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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."
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.
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.
... 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.
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.
As far as I have read, this has yet to be proven.
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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.
Or so I'm told... :)
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
you must be from Indiana...
I wrote a Pi calculating program, that worked in base Pi. It didn't take long at all to compute pi, and it is a great source of random binary. The answer I got was "10". Now, simply take one of those digits 8 times, and you have a completely random byte.
Mod parent down, he needs to take a basic number theory class. It has not been proven that pi is normal. It has been proven that there are all kinds of infinite sequences which are not normal. Random is not the same as normal
OMG!!! You mean Pi knows my SSN??? It must be a terrorist! We have to do something! (Maybe it knows where WMDs are, too)
"I was under umpression that truly random data should be completely uncomressible."
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.
That is a true and fun little fact, but it is nothing special to pi. You can do that with any irrational number, i.e. sqrt(2). Anyway, this story is ridiculous, noone pay attention to it. They did (from the article) 2 or 3 tests, the most significant appearing to be dividing 100 million digits into blocks of 10, plopping a decimal in the front. They then grabbed these blcoks in groups of 3 for x,y, z coordinates. They mapped these points in an imagnary cube and then graphed their distribution in the cube. From this they concluded that the other RNGs are more random. That is an extremely false conclusion. Arguing that one distribution is more random simply because it covers more of the cube or it's distrbution is more of a bell curve is just plain stupid so I really hope I missed some important fact when I read the article. Random is random and there is no rule saying that randomness is only random if it is distributed evenly or forms a bell curve (any such constraint would go against the nature of being random). Most RNGs try to distribute digits in a even manner because for cryptography purposes it is important, but is pointless when trying to deal with true sources of randomness. The fact that there is any such predefined distribution obviously shows that it isn't random (thus they are called pseudo-random), but arguing one algorithm generates a bell curve and another doesn't so the first one is better is just a dumb argument when dealing with random numbers. I hope a few mathematicians chime in and either blow my argument out of the water or confirm what I said.
Regards,
Steve
be careful what you prove next or zebra crossings might become dangerous.
Warning: Opinions known to be heavily biased.
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
Even quantum physics, although theoretically 'random', is generally predictable and reliably recreatable for a large T distribution over time.
If you want truly unpredictable, unrecreatable, random numbers - let my wife balance your checkbook.
Glonoinha the MebiByte Slayer
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!
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
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
No... 1 is one in all bases, by definition. 10 is the number of the base in all bases, again by definition. In base ten with represent ten as 10. In base pi, pi is 10.