
Re: PI: infinite and contains everything?
Posted:
Sep 6, 2017 6:37 PM


On 06/09/2017 20:36, pamela wrote: > On 22:03 5 Sep 2017, Markus Klyver wrote: > >> Den tisdag 5 september 2017 kl. 18:56:58 UTC+2 skrev Mike Terry: >>> On 05/09/2017 17:30, Markus Klyver wrote: >>>> Den torsdag 24 augusti 2017 kl. 19:54:31 UTC+2 skrev Peter >>>> Percival: >>>>> pamela wrote: >>>>>> I was watching a drama on the telly where a teacher >>>>>> explains to his class that the number PI goes on for ever >>>>>> and never repeats itself. That seems true enough for >>>>>> irrationals like PI. >>>>>> >>>>>> He went on to claim that this means >>>>> >>>>> No it doesn't. What you say next is so if pi is normal base >>>>> 10 (supposing the the digits of pi and various integers >>>>> referred to are base 10). Mere irrationality won't do. See >>>>> https://en.wikipedia.org/wiki/Normal_number. >>>>> >>>>>> all other numbers (presumably he means integers) will >>>>>> appear somewhere in the sequence of digits of PI. So my >>>>>> phone number or the works of Shakespeare (represented as >>>>>> digits) would be in PI somewhere. >>>>>> >>>>>> Is this true or is it dramatic licence? >>>>> >>>>> >>>>> >>>>>  Do, as a concession to my poor wits, Lord Darlington, >>>>> just explain to me what you really mean. I think I had >>>>> better not, Duchess. Nowadays to be intelligible is to be >>>>> found out.  Oscar Wilde, Lady Windermere's Fan >>>> >>>> In a normal number, any finite sequence of digits will >>>> appearc almost s urely, yes. We don't know if Ï? is normal, >>>> but being a normal number isn't that special for a real >>>> number at all. Almost all real numbers are normal, and so >>>> "everything is contained" in almost all real numbers. >>>> >>>> https://groups.google.com/forum/#!topic/sci.math/ivNvUw_08Ew >>>> >>> >>> In a normal number, any finite sequence of digits DOES APPEAR. >>> Above, you add the expression "almost surely" which is not >>> required, and will probably mislead some readers. >>> >>> Regards, Mike. >> >> Almost surely mean the probability is 1. > > Doesn't it mean the probability is almost 1? >
No, it means the probability is 1. It has a technical meaning in probability theory. I'll give you an example...
A man (or woman) with £5 goes into a casino, and sits down at the roulette table to play the following customised game: on each spin, £1 is staked on the number 3. If number 3 comes up, the gambler wins £1 (plus gets his/her stake back) otherwise the stake is lost. The man/woman plays this repeatedly. [OK, obviously this is a crooked casino, and the naive gambler doesn't realise he/she is being ripped off! We don't expect this game to last very long...]
We wonder  assuming the game has no time limit so it is played over and over indefinitely, will the gambler run out of money? (I.e. will the gambler's account, which started at £5, and each spin of the wheel goes up or down by £1, go all the way down to £0?)
Mathematically this is an example of a (biased) random walk, and it's fairly clear to us that the gambler is not going to last very long :( Probability theory can be used to analyse the situation, and unsurprisingly the conclusion turns out to be that the probability of the gambler losing all his/her stake is 1.
Note that the probability is not 0.9, or 0.999 or "almost 1". If we altered things so the game only had a fixed number of spins, say 100 or even 1000000 spins, then there would be a (small) nonzero probability that the gambler survives without going bust. But with an effectively infinite sequence of spins, the probability of the gambler going bust is 1. (Actually this is all quite complicated when you get fully into it, so I won't even try to justify why this is the case, but at least it should seem plausible to you...)
So for this game, Probability (gambler goes bust) = 1
But.... this is NOT saying that there is NO (infinite) sequence of spin outcomes where the gambler avoids going bust. Far from it! For example, the number 3 comes up on the 1st spin, then again on the second spin, and just keeps on coming up every spin. And there are infinitely many basically similar (unlikely) variations. It's just that when we mathematically work out the probability of ANY of the "surviving outcomes" occuring, the probability is zero.
So there is a bit of a language problem here  mathematically we don't want to say "there's no way the gambler can avoid going bust" or similar, because in fact there do exist outcomes where he/she does so. Instead, for situations where even the conglomeration of all these "survival" outcomes looked at together still only has aprobability of zero, mathematicians say "the gambler goes bust ALMOST SURELY". (Or "almost certainly" or similar.)
So that's the technical way the phrase "almost surely" is used.
In the case of a "normal" number, the definition of "normal" ensures that its decimal digits include any specific digit, e.g. "7" infinitely often. I.e. if it didn't, the number wouldn't qualify to be called a normal number. No probability involved here, so we wouldn't say that "7" occurs in the numbers decimal expansion "almost surely".
(Also with regard to what others have said regarding your original question, bear in mind that with all this talk of "normal" numbers, nobody has so far proved that Pi is in fact normal! :) Although, this claim seems highly plausible to the point where we would be suspicious if someone claimed to have proven the opposite...)
Hope this wasn't too long for you, Mike.

