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At least in physicist's thinking, information, vaguely, is something that allows one to select a subset from a set.

Say, a system can be in states A and B, we have done a measurement on it (extracted information), then it is in either A or B. Now we are able to say, in which state among all the possible states is the system in.

Now, consider numbers, say number e. One can write many digits to specify the boundaries of an interval to which e belongs. In fact, arbitrarily good precision can take arbitrarily large amount of information to specify the decimal representation of the number.

Now, however, one can also write an expression for e, say $e = \sum_{n}\dfrac{1}{n!}$, and it seems like e is defined to an arbitrary precision straight away. That is to say, here we have something, for which we would have needed infinite amount of information.

The question then: does this expression contain infinity of information/any information at all? Or, can information be defined at all for expressions?

One might surely argue that given an expression, one still has to perform infinite number of evaluations to obtation a decimal expression of the number. But then a number of other questions would arrive: "Is it evaluations of faculties and additions then that produce information?", "Is the information only about decimal representations of numbers, but not the numbers themselves?", and perhaps many more.

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Perhaps you want this? en.wikipedia.org/wiki/Kolmogorov_complexity –  Potato Oct 11 '13 at 13:52
    
@Potato: Indeed looks very relevent. Thank you (10^10 times)! –  Alexey Bobrick Oct 11 '13 at 13:57
    
This is a very deep question. – Someone could write down a 50 places decimal number like $2.71828\ldots9$, and you might think this number contains a lot of "information". But this information is absolutely worthless unless you are told that this number is very near some number holding big secrets. –  Christian Blatter Oct 11 '13 at 14:30
    
This will answer your question math.stackexchange.com/questions/20566/prove-there-are-no-hidden-messages-in-pi/ , if you have any more specific ones after reading that please refine your question. –  Abdulh Khazzak Gustav ElFakiri Oct 20 '13 at 12:38
    
Dear @AbdulhKhazzakGustavElFakiri, your link is really interesting and somewhat relevant to my question. However, it is far from obvious to me, how exactly does it provide the answer. Can you please kindly state it more explicitly? –  Alexey Bobrick Oct 20 '13 at 21:46

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