It is conjectured that pi is "normal" in the sense that the digits occur with equal liklihood. Making that statement precise is exceedingly hard, and not very enlightening.
EDIT: philh points out that the full definition of "normal" says that every block occurs equally often (a concept that needs care to make formally correct) and that does imply that every finite sequence does occur.
pi is known to be transcendental. It is not only not rational (cannot be expressed as the ratio or integers (although 355/113 is close and you can get as close as you like by using bigger numbers)) but it is also not the root of any polynomial with integer coefficients (rationals being a special case).
The digits of pi cannot become periodic.
None of the above is enough to show that every finite number occurs "eventually". As an example, consider the number:
Here the digits 1 through 3 occur equally often, it never becomes periodic, it is transcendental, and yet the sequence 321 will never occur. Now instead of using 1 through 3, generalise to using 1 through 9.
It is generally believed that the digits are "effectively random" which means any finite sequence will occur with probability 1, but the observations made above are not enough to ensure that. (EDIT: although the full definition of "normal" does, rather thanthe limited version I originally put - thanks to philh for the correction).
(edited a typo 133 -> 113 : thank you for the correction)
>It is conjectured that pi is "normal" in the sense that the digits occur with equal liklihood.
That's only part of it. You need that every block occurs as often as every other block of equal length. So every digit occurs as often as every other digit; every pair of digits occurs as often as every other pair; every triple occurs as often as every other triple; and so on. So the number you give is not normal in base 3.
A consequence of normality is that every finite sequence of digits eventually appears.
This isn't true in a strict sense, is it? The digits of pi aren't random, and for numbers of arbitrary size, it seems that you could just keep asking for a longer numbers until you can't find a match any more.
Ah, we know that none of the numbers of the form pi + x can found in pi.
But I still wonder whether there exist numbers of finite length that can't be found within the digits of pi. Any hardcore mathematicians in the house?
If the conjecture is true then any finite sequence of numbers can be found in the digits of pi. As others have noted, infinite sequences are not guaranteed or even likely to appear. As zackattack points out, for example, the decimal expansion of 1/9 does not appear in the digits of pi.
I'm not a mathematician, but I read a lot of time ago that since pi is infinite but not periodic, you shold be able to find in it any sequence you want, given infinite time to search in it.
The statement
(infinite && not periodic) ==> (any finite sequence is in it)
is not true. \a number could have a decimal expansion that is not periodic but does not include certain digits.
E.g: write Pi in binary and consider the same sequence of 0's and 1's as a number in decimal
Wikipedia lists Khaled, Algerian raï musician; Richard Ramirez, American serial killer; Tony Robbins, American motivational speaker as being born on 29 February 1960.
Oddly searching for 29021960 doesn't return a result. But searching for 2902196 does and shows there's a zero after it.
ISO format is yyyymmdd, not ddmmyyyy. First recent ISO date in pi is 19530921 at the 417th decimal place. All Wikipedia has for that date is the birth of a musician and recording engineer called Andy Heermans.
It is conjectured that pi is "normal" in the sense that the digits occur with equal liklihood. Making that statement precise is exceedingly hard, and not very enlightening.
EDIT: philh points out that the full definition of "normal" says that every block occurs equally often (a concept that needs care to make formally correct) and that does imply that every finite sequence does occur.
pi is known to be transcendental. It is not only not rational (cannot be expressed as the ratio or integers (although 355/113 is close and you can get as close as you like by using bigger numbers)) but it is also not the root of any polynomial with integer coefficients (rationals being a special case).
The digits of pi cannot become periodic.
None of the above is enough to show that every finite number occurs "eventually". As an example, consider the number:
0.123 112233 111222333 111122223333 111112222233333 ...
Here the digits 1 through 3 occur equally often, it never becomes periodic, it is transcendental, and yet the sequence 321 will never occur. Now instead of using 1 through 3, generalise to using 1 through 9.
It is generally believed that the digits are "effectively random" which means any finite sequence will occur with probability 1, but the observations made above are not enough to ensure that. (EDIT: although the full definition of "normal" does, rather thanthe limited version I originally put - thanks to philh for the correction).
(edited a typo 133 -> 113 : thank you for the correction)