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I was told that the digits of $\pi$ are random (or at least nearly so). Would $\pi$/2 etc. also have that property? Which other numbers have that property? In case there are a vast number of them, do they have some common properties such that one could computationally randomly pick one out of a class of such numbers for use? If one combines two such numbers digitwise $\mod 10$, is it always true that the result can only be better random and never worse?

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See en.wikipedia.org/wiki/Normal_number. –  joriki Sep 27 '12 at 15:26
    
You way be interested in this Wikipedia article on normal numbers. The question you are asking is about simply normal numbers to the base $10$. –  André Nicolas Sep 27 '12 at 15:28
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Actually, those numbers that are most easily proved to be base-10 normal (like Champernowne's constant) are far from what one might really call random ... –  Hagen von Eitzen Sep 27 '12 at 16:08

3 Answers 3

$\pi$ is conjectured to be a normal number, which means that all digits and all combinations of digits appear just as often as you'd expect if you were choosing digits randomly, at least asymptotically. Almost all numbers share this property, though it's hard to prove that a given number has it. Numbers known to be non-normal include rational numbers.

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Do all irrational numbers have that property? If yes, could you please give a reference? –  Mok-Kong Shen Sep 27 '12 at 15:31
    
From the Wikipedia page: "It has been conjectured that every irrational algebraic number is normal; while no counterexamples are known, there also exists no algebraic number that has been proven to be normal in any base." –  SiliconCelery Sep 27 '12 at 15:40
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@Mok-KongShen: Not all irrational numbers have that property. An example that does not would be $0.110100100010000\ldots $ or any irrational decimal that has no $7$'s But almost all of them do. –  Ross Millikan Sep 27 '12 at 15:42
    
I think transcendental numbers like pi and even other irrational number appear to be "normal". I don't know how you could really show it though since it is a matter of looking at an infinite sequence. Rational numbers are know not to be normal because for example 1/3 is just the repeating decimal number 0.3333333333... Other rationals will eventually repeat destroying any chance to be normal. –  Michael Chernick Sep 27 '12 at 21:10

On your final question

If one combines two such numbers digitwise mod 10, is it always true that the result can only be better random and never worse?

the answer is no if combining digitwise means interleaving: if $\pi$ has random digits then so too does $10-\pi \approx 6.858407\ldots$, but their combination is decidely not random in that you can predict half the digitis from the previous digits.

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The digits of $\pi$ are, of course, deterministic and so not random. But looking only at the digits, they are nonrandom in an important sense: they have low Kolmogorov complexity. There are short algorithms that can (given enough time) generate as many digits of $\pi$ as desired, and so the digits of $\pi$ are highly compressible: instead of writing out the first billion digits, taking on the order of a billion bytes, you can write a program for generating pi to a given number of digits (taking a few kilobytes) and then the number one billion, taking a few additional bytes. Truly random numbers cannot be compressed in this fashion.

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