Sunday, March 14, 2010

Happy Pi Day!
Tom Smith


What I wonder is, must pi contain any arbitrary set of integers (say, 098759128759875498174918419240912509091) somewhere to the right of its decimal point? When I try to think about this, I get that unpleasant sensation one gets when trying to do an exercise with weak, atrophied muscles. Though unlike weight lifting, I suppose number theory is not known to cause permanent injury.

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Comments

"number theory is not known to cause permanent injury": Smith's Conjecture.

Posted by: dearieme | Mar 14, 2010 12:28:06 PM

in general, irrational numbers may or may not contain all finite sequences in their decimal expansions.

The number .10100100001000000001. . . (always just append twice as many 0's as in the previous segment and then a 1) is irrational (we can see this as the decimal expansion doesn't repeat), but doesn't contain (e.g.) the sequence "314".

That irrational numbers may contain all finite sequences follows pretty easily from the fact that finite sequences are enumerable. In fact, any number whose decimal expansion has this property (i.e., contains all finite sequences) is necessarily irrational.

The question of whether pi contains all arbitrary sequences is open. In fact, it's not even known whether pi contains an infinite number of 7's in its decimal expansion! My sense is that this - like the Collatz Conjecture - is one of those simple-to-state problems that will remain open for 1000 years.

Posted by: EdenH | Mar 14, 2010 12:55:42 PM

The number pi is not only irrational, but transcendental. There are lots of irrational numbers that are not transcendental, but not vice-versa.

It is intriquing to think that all of the Constitution, Mein Kampf, the Bible and Shakespeare are contained in coded sequence in the decimal expansion of pi and e. I wonder if smuggling them into Germany, Iran or Saudi Arabia would be punishible.

Posted by: jimbino | Mar 14, 2010 2:23:12 PM

Well, if it is an open question whether pi contains all arbitrary sequences, I take it is an open question whether *any* transcendental numbers contain all arbitrary sequences? And if that is the case, we don't really know whether the Constitution, the complete Sherlock Holmes etc. is encoded into pi.

Posted by: Tom Smith | Mar 14, 2010 3:04:05 PM

i cant see everything,what is the bad?

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