Pi: Less Random Than We Thought 416
Autoversicherung writes "Physicists including Purdue's Ephraim Fischbach have completed a study comparing the 'randomness' in pi to that produced by 30 software random-number generators and one chaos-generating physical machine. After conducting several tests, they have found that while sequences of digits from pi are indeed an acceptable source of randomness -- often an important factor in data encryption and in solving certain physics problems -- pi's digit string does not always produce randomness as effectively as manufactured generators do."
This just in: (Score:2, Funny)
Re:This just in: (Score:4, Funny)
Re:This just in: (Score:2)
because it ain't random (Score:3, Funny)
Re:because it ain't random (Score:5, Funny)
Re:because it ain't random (Score:4, Insightful)
So I could take, for example, every 14th digit in Pi and that would make a good random string of numbers between 0 and 9.
Re:because it ain't random (Score:4, Informative)
Re:because it ain't random (Score:4, Insightful)
Re:because it ain't random (Score:2)
But if you did it again it wouldn't be as random as a random number generator.
Re:because it ain't random (Score:4, Informative)
That's why most random number generators let you specify a seed value. As long as you use the same seed value, you get the same sequence back. If you want a new sequence every time, peg your seed value to some number that varies, like the current time...
Wow: a spoon isn't a good fork! (Score:2)
Re:Wow: a spoon isn't a good fork! (Score:2, Funny)
Re:because it ain't random (Score:2)
Re:because it ain't random (Score:3, Informative)
As a side note, the current record by the Kanada lab is 1.2411 trillion digits, their previous record was a bit above 200 billions.
Re:because it ain't random (Score:2)
and this has what to do with random? (Score:3, Insightful)
Re:and this has what to do with random? (Score:3, Informative)
It's just arrogant to believe that pi goes on forever just because we want it to. When we find that pi comes to an end, all those mathematicians will look really fucking stupid.
Bollocks yourself. If pi can be expressed as a finite series of digits, then i
Re:because it ain't random (Score:2, Funny)
And drum roll please:
3 !
Re:because it ain't random (Score:3, Funny)
Re:because it ain't random (Score:5, Insightful)
Yes, that's right...
and therefore cannot be exactly determined
Er, that depends on what you mean by "exactly determined". Do we need to know the digits in decimal expansion (base 10) to "determine" pi? How about saying that pi is exactly "1.000" in "base pi"? IMHO, whether or not a number can be exactly determined is independent of whether its decimal expansion is known. By your logic, sqrt(2) cannot be exactly determined, as it is an irrational number and has infinitely many digits (and they aren't periodic, unlike 1/3=0.33333333333... which also has infinitely many digits). But I am not entirely comfortable with saying that sqrt(2) cannot be exactly determined. After all, we know exactly what it is -- the positive number whose square is two.
I expect e and the square root of 2 to be better choices
WTF? How is e a better choice? It is also a transcendental number, just like pi. And sqrt(2) isn't even transcendental!
Re:because it ain't random (Score:3, Informative)
Re:because it ain't random (Score:3, Insightful)
Except that usually you use integers as number bases...and for good reason. I can't show you what 1 apple in base PI looks like without fractions. I'd hate to have 1 Pi fingers to count on etc. It gets tough when you count with fractions.
We're surprised? (Score:2, Interesting)
Uhh.. we're surprised? Pi can be described by numerous simple iterative formulas. When we do that with especially built algorithms we get pseudo random numbers.
I'd expect pi to be much worse than a PRNG.
Computing any digit of pi (Score:5, Interesting)
GOD HAS 16 FINGERS! (Score:3, Funny)
and thus God disappears in a puff of logic (Score:4, Funny)
Re:Computing any digit of pi (Score:2)
Re:Computing any digit of pi (Score:5, Funny)
Re:Computing any digit of pi (Score:2)
Re:Computing any digit of pi (Score:2)
Re:Computing any digit of pi (Score:3, Informative)
Re:Computing any digit of pi (Score:5, Informative)
I don't believe that's true. Pi is a transcendental number [wikipedia.org], which pretty much precludes it being periodic in an fractional base.
(Assuming by "fractional" you mean "rational", the ratio of two whole numbers. Sorry to be picky; I'm just trying to be complete.)
You can compute arbitrary digits of pi in hexadecimal (and binary and octal and any other 2^n base), but as far as I'm aware there isn't any corresponding algorithm for decimal numbers. I'm not certain it's been precluded, either, but I'm fairly certain you won't find pi to be periodic in any fractional base.
If there does exist a proof that you can't do it in decimal, I suspect it will involve the fact that there exist fractions in base 10 that don't have terminating representations in base 16 (e.g. 1/5). That'll make it hard to apply the algorithm from base 16 back to base 10; a one-bit change in the base 16 representation will have dramatic effects all over the base 10 representation.
(I'm not a mathematician, but I used to be, which is why this post is so maddeningly vague. I hope somebody gives you a better answer than I just did.)
Re:Computing any digit of pi (Score:3, Funny)
No, he means fractional. It's exactly equal to 1.0 in base Pi.
I had great hope when clicking on the link... (Score:5, Insightful)
> Pi never scored less than a B on the tests, and in one case outperformed all the RNGs, which in addition to mathematical algorithms included a device that uses turbulence in a fluid as its source of randomness. But in most cases, pi lost out to at least one RNG, and in several it finished decidedly in the middle of the pack.
Obviously. There is no reason that pi would beat every RNG out there on a sample of numbers. It should just be slightly ahead the pack (if some RNG are bad), or just in the middle (if all are good).
> "Our work showed no correlations or patterns in pi's number set - in short, pi is indeed a good source of randomness," Fischbach said. "However, there were times when pi's performance was outdone by the RNGs."
Well, there is a reason why mathematicians consider that statistics are not a branch of mathematic. And such article are a proof of it.
pi output on the statistical tests were correct (if they werer not, then it would be an important news, as it would imply correlations). The fact that some other RNG generated "better" output for the (relatively) small sample they used is meaningless.
Slightly different view.. (Score:5, Insightful)
Are you sure it's random? (Score:5, Funny)
Number Generator Troll: "Nine Nine Nine Nine Nine Nine"
Dilbert: "Are you sure that's random?"
Accounting Troll: "That's the problem with randomness: you can never be sure"
Re:Are you sure it's random? (Score:3, Funny)
"The string 999999 was found at position 762 counting from the first digit after the decimal point. The 3. is not counted."
Not Random (Score:2)
Of course a source you know to be random will be more random than Pi which is still arguably not random.
Re:Not Random (Score:2, Informative)
Pi's digits also pass certain tests for non-correlation.
OOGG wish correct additional mistaken assumption of yours:
Knowledge or proof of FACT does not CHANGE fact. ONLY CHANGE OUR KNOWLEDGE.
Digits of PI either "random" or "not random" depending on definition
Re:Not Random (Score:2)
Completely Unsurprised (Score:4, Interesting)
When you cite for example a deviation from a Chi distribution, then there is probably some connection between Chi and Pi that doesn't seem obvious from how Chi is calculated, but is there non-the-less.
I am not a mathematician (though I did work at Wolfram Research for ten years). I look forward to seeing real mathematicians take on this.
Re:Completely Unsurprised (Score:5, Interesting)
On the other hand, as you say, they're essentially "pseudorandom" in the sense that they can be computed by a deterministic program.
What you're groping for here is called "Kolmogarov complexity", or sometimes "Kolmogarov- Solomonov- Chaitin complexity" which can be defined as the length in instructions for some fixed machine of the shortest program that can compute an output sequence. If, without loss of generality, we choose something like a conventional machine, you can think of this as the length of the shortest program in bits.
What's kind of amazing about it is that there is a supremely elegant and simple proof that there are "really" random sequences in the sense that there is no program that can compute and output a sequence random sequence R that's any shorter than "print R". This is what you're looking for in the sense you're talking about "randomness".
(The proof comes directly from the fact that there are more bit sequences of length n than there are sequences of length (n-k) for k>0. Thus there must exist sequences of length n which can't be computed by a program of length (n-k).)
This leads to all sorts of cool stuff, including things like a unification of Gödel's Proof, Turing's Halting proof, Hilbert's Tenth Problem, and chaos theory.
To learn more, Google for "algorithmic information theory" and "Gregory Chaitin".
Re:Completely Unsurprised (Score:3, Interesting)
And provably optimal AI [idsia.ch]. Truly fascinating stuff.
Re:Completely Unsurprised (Score:3, Informative)
Pi has not been proven normal.
http://mathworld.wolfram.com/NormalNumber.html [wolfram.com]
There must be a bug in my implementation... (Score:3, Funny)
Re:There must be a bug in my implementation... (Score:2)
I think there's a conspiracy here.
Re:There must be a bug in my implementation... (Score:2, Funny)
Re:There must be a bug in my implementation... (Score:2)
Pi experiments and random numbers (Score:4, Interesting)
I was wondering, maybe not more than an hour ago, why not get a TV card and gather randomness from there? There are lots of channels on TV, and they have both a video and an audio component. You could set the thing up to change channels at random intervals, and gather things like the color of random pixels at random times, the frequency of random sounds, etc. Perhaps you could use a radio card to do something very similar with the radio. That, combined with entropy from the keyboard, mouse, the time between interrupts of various kinds, the contents of various processor registers or random memory locations, or whatever, should provide basically a random pool that is so random, you'll never have to worry about security problems with relation to them.
Speaking of which, there are ten digits used in our radix 10 notation; if you want to store a character string in a strange format, you could conceivably store two digits in one byte, because four bits are enough to describe all ten digits, leaving plenty of room for things like a decimal point or a negative sign. I'm saying this because it's not too terribly expensive these days to get a terabyte of storage. If you store, on this terabyte, nothing but digits from pi, in this space-saving format I'm describing, you could store 2,417,851,639,229,258,349,412,352 digits from pi. You'd need some kind of cluster, like PI@home, to compute all those digits. Once computed, who said you can't use pattern-matching algorithms to see if there isn't some kind of pattern? I still believe that somewhere in there, there is a pattern, though it is very large. Hell, who said you can't get an exabyte of storage and do this? If anything, it could become one component in a random number generator that simply never repeats itself.
Re:Pi experiments and random numbers (Score:2, Troll)
Wow! What an unusual format. Lets think of a name for it. Hmmm... it's in binary, but it's encoded in a decimal form... I know, lets call it Binary Coded Decimal! It even has a catchy acronym (BCD) that fairly rolls off the tongue. Wow! Maybe we could get so
Re:Pi experiments and random numbers (Score:2)
Re:Pi experiments and random numbers (Score:3, Interesting)
Just dont tune in a channel and listen to the noise/take only the least significant bits.
Should work more reliable.
Re:Pi experiments and random numbers (Score:2)
You would think that someone would need a lot of points to do that, wouldn't you?
I wonder if the metamoderation system actually changes the score if someone clicks "unfair"... In other words, suppose I have 50% funny, 50% troll, and someone metamods the troll mod as unfair; what happens? Does the original troll modder get mod points less often?
Did Sagan See This? (Score:4, Funny)
Or so I'm told... :)
Re:Did Sagan See This? (Score:5, Interesting)
What's worse is that somewhere else is NTS video of the same hand, writing "I don't exist after all, yours sincerely, God."
(I leave the proof of this as an exercise for the interested student.)
Re:Did Sagan See This? (Score:3, Funny)
Re:Did Sagan See This? (Score:3, Funny)
Re:Did Sagan See This? (Score:4, Funny)
10.000000......
More on pi and randomness (Score:4, Informative)
ScienceNews article (2001) on Randomness of Pi's digits [sciencenews.org]
Interesting work from Johan on Testing the a-periodic randomness of and comparing it with a Quantum Mechanical source. [versatel.nl]
But are the digits truely random ? In 1996, NERSC [nersc.gov] Chief Technologist David H. Bailey, together with Canadian mathematicians Peter Borwein and Simon Plouffe, found a new formula for pi. This formula permits one to calculate the n-th binary or hexadecimal digits of pi, without having to calculate any of the preceding n-1 digits. This formula was discovered by a computer, using Bailey's implementation of Ferguson's PSLQ algorithm
chaos (Score:3, Interesting)
I think "random" has a misleading connotation. Just because something is highly unpredictable, it is not necessarily without pattern. We take "random" to mean something that cannot ever be predicted, that follows no pattern. But attractor fractals and many areas of Chaos Theory have proved that there are patterns that defy the human pattern recognition faculty (or at least require the use of a pencil, calculator, super-computer, etc.).
In other news ..... (Score:2, Funny)
Physicists have completed a study comparing the randomness in Darl Mcbride's brainwaves to that produced by 30 typing rats. After conducting several tests, they have found that while sequences of digits from Darl are indeed an acceptable source for randomness, Darl's digit string does not always produce randomness as effectively as rats if the rats are using unixware.
Rnadomn (Score:3)
Re:Rnadomn (Score:2)
No, random is when there is a equal chance that either of the elements are picked, and there is no way to determine what was picked afterwards.
Re:Rnadomn (Score:3, Insightful)
Technically speaking if we had enough infomation nothing would be considered random.
That might be true, except for the heisenburg uncertainty principle. In short it says you can never determine both the exact position of a particle and its momentum. The essential problem is that measurement of either of these properties disturbs the thing you're trying to measure in an unpredictable way.
The end result is that you can never have enough information. Randomness isn't a lack of understanding, it's a fund
Who said pi was *supposed* to be random?? (Score:3)
The only reason anyone could think it would be a good indicator of randomness is because its complexity goes beyond the comprehension of man or machine. I'm not a professional mathematician, so there's not a lot about the nature of pi I can comment on, but it seems to me that in being an ordered number that describes a physical phenomenon, pi has about as much chance to produce randomness as counting the number of leafs on clovers.
Re:Who said pi was *supposed* to be random?? (Score:2)
In lower level statistics courses, to pick random numbers, you use a line from a random number table. These numbers were generated using some pseudo-random methodology like observing a process that's nearly random.
Now, Pi comes into this because it's an irrational, so it has a non-repeating decimal expansion. You can use that decimal expansion to cherry pick number
Mafia number theory (Score:2)
Duh... (Score:2)
Pi is not random (Score:2)
Duh.
Microsoft Solitaire totaly random! (Score:2)
Does the concept of anything "random" really exist or is it just a word synonymous with obfuscation?
Re:Microsoft Solitaire totaly random! (Score:2)
Forum on Random number genration [halfbakery.com]
I detect a disturbance in the force... (Score:2)
This has bothered me since I first ran across it in a colloquium when I was in grad school (in math) in the early '90s. It's more a matter of symantics than anything else, but it still bugs me becuase, like the difference between the words "secure" and "vulnerable", it leads to a lot of confusion.
Let's say I ask you if any old sequence of numbers (like this one) is random or not:
1,0,1,0,1,0,....
The correct answer is that you can't tell me unless I also tell you HOW the sequence was generated: Did
In school (Score:5, Funny)
Admittedly, he wasn't a math teacher though...
The randomness of 1/4 (Score:2)
It's a Firefox 1.0.3 on a Mandrake 10.2.
From a mathematician ... (Score:3, Insightful)
(2) If you REALLY want randomness (with impossibility of prediction, and unreplicability of the sequence) - go and count events in a radiactive decay experiment. (More precisely, count waiting times for each successive decay - they follow an exponential distribution). (I think fourmilab has a 1-time rnadom number generator linked up to a geiger counter - don;t remmeber the URL any more).
(3) Why do mathematicians find "randomness" in digits interetsing? The reasons are similar to why people prove theorems about "how randomly are the primes distributed among the integers". It says something about the structure of the primes. I am not a number theorist - so I cannot give explicit results.
Al-Kashi, a cool mathematician (Score:5, Interesting)
In the first half of the 15th century the Persian mathematician Al-Kashi calculated pi to 14 places. It would be over a hundred years until a European calculated it to 9 places. But that's not what makes Al-Kashi cool, the Arabs where so much better at math in that period. What made him cool was that he stopped. He observed that, with his pi, the calculation of the circumference of a circle with a radius twice the size of Earth would have a margin of error smaller than a "horse hair" (a Persian unit). Problem solved, next problem. Meanwhile, people are still today using computers to get pi to _hundreds_of_billions_of_decimal_places!! As if there's something unique about pi because it's irrational and transcendental, when this is in fact true of the vast majority of all real numbers. Here's to Al-Kashi, a sane man and a pragmatic!
Chudnovsky Brothers, a cool mathematician (Score:3, Interesting)
In case, you find that interesting, here is a more recent article on their exploits.
Capturing the Unicorn [newyorker.com]
Re:Al-Kashi, a cool mathematician (Score:5, Funny)
Lazy bastard.
Re:infinitely improbable (Score:4, Insightful)
Re:infinitely improbable (Score:4, Funny)
Re:infinitely improbable (Score:4, Informative)
Re:infinitely improbable (Score:5, Informative)
Re:infinitely improbable (Score:3, Interesting)
http://www.angio.net/pi/piquery [angio.net]
Re:infinitely improbable (Score:4, Funny)
Re:infinitely improbable (Score:2)
In other news, Pi has been refused the status of prisoner of war, since it's just a number...
Re:infinitely improbable (Score:3, Insightful)
Re:infinitely improbable (Score:3, Informative)
Re:can pi be described in one line of perl? (Score:2, Insightful)
Re:Why pi has no exact value (Score:2, Funny)
Polar coordinate can also use pi: circle of radius pi, arc length of arc subtending angle of pi radians, etc.
OOGG recommend you not change major to math. Otherwise, GPA likely much less than pi.
Re:Why pi has no exact value (Score:3, Insightful)
What I mean by metric is that, after change of coordinates into the polar system, points are still the same distances apart from each other (the mapping from the cartesian plane to the polar plane is an isometry). Therefore, any circl
Re:Truly random (Score:3, Informative)
Technically it is *chaotic* data that is not compressible. Since random data is almost always chaotic, people tend to play loose with the terminology. But random data can happen to be ordered very well, in which case you can compress it.
"Random" is a feature of the method by which a number was created.
"Chaos" is a feature or the number itself, regardless of how it was created.
Re:Truly random (Score:2)
I can make a pseudo-random number generator which generates a very big uncompressible file (with any entropy encoding method), but then you could say - I can compress it by finding your program by brute force.
Our mathematical system (with which you wrote your formula for pi) is only one of infinite systems with which we can describe sequences of numbers. If I make a compiler which generates a random-number generating program, when the input to the compiler
Re:I'm not too suprised (Score:2)
The control group wasn't all PRNGs. Most hardware random-number generators are indeed not pseudo-random: they use quantum effects or something to generate truly random numbers.
Pi isn't "random" in the sense that it isn't a chance physical constant like, say, the gravitational constant. It's the only ratio of circumference to diameter possible in a Euclidean geometry. It's half the period of the solution to d^2 y/dx^2 = -y. Quantum effects, however, are God rollin
Re:I'm not too suprised (Score:2)
Re:I'm not too suprised (Score:2)
One can precisely compute arbitrary digits in pi, it seems -- given awareness that it's pi, because it's a constant and not a random process.
Re:I'm not too suprised (Score:2)
But, it's also everything in the sense that if one would encode a language like English into numerals and then if one could somehow zoom to the appropriate sequence within Pi's digits, one would find, for instance, the Declaration of Independence.
Re:I'm not too suprised (Score:5, Funny)
If you want truly unpredictable, unrecreatable, random numbers - let my wife balance your checkbook.
Re:How is something (Score:3, Interesting)
Re:is it just me or.. (Score:2)
For 99+% of uses, pseudo-random numbers are perfectly acceptable. For the remainder, there's likely to be enough money to build a true hardware-based random number generator. Or if there isn't enough money, then clearly the project doesn't need truly random numbers that much.
Re:My crackpot PI theory (Score:4, Insightful)
This is untrue. The most common fallacy about random numbers is that they need to "appear" random.
Of the list of numbers,
734901253789
666666666666
123456789012
Which is random? One answer is that all of them may be random. There is no reason why 1234 is any less random than 7305. A truly random number with infinite digits will absolutely repeat any sequence of numbers you can think of of any length whatsoever.
Think of it this way: If you have a true random number generator, spitting out a digit every second, and you see it spit out:
1...2...3...4...
then can you predict what the next digit will be? If it is truely a random number generator, the answer is no, you can not. However, the next digit has a 1 in 10 chance (0..9) of being a 5, so it is possible. If you reject 1...2...3...4...5 as possible sequence, then you have instituted a rule restricting the possible outcomes of the random number generator--and have therefore reduced it's effective randomness. Rules defeat randomness, so 12345 is as valid a random number as any other sequence of five digits.
Jim