Ah, yes, this is related to the open problem of the normality of Pi. To quote wiki,
In mathematics, a normal number is a real number whose digits in every base show a uniform distribution, with all digits being equally likely, all pairs of digits equally likely, all triplets of digits equally likely, etc.
It is unknown if Pi and any other important constant in mathematics, like e or gamma, are normal.
Said otherwise, in a normal number, the digit "3" will appear with a frequency of once every 10 digits, on average. Same thing goes for 1, 2, 4, 5, etc. But if it is so, then in a normal number there is a probability of 1/100 that any specific pair of digits, like "12", "67", etc, appears in its expansion, that is, again, any specific pair of digits will appear with a frequency of once every 100 pairs of digits, on average.
Generalizing, in a normal number, any specific n-tuple of digits will appear on average once every n-tuple.
For more information, see wiki's open question section of its Pi article and its normal number article
http://en.wikipedia.org/wiki/Pi#Open_questions http://en.wikipedia.org/wiki/Normal_number
Edited by deimya (01/22/09 09:57 AM)
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Since '1' recurs many times in pi, and so does '12' most probably, and so does '357' and pretty much any other mundane collection of integers.
but just because they pop up again and again in different places, this is not known in mathematics as 'repetition'. perhaps it is known as 'repetition' in some aesthetic study of the presence of numerals. but as far as the number/ration 'Pi'.. all those 'repetitions' of arbitrary sequences of numerals mean nothing at all.
If I get the numer 4.576839576023576111
and point out that the '576' 'repeats itself' I am not saying anything constructive in terms of mathematics.
if however, the number kept going 4.576839576023576111576839576023576111576839576023576111 then I could start looking to see if this repetition after the decimal reduces to a fraction of integers (thus a rational number).
merely looking at a string of numerals and seeing that some of them repeat, or that groups of them repeat, is just pointless fun/timespending.
When your maths teacher said that pi never repeats, he didnt just mean that the whole string from '3.14 etc' actually repeats itself because that would be stupid to have a decimal point after a decimal point. he also didnt mean specifically that pi doesnt repeat from the '3' again.
he meant that no matter how far you go into pi you will never come across a section that you just read before that section. if this was the case then it would potentially be possible to scale the decimal point down to the moment the repetition starts and turn the number into a whole fraction between two integers.
I will take my own stab at getting to the root of this thread:
IT does seem reasonable to say that if anything continues infinitely, it will HAVE to eventually repeat itself. simply because it continues forever and changes forever so therefor it will not stop changing until it reaches a point where it repeats its earlier segment.
in this way, it is reasonable to say that for any set of numerals that you pick out of pi, they will repeat again at some point.
But this is completely different topic to what your maths teacher was talking about
it is completely inconsequential, a trivial truth that we know to exist but we will never actually whitness unless by slim chance and even then we can do nothing but nod that certain strings of numerals eg '1' or '55' or '83219' occur more than once in the infinite sequence of pi
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