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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."
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
... 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.
As far as I have read, this has yet to be proven.
Switch back to Slashdot's D1 system.
$pi = 4 * atan 1;
It's not that Pi is random or was ever though to be. But you can generate random (or not so random according to the article) numbers by picking out single digits from Pi.
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.
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
I don't see why one should expect Pi to be the ultimate in mathematical random number generation. Its chaos comes from the fact that it is an iterative function; why should we assume that this particular iterative function generates more chaos than others? That would be too convenient.
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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.
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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
I'm going to take "not exact" to mean transcendental, which is probably the strongest "weird" condition that's commonly accessible. In that case, it has nothing to do with our choice of coordinate system, rather our choice of metric and, even more fundamentally, number system.
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 circle still has the same length (circumference) and diameter, so Pi still has the same value.
If we define Pi by infinite series rather than the ratio of circumference to diameter of circles in Euclidean geometry, however, it's still transcendental because of the real number system. Transcendental, for the non-math people, means it's not a solution to any polynomial equation with rational (i.e., p/q, where p and q are integers) coefficients. So, for example, sqrt(2) is irrational, but it's not transcendental, because it's the solution to x^2 - 2 = 0. Pi is transcendental because of the properties of the real numbers as a field, which takes the problem even deeper than the geometry.
Now, I think what you might've been getting at is we can go back and look at our number system and define the Pi-rationals (rational multiples of Pi) as our new rationals, because our choice of the "regular" rationals was arbitrary. However, if our circle has Pi-rational radius and diameter, their dividend is going to be in the "regular" rationals. This is impossible, so at least one of the two must be irrational in our new number system. Therefore, you still have an irrational number involved somewhere in the process, no matter how you slice it.
I love /. for the tech coverage but the math articles tend to suck. Why is that? And yes, I am a mathematician.
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 fundamental part of the universe.
AccountKiller
141
(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.
Pi is a transcendental number
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!
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
Plenty good enough for most of the great engineering of the 20th century. Another example of practical and "good enough".
How about saying that pi is exactly "1.000" in "base pi"?
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.
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