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Pi: Less Random Than We Thought
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."
PI is exactly three.
Pie....oh, wait! Pi! Sorry :)
Gee, they found that pi wasn't random. Imagine that. Maybe someday we'll even be able to predict the value of pi.
I'm an American. I love this country and the freedoms that we used to have.
It's always cherry, apple or blueberry...
The parent was definately not random and completely predictable in the ./ sequence of events.
No matter where you go , there you are.
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.
since computer programs aren't random, the encryption and decryption process is not random, and attempts to crack programs are not random. If the programs surrounding a Pi encoded message were truly random, then Pi might be more suitable than the program generated psuedo random numbers.
I Want To Believe
If you calculate pi long enough you'll come up with everything of any significance. Try downloading a pi generating program, such as PiX and requesting enough digits to run for a few days, then open the file in a basic text editor and search for things like your social security number, your phone number, etc. Part of it being infinate is comming across every possible sequence.
now that would be compression
Given that its possible to compute any digit of pi without computing the preceding digits its not surprising that the digits have structure. The bizarre part of this algorithm is that computes digits in hexadecimal.
Two wrongs don't make a right, but three lefts do.
"less random"?
... but it seems a shitty research, based on the article:
> 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.
Y'know I would have thought this fact would have made it into at least some religious text books.
Deleted
Doesn't it bother anyone that any software random number generators are merely pseudo-random? How do check randomness of Pi by comparing it to a pseudo-random set of numbers..?
Did you know that "FTW" ("for the win") is a direct translation of "Sieg Heil"?
No I can't prove it, but there is no proof that Pi was random in the first place. This is just an assumption.
Of course a source you know to be random will be more random than Pi which is still arguably not random.
"And we have seen and do testify that the Father sent the Son to be the Savior of the World"
1 John 4:14
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.
Letter To Iran
...of pi. It's not random at all, I always get 3.14159....
Doesn't it make you feel good to know that our freedoms are protected by politicans, lawyers and journalists.
Sure, that's what its value is this time but if it turns out to be random after all then next time it might be 7.23129873672167367182637678 exactly.
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.
Suddenly we'll run across an enormous line of billions of ones and zeros.
"chaos-generating physical machine"
The fact that this exists, whatever the hell it is, is pretty cool. (I don't want to read the article; I prefer to imagine it otherwise)
"excellent..."
Beats moon pies!
Or so I'm told... :)
I'm no math major, but it's fairly pointless that people marvel over pi having no exact value. It's because we use a cartesian plane system to measure it. It's like in Calculus with finding the area under a curve. It can't be accurately done to an exact amount using an x/y system. Using a polar graphing system on the other hand does give an exact value of pi and other curves since it uses 'round' units....
In undeveloped countries, the consumer controls the market. In capitalist America, the market controls you.
If pi were truly random, there would be a better chance of me passing my math course.
Help a poor college student. Send a couple cents via paypal to chucks86@gmail.com
they say: pi's digit string does not always produce randomness as effectively as manufactured generators do.
I say: apparently the deep sequence of pi digits in base ten is less effective at predictably producing random sequences than something that is supposed to produce randomness predictably, therefore methods with pi are less predictable, and therefore truly more random.
http://tinyurl.com/4ny52
ScienceNews article (2001) on Randomness of Pi's digits
Interesting work from Johan on Testing the a-periodic randomness of and comparing it with a Quantum Mechanical source.
But are the digits truely random ? In 1996, NERSC 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
I was under umpression that truly random data should be completely uncomressible. If that's the case, then PI doesn't quailify because n-th hex digit of PI can be expressed using the following formula:
pi = sum (4/(8n+1) - 2/(8n+4) - 1/(8n+5) - 1/(8n+6))*(1/16)^n
That's a really good compression, if you ask me. What am I missing here?
"You mortals are so obtuse." -Q
Fractals, which resemble nature, are not random though they appear to be. Therefore, I've often considered all the universe to be one giant, multi-dimensional fractal.
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.).
Esoteric reference.
Your tax dollars at work.
"Lord, grant that I may always be right, for Thou knowest that I am hard to turn" -- A Scots-Irish prayer
You can't mathematically prove randomness simply by comparing two numbers. You have to know something about how they were generated. A perfect RNG could spit out 50 billion 2's in a row. Highly unlikely, but technically possible. Mathematics does not deal with any level of certainty other than absolute certainty.
So if an apparently random number can be created by a program that does nothing but spit out that same number every time, and a perfect copy of my telephone book can just happen to be spit out by a true RNG, a determination of randomness is obviously an investigation into the method of creation, not the result.
Now if you are talking about chaos, that is a whole different story. Even a completely non-random number can have perfect chaos, and a completely random number can have no chaos at all (like those 50 billion 2's).
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.
I just can't imagine what you are thinking.
That would mean that a cracker would know that every encryption key generated utilized the same random input at time of the key creation. And that would make it non-random, even if the sequence "passes the test" when analyzed by their algorithim.That would make the encryption just a tad more effective than ROT13.
The scientists' randomness analysis is flawed and way too simplistic: A naturally occuring scalar approximation of a constant tests as good randomness. I suggest they recruit a brilliant economist weigh in on their algorithim.
One good example: analysis of digits occuring in accounting books can reveal cooked books with a high degree of accuracy. The specific cooked numbers are not revealed, but the occurences (but not frequency) of digits in the books is understood to be non-random. Don't ask me to explain why. Google for it, there's papers on it.
Point being, the article states they are only testing for the randomess of the sequence as it compares to chunks of itself, but all those chunks are a fixed set of digits, and are unchanging. No one believes that's good enough for RNG do they? And you'll still need a RNG to pick which 10digit sequence of PI you will employ ;-)
Anyone seen my low uid? last seen 10 years ago while panning the #@$# out of Taco's 'web based discussion system'
Isn't randomn just something we can't understand ?. Technically speaking if we had enough infomation nothing would be considered randomn. I guess with encryption you pick something thats pretty damn complex and then hope that your competitors agree with you.
I know this has been said before, but perhaps not in this way. Pi is a number that represents a (ideal) physical phenomenon. Yes, it's complex and (probably) infinite, but it still is a numeric representation of an exact property. To me, that automatically presupposes that by its nature it's ordered.
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.
The Internet is generally stupid
Holy crap. Careful -- the NSA might be using that soon. Great!
I just finished reading the digits one by one, and I can vouch that they are random. Take my word for it. I swear!
24 is the highest number, you bunch of fazools. Fuhgeddaboutit.
Shop as usual. And avoid panic buying.
be careful what you prove next or zebra crossings might become dangerous.
Warning: Opinions known to be heavily biased.
At the risk of losing karma, what an obvious statement!! Pi is a mathematical number used to calculate certain things such as circumference OF COURSE ITS NOT RANDOM, if it was then we wouldn't be using it for so many important functions
made in Michigan in 1897.
It says pi=3.
I have discovered a truly marvelous proof of killer sig, which this margin is too narrow to contain.
I an amazed that software custom designed for a task is better at that task than something not!
Engineering is the art of compromise.
It's the same every time.
Duh.
"After conducting several tests, they have found that while sequences of digits from pi are indeed an acceptable source of randomness..."
So what if the random number generators they compared Pi against were utilising the randomness displayed by Pi as their source?
Also, is there such a thing as randomness given that even in randomness, patterns could be seen over time, albeit an incredibly long time?
Scroll down the article page..
ABSTRACT
A study on the randomness of the digits of ¼
Shu-Ju Tu and Ephraim Fischbach
We apply a newly developed computational method, Geometric Random Inner Products (GRIP), to quantify the randomness of number sequences obtained from the decimal digits of ¼. Several members from the GRIP family of tests are used, and the results from ¼ are compared to those calculated from other random number generators. These include a recent hardware generator based on an actual physical process, turbulent electroconvection. We find that the decimal digits of ¼ are in fact good candidates for random number generators and can be used for practical scientific and engineering computations.
and here I was using MS Solitaire to generate my random numbers. :P
Does the concept of anything "random" really exist or is it just a word synonymous with obfuscation?
irrational does not mean random, it means that, not only is the decimal repeating, but there is no decreeable pattern in the digits. I think that article meant to say that Pi is more patterned than some random number machine's output.
I am the Alpha and the Omega-3
The article is shit.
If Pi is at 1 sigma off the mean for some test then it isn't news until more experimentation is done. They have to use 16 times more digits. If Pi is now 4 sigma off *then* it would be news. Otherwise the null hypothesis holds: Pi, Just as Random as We Thought.
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 I use a really poorly designed algorithmic pseudorandom generator? Did I flip a coin? Did I draw numbered balls from a hat? The method decides if it is random, not the outcome.
If the trials that generate each element in the sequence are random, than such sequences as 1,0,1,0,... or 314159... must be possible because the definition of randomness requires that each trial be independently as capable of generating your favorite digit as any of the others. Indeed, looking at it from the other side, any generator that is *not* *capable* of generating such sequences cannot be classified as "random" because these sequences have been, by design or fiat, disallowed.
I just think that the word 'random' is the wrong word to describe what they are studying here because it contradicts the standard definition given to every math student in Prob&Stats 401. They would be better off calling it something else.
"Lawyers are for sucks."
- Doug McKenzie
"Anyone who attempts to generate random numbers by deterministic means is, of course, living in a state of sin." --
To put a witty saying into 120 characters, jst rmv ll th vwls.
I don't agree with the term unknowable. Pi is certainly knowable. It just can't be expressed as a finite string of digits after a decimal point. But even if it were unknowable, that doesn't mean it is random. There are many algorithms in mathamatics that produce infinite series, but that doesn't mean they are random. Look at fractals for one example. A very simple math formula can produce an infinite and extremely complex mathmatical result, but even though that result is infinite it is certainly not random. Nor is Pi.
I'm an American. I love this country and the freedoms that we used to have.
I had a teacher who insisted that Pi is exactly 3.14, and that the radiation after nuclear explosion decays by a factor of 2 in exactly 5 hours.
Admittedly, he wasn't a math teacher though...
The real issue with statistics is that people who use them generally do not understand them. I get irritated with people all the time when people "prove" some statement. Statistics shows that a sample of the populace has some correlation within some bound that is likely to be true some percentage of the time. So, the real question is: what was the bound and what percentage of the time was the randomness within that bound. If PI's bound exists outside of the statistical error of the bounds of the other tests then one could say that PI is less random; however, it sounds like they indeed found a few tests where PI "beat" the other tests. In other words, the bound PI was within the statistical error of the other tests, but the computed mean was occasionally better. But, occasionally better is to be expected some percentage of the time. If it is with in that number of times, it is as you say, a meaningless conclusion. Statatics within the bounds of error are completely equal. Probability is math, but it is also just very probable that it is used wrong.
Bel, the mostly sane.. "Of course I can't see anything! I'm standing on the shoulders of idiots." -- Me
Before all you CS people jump on my shit, I know there are "practical" applications for number theory, but there are better tools to be found in number theory's bigger brothers: algebra, elliptic curves, commutative algebra, algebraic geometry, etc.
Anyone who isn't confused really doesn't understand the situation.
It's a Firefox 1.0.3 on a Mandrake 10.2.
If someone gave me pie right now I wouldn't care how random it was.
Pi can be simply calculated in base 16 according to the following formula:
h tml )
Pi = SUM(for k=0 to infinity) { 16^(-k) [ 4/(8k+1) - 2/(8k+4) - 1/(8k+5) - 1/(8k+6) ] }
(source: http://www.math.hmc.edu/funfacts/ffiles/20010.5.s
It was discovered in 1995.
(So much for randomness)
I love /. for the tech coverage but the math articles tend to suck. Why is that? And yes, I am a mathematician.
I'd give this letter an e.
My mathmatical theory is a little rusty now... I haven't been in the "theoretical" domain of things for some time...
But can someone explain to me how a string of numbers can be qualitatively judged as more/less/better/worse random than another? I always thought random was exactly that... and a random number generator could spew out 6 "8"s in a row, and well - there is a statistical probability (albeit low) that that would indeed happen - so as long as it doesn't happen every 30 characters/numbers... blah my brain is going to hurt momentarily.
Pi is also known as 22/7.
Debugging? Klingons do not debug. Bugs are good for building character in the user.
Here's something I've always wondered about:
Is it true that the digits of every trancendental number (pi, e, etc.) eventually become pseudo-random in the limit as you keep generating more digits?
And what about the pseudo-randomness of irrational numbers? For example, if n^(1/m) is not an integer (for positive integers n and m), then does that always mean that the digits of n^(1/m) eventually become pseudo-random in the limit as you keep generating more digits?
Actually, I'm most interested to learn if any interesting counter-examples are known. I.e. are there any known trancendentals (or irrationals of the form n^(1/m)) that "fail" the standard tests for pseudo-randomness?
Clearly pi is not random in any absolute sense. One can give an algorithm to calculate it. Of course if you don't have the benefits of a hardware random number generator you are stuck choosing some psuedo-random source but the idea that there is some universal hierarchy of psuedo-random algorithms is just downright idiotic.
Whether a particular psuedo-random algorithm is good or not depends quite heavily on what it is being used for. The requirements for a psuedo-random number for a monte-carlo simulation are much different than those needed for cryptography (in one you want an even distribution over some relevant space while in the other it is usually unpredictability that is desired). Moreover, even within one type of application the specific algorithm being used can make a huge difference. For instance if I am trying to use monte-carlo integration on some nice smooth polynomial the standard psuedo-random technique of feeding back the output of a mod operation to itself will work quite well but if you are trying to integrate a function defined by a similar sort of recursion on mods it may perform horribly.
So there simply can't be any answer of what psuedo-random algorithm is better or worse. Of course one might still ask what would be a good choice for the standard psuedo-random algorithm in an OS given most people's software choices. For this application you want some source that won't displat a pattern relevant to normal computation. Thus, since trigonometric and other pi related functions are very commonly used pi is probably a fairly dangerous choice even if it works well on test cases.
In other words any number or algorithm that like pi pops up in many computer applications is going to be a bad choice. Not because you can run some ridiculous test which will rank its 'randomness' but because it is much more likely to be relevant to the program someone is running. In other words pi is a bad pragmatic choice because someone is alot more likely to use monte-carlo techniques to calculate pi or something to do with pi than with some recurrent mod generator (or something more sophisticated but not so common).
Thus from a pragmatic point of view these tests are just dumb. They would be better off running an array of common software and seeing how it performs. From a mathematical point of view there is nothing to be said as all of these sources (except the hardware one) aren't random at all so the only question is pragmatic.
If you liked this thought maybe you would find my blog nice too:
...I wanted PI and used a random number generator and my radian measurements were all out of wack. I wanted random numbers so I used PI but it all came out the same every time.
Well isn't that the darndest thing.
Next time I'll just use a pseudo random number generator for those random numbers and leave PI for all them important rads.
Now if I can just figure out how to write an encryption algorithm using famous celebrity names, I'll be all set.
there's no such thing as random anything in this universe.
even einstein said "god doesn't play dice..."
the gist of that was that everything is predictable given enough understanding.
humans are not yet sophisticated enough to comprehend the nuances and sublties of the reality we live in.
give it a few millennia. we'll make some progress or die trying.
Science : Proprietary , Knowledge : Open Source
(1) "Pi is not random becuase I have a formula for its digits" is nonsense. Randomness is not the inability (or impossibility) to predict (at least in this situation). Randomness refers to statistical properties of the sequences. For ex. no correlation between conseq. digits, no corr. betweteen conseq. pairs of digits and so on brings a sequence closer to randomness.
(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.
I must tell you a story.
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!
Is there any reason to suspect that triplets of ten digit sequences of pi might be correlated? Why should someone use tax dollars to investigate? What on earth does "this random sequence is more random than that one" mean?
It all looks pretty random to me.
mt
the fact that PI can be represented by a formula does not mean that there is any correlation bitween digits in its decimal representation. And, no, you are wrong - it is possible to calculate binary and hexadecimal digits in isolation, but not decimal.
Accounting Troll: "Over here we have our random number generator"
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"
Recycle PCs and build a wireless community network www.hillsborough.org.nz
http://3.1415926535897932384626433832795028841971
Of COURSE Pi isn't random, it's a transcendental number with no end. And I have a crackpot theory about that.
Most representations of Pi start like 3.141592653589793... but Pi will run on to an infinite number of digits. HOWEVER, somewhere, someplace way WAY down the sequence, it will certainly start repeating the ENTIRE sequence again, like 3.14159....3141592653589793....
Sure you'll find a short number of substrings of Pi's digits, but if you could truly calculate it out to infinity, surely at some point you'd find Pi is a repeating number.
A truly random number would never repeat a sequence no matter how many digits were in the sequence.
Some math-head, please check my conjecture. If it isn't based on a totally bad premise, please accept it as "Sakusha's Conjecture" and I await its proof sometime in the next century or two.
n/t
C'mon people, it's funny...
The article says they also tested against "one chaos-generating physical machine".
This reminds me of a Dilbert comic, which I believe was also mentioned on the "Shuffle playing favorites" story:
[Tour of Accounting]
Accounting Troll: "Over here we have our random number generator"
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"
Honestly, how can you call something "more random" than something else? The whole concept of randomness is that you cannot predict the next number. Of course, if you have the right algorithm and know the seed, then you can predict all of the numbers that a "random" number generator will generate, just like if you know the position of pi, you can tell what number it will be. There really isn't much of a difference here.
Now, a "really random" number would be generated by taking the time of day down to the millisecond, using that to seed one generator, then use it to seed a different one...ad infinitum. But even then, these could be predicted.
The only real way to have a random number is to get it from something that we truly don't understand, such as the human brain. Tell everyone in a room of 100 people to pick a number, then have a person pick another person in the crowd to pick a number between 1 and 100, and that will be the number of the person to go to for the random number.
Other than that, you're pretty much stuck for a truly random solution. As for using it for encryption, since Pi has an infinite number of decimals, you get an infinite number of keys or whatever, so no matter what random number generator you use, Pi gives you a (relatively) easy way to get long strings of random numbers. It doesn't really matter "how random" it is, as long as it's random at all, or can't be easily cracked.
http://users.aol.com/s6sj7gt/picode.htm
:)
Quite an entertaining read
When you search for 666, you get it in the 2,440th place. Which can only mean...the world will end in the year 2440!!
Pi in base Pi is not a good source of random digits. In particular, Pi in base Pi is:
I suspect that random sequence quality level sited in the paper would apply to digits of Pi in any non-transcendental base: such as base 10. Many transcendental bases, such as base e, should exhibit the a similar quality level to that sighted in the paper. However I suspect that any base that was a result of a polynomial function of Pi might not do so well.
chongo (was here)
From TFA:
> pi cannot be expressed as a ratio of two whole
> numbers, and its apparently endless string of
> digits is sometimes expressed as 3.14159...
Does the author have a clue what he's talking about? The decimal representation of isn't "apparently" endless - it's actually endless, or else pi could be expressed as a ratio of two whole numbers.
It's hard to see the value of a report like this. It's acceptably random, but not as random as it could be? What is that supposed to mean?
Given that the article says glaringly incorrect things such as, "But no one has ever found evidence that calculating finer and finer values of pi will ever reveal an end to the string or that there is any regular pattern to be found within it" (pi has been proven to be transcendental, so the first part of this is clearly wrong; it's known that the sequence cannot end) it's a bit hard to take their confusingly stated thesis seriously.
http://mathworld.wolfram.com/NormalNumber.html has a good description
Not definable: PI is clearly definable -- it is the ratio of a circle's circumference to its diameter.
Not calculable: PI is clearly calculable -- that fact that we can calculate its value to an arbitrary degree of precision is proof of that.
PI is just about as tractable a number as you could ask for. It's easy to analyze and use. If someone really wants to deal with a tough number, ask them to try and calculate the digits of Chaitin's Omega for the Lambda Calculus.
From the article:
But no one has ever found evidence that calculating finer and finer values of pi will ever reveal an end to the string or that there is any regular pattern to be found within it.
I'd say the ancient proof that pi is irrational counts as pretty strong evidence that we will never find an end to the string. Duh.
For example, lots of large numbers follow Benford's Law. Excerpt: "Benford's law states that in listings, tables of statistics, etc., the digit 1 tends to occur with probability ~= 30% , much greater than the expected 11.1%" The probability distribution is logarithmic; the probability of a digit D is log10(1 + 1/D). This is a way the SEC checks filings for fraud. If the numbers are too evenly distributed, there's a good chance of fraud. Obviously if you know about this law you can spoof it to some degree, but it was an effective tool for a while (still probably is for some not so smart firms).
"Nature doesn't care how smart you are. You can still be wrong." - Richard Feynman
...the generation of random numbers is too important to be left to chance.
I am sorry I can't recall the exact title of the esssay but, Isaac Asimov once wrote something similar stating that we can use the decimal places of pi we have right now to calculate the circunference or radio of the whole galaxy (I think he could have wrote the "universe" but I really do not want to state it as I am not 100% sure) with an error of 1 inch.
As you say, can you imagine? so what TF is the use of calculating the 192301931029301293120 decimal position of pi??? it is like making a computer calculate the same position for 1/3, NO USE!
Ubuntu is an African word meaning 'I can't configure Debian'
How does one measure randomness in the first place? Can something (mathematically) be more or less random than another?
Plenty good enough for most of the great engineering of the 20th century. Another example of practical and "good enough".
Pi is giving digits that relate to the ratio of the diameter of a circle to it's circumference. That's a very definite thing.
The other generators are returning, well, noise.
I don't know the meaning of the word 'don't' - J
So they have only taken a very small number of the digits of pi. Would they get the same result if they took the first googleplex (10^10^100) of digits? Even that would prove nothing about the randomness of pi's digits.
Nice computer science, but poor pure mathematics.
There are four sorts of people in the world: fools, lunatics, idiots and morons. - Umberto Eco, Foucaut's pendulum.
Mmmm... Pie...
I don't know the meaning of the word 'don't' - J
I remember first learning about the idea of Kolmogorov complexity, and thinking about how incredibly one-way it is. Given a seemingly random decimal sequence, is there anything better than a brute-force search to determine that it's actually sqrt(5pi)^(pi-1) or some tiny construction like that?
And, of course, there can exist no algorithm to even determine the Kolmogorov complexity of an arbitrary string. There's no way to pull order from chaos, at least not in a general sense.
Interesting from an algorithmic point of view, if nothing else. The Kolmogorov complexity of, say, e or is very low, expressible in relatively simple mathematics (no funny functions, just infinite sums or integrals) in under twenty characters.
--grendel drago
Laws do not persuade just because they threaten. --Seneca
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In case, you find that interesting, here is a more recent article on their exploits.
Capturing the Unicorn
"sweet dreams are made of this..."
How is one able to say something is or isn't "random". All "randomness" we know(or don't) is applied probability. If something is thought to be "random" because it returns a different value than before that is actually less random in the sense that you can predict that number has a less chance of being chosen next time. This presents a problem in that all randomly generated numbers can be generated again if the conditions are right.
After all you probably know that the first digits are three, one, and f--whoa whoa whoa, you tryin' to get a test answer o' sumthin?
You can hold down the "B" button for continuous firing.
...only ignorance. That's why I hate statistics (and probabilistic computational complexity classes), because it admits human ignorance. We must know it all, dammit!
how do you measure "randomness"?
and if you can measure it, how can it possibly be random?
Apparently the point is to look for patterns beyond the point where digits are no longer significant enough to have any possible bearing on the physical universe. Though I can't vouch for the 100 millionth digit and thereabouts, it seems like they should have used the arbitrary digit formula and compared randomness with respect to distance from zero, power of power of .. of pi, or somesuch. So where is that point, and would perfect randomness in pi produce anything visibly obvious in our universe?
Would not any perfectly random number have to have subsequences of digits with imperfect, variant randomness? Otherwise, at any point, its randomness could be predicted infallibly, making its randomness, well, unrandom. Therefore, is this not honestly a confirmation of pi's randomness?
We find that the decimal digits of pi are in fact good candidates for random number generators and can be used for practical scientific and engineering computations.
So it seems paper authors do not want to say anything about pi's randomness and we will get in a second to why. They instead say: This study probably says more about our commercially available random number generators than the nature of pi.
Now that we established the fact that the headline is just sensationalism and not what authors intended to say, let's see where confusion comes from. (Most people can stop reading here.)
The authors generated some random sequences and looked at a variable that desribed the randomness of those. By construction, any measure of randomness of such sequence is itself a random variable obeying some unknown distribution. So the scores would themselves be random, therefore the rank of any RNG (including pi) is itself a random number. So the authors found that pi digits were ranked not on top of the list. But gee, the rank itself is random, so even if pi digits were ranked the worst of all, that doesn't tell us ANYTHING.
And as a final point, were the authors really to discover any non-randomness in pi digits, that would have effectively proven pi to not be a normal number, and that is something that has been waiting for a while for someone to sort out and would be a serious scientific achievement. So for all we know, pi digits are a perfect random generator and the paper does not disprove that.
Human history becomes more and more a race between education and catastrophe.
H.G. Wells, "The Outline of History"
Now I even I would celebrate in rhymes unapt the great immortal Syracusan, rivaled nevermore, who in his wondrous lore, passed on before, gave men his guidance how to circles mensurate.
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- aqk
F U
test
...and Carl Sagan guessed wrong when he said it would be base 11 that we'd find the message in Pi, but he'd probably get a kick out of it anyway and it's no worse than many guesses held up as proof of seeing into the future.
It's a small world and it smells funny; I'd buy another if it wasn't for the money; Take back what I paid (SoM)
It's simply:
3.
1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999 and so on.
The road to hell is paved with good intentions.
He's absolutely right http://mathworld.wolfram.com/NormalNumber.html
The road to hell is paved with good intentions.
Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin.
--John von Neumann
What do you mean my sig is repetitive? What do you mean my sig is repetitive? What do you mean....
Recalls this
I mean, sure the first billion numbers are only a B, but there's some real good A-grade randomness down around the forty-three trillionth.
J.
You're only jealous cos the little penguins are talking to me.
When one tests for randomness, pseudo random generators can often get scores higher than true randomness. One example of this is cyclic feedback registers, where all numbers will be used exactly one time before they repeat.
So, Pi getting a lower score may very well be because Pi is more random, not less.
Kim0
The most pressing open question about is whether it is a normal number -- whether any digit block occurs in the expansion of just as often as one would statistically expect if the digits had been produced completely "randomly". This must be true in any base, not just in base 10. Current knowledge in this direction is very weak; e.g., it is not even known which of the digits 0,...,9 occur infinitely often in the decimal expansion of .
Bailey and Crandall showed in 2000 that the existence of the above mentioned Bailey-Borwein-Plouffe formula and similar formulas imply that the normality in base 2 of and various other constants can be reduced to a plausible conjecture of chaos theory. See Bailey's web site for details.
There are four sorts of people in the world: fools, lunatics, idiots and morons. - Umberto Eco, Foucaut's pendulum.
There are two issues which are distinct:
(1) uncertainty principle, otherwise known as non-commuting quantum mechanical operators. As that, it is very likely to be an immutable fundamental part of QM. Nothing random about it however.
(2) the 'projection' of mixed states to eigenstates of classical observables and a "random" choice thereof. Now, this may not be "right" in a fundamental sense. If you had full Laplacian knowledge of the wavefunction of the Universe and integrated it as an initial value problem, where does the "random" come in? That is an added "hack" to fundamental QM.
I personally believe that it is nothing other than the practical impossibility of observing all the quantum mechanical phases of all the particles in a macroscopic (classical) device that us humans use to measure things with. But this randomness then has the same theoretical structure as "random" in a roulette wheel. Is roulette random? No, it obeys classical Newton's laws of mechanics extremely well. But it has a very large Lyapunov exponent (measure of divergence of initial conditions) with respect to the final ball position. Most humans can't integrate forward from the initial conditions (spin of the wheel, initial velocity from the croupier) to any level of profitability. Unless you have a computer on you. (which people have done, and which is now banned in casinos).
QM: Lots of quantum mechanical roulette wheels spinning very fast, and then you interact it with an object with 10^23 things of its own. Physical chaos not "intrinsic random" but damn good approximation thereof.
What about the golden ratio (sqrt(5)+1)/2, which is the most irrational number? It gets that name because continued fraction expansions are used to find the best rational approximation to a number (such as approximating pi by 22/7), and it can be proven that the golden ratio has the slowest converging continued fraction approximation.
Just figure out the probability of producing an infinite improbability drive and plug that into your handy finite improbability drive. Jeez even a janitor could figure that out...
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Can you be Even More Awesome?!
The interpretation of these results is fundamentally flawed. All number sequences, including "00000" (etc) are fully random. The study to which is being referred discusses the distribution of digits and digit sequences, which is a fundamentally different issue.
Pi was never thought to be random; in fact the idea that Pi is random is nonsense. However, the distribution of Pi's subsequences has long been under question, and in fact nobody has thought Pi's distribution was particularly even for quite some time.
The issue at hand is that there's no known way of addressing the characteristics of the complete sequence. We can talk about the distribution of the first N digits of Pi, but not of Pi itself. This study is just a confirmation of the expected results over the first chunk of the number.
It doesn't really say much, in effect.
StoneCypher is Full of BS