Re: Re: Creating a Random Function to Select an Irrational

*To*: mathgroup at smc.vnet.net*Subject*: [mg102243] Re: [mg102210] Re: [mg102176] Creating a Random Function to Select an Irrational*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Sun, 2 Aug 2009 06:01:00 -0400 (EDT)*References*: <200907310957.FAA19545@smc.vnet.net> <200908010801.EAA07130@smc.vnet.net> <2B9B5B52-EA39-418D-AE5F-269570E18AC0@mimuw.edu.pl>

On 1 Aug 2009, at 20:54, Andrzej Kozlowski wrote: > > > As an example: we could generate irrational numbers approximately > uniformly distributed in the interval {0,1} by a formula like this: > > Rationalize[RandomReal[{0, 1}], 0] + Sqrt[2]/10^18 > > the numbers so obtained are certainly irrational and approximately > uniformly distributed. Looking at their digits is not likely to > reveal any "regularity" but of course we know that they the > generated sequence has a "regularity": the difference of any two > numbers is always rational. This was certainly incorrect: one will of course see the regularity if one looks at enough digits. Andrzej > There are obviously infinitely many ways of generating similar > sequences and they are obviously all equally useless. > > A more interesting (in my opinion) way to generate a random real > number is by specifying its digits as follows: > > r[0] = RandomInteger[{0, 1}, 1]; > r[n_] := r[n] = Append[r[n - 1], RandomInteger[{0, 1}]] > > This defines a unique real number, almost certainly irrational, > whose digits can be computed (in principle) to arbitrary precision, > e.g. in this case > > N[FromDigits[{r[120], 0}, 2], 100] > 0.5216581269311984505815006839069293336802568327655056358867877566088653250230499323902844110989462934 > > In reality this number will be "irrational" only if the random > number generator is truly random, otherwise the series of digits may > start repeating at some point and the number will turn out to be > rational. > > Related to the above but more interesting and deeper examples of > random irrationals are Chaitin constants (http://mathworld.wolfram.com/ChaitinsConstant.html > ). > > Andrzej Kozlowski > >> >> (The good and bad news may seem contradictory, but they are not.) >> >> So, the only way to get irrational numbers in Mathematica is using >> symbols >> (Pi, E, for instance) and expressions you know are irrational, such >> as >> Sqrt[2]. Your link shows a few other ways. But it's hard to call >> creating >> one of these "random". >> >> You could make a long list of irrational numbers and randomly >> sample them, >> perhaps adding several together... but I don't see what you could >> prove or >> accomplish, with such a procedure. >> >> Bobby >> >> On Fri, 31 Jul 2009 04:57:40 -0500, BenT <brtubb at pdmusic.org> wrote: >> >>> Although Mathematica has built-in functiona to obtain random >>> integers >>> and real >>> numbers, I need to select a random irrational number. At this >>> webpage, >>> >>> http://mathworld.wolfram.com/IrrationalNumber.html >>> >>> several definitions are given for known conditions to create >>> irrational numbers, such as >>> >>> Numbers of the form n^(1/m) are irrational unless n is the mth power >>> of an integer. >>> >>> Can anyone define a function to allow a similar capability as >>> Random[] >>> in selecting a "member" from the above defined "set" of values, or >>> any >>> other of the definitions listed on the same webpage? >>> >> >> >> >> -- >> DrMajorBob at bigfoot.com >> >

**References**:**Re: Creating a Random Function to Select an Irrational***From:*DrMajorBob <btreat1@austin.rr.com>