole number for any pair of whole numbers. If we convert words to numbers in the manner, or a similar manner, provided above it would prove to be very difficult. the problem here is that it is only unique to the unique message, but the method is the same. once the method is cracked (which I would love to see done) then all messages are compromised. It is my understanding that high-security encryption is different in the sense that the encryption method per message is different, and so to glean one not all messages are compromised.
Erislover, this is silly. We had a thread about cryptography not that long ago so let’s not hijack this thread and turn it into cryptography. Your system doesn’t even begin to being remotely close to useful. I am not going to go into the whole thing because it is utterly irrelevant. The point is there is nothing special about PI and any other of millions and millions of numbers can be used just the same if not better. If you want to discuss encryption I encourage you to learn the basics and then we can discuss it. I have always had an interest in it as you can see in the past thread.
To successfully counter what I am saying you need to prove not that PI can be used for any particular purpose but that it can be used better than any other number. In your example substitute sqrt(13) and it works (or rather, doesn’t work) exactly the same. The question, in case you have forgotten, is “what’s so special about PI?”
The answer is still: nothing other than it’s a round number.
Actually, getting back to what I was saying about normal numbers and how they contain all possible finitely encrypted messages, and how almost all the real numbers have this property, the fact is that no absolutely normal number has ever been found (“absolutely normal” meaning normal in any base). It’s always possible that pi could be the first absolutely normal number to be discovered, which would make it pretty special, I think.
The point is this: How ubiquitous is sqrt(13)? How many different fields have to deal with it? How entwined into nature is sqrt(13)? You don’t have to look deeply into math or physics to see examples of the importance of pi. But the deeper you look, the more you see. There are special numbers, whether anyone likes it or not. 1 is a special number. So is 0. And pi. And e. And sqrt(5). But there are numbers that are just mundane numbers with no special mathematical significance, other than that they exist.
Remember the OP? Practical Application
pi may be very mathematically interesting and pop up all over the place, but these don’t prove the usefullness of knowing the quadrilliionth+1 digit. Some have said cryptography is a useful application, Sailor (IMHO) has decimated this contention.
Let’s not lose the OP for all the digressions.