Can it be proven that any finite sequence of digits we can define, appears somewhere in the (say, decimal) expansion of pi? For example, that the string "1234567890" or "5555555555" appears somewhere? Or can it be proven that strings can be specified that do not appear?

Originally posted by JS357 Can it be proven that any finite sequence of digits we can define, appears somewhere in the (say, decimal) expansion of pi? For example, that the string "1234567890" or "5555555555" appears somewhere? Or can it be proven that strings can be specified that do not appear?

Since the decimal representation of pi is an infinite sequence of digits
and that those digits are random.

I think it follows that any string of digits you care to specify is
imbedded there somewhere.

Which leads us to the conclusion that everyone's phone number is in there.
And also the complete works of Shakespeare!

This is a common misconception. It has not been proven that every string of digits appears in the decimal representation of pi. Such a number is called disjunctive or rich (in base 10). It is not known that pi has this property and nobody really has any idea how to establish that. However, it is a widely believed conjecture.

There are many questions about properties of particular real numbers that are just beyond any proof techniques mathematicians have produced. Some of them are famous, and the one you're asking is one of them. Many famous questions regard the rationality of certain numbers. For example, the idea that e+pi could be a fraction seems silly, but nobody knows how to prove that it isn't.

For large numbers it can even be impossible to tell if they are whole numbers! For an example of such a number, see here: http://math.stackexchange.com/questions/13054/how-to-show-eee79-is-not-an-integer

For a puzzle, can you find a number you can próbę is disjunctive in base 10? Recall that any infinite string of digits after a decimal point gives you a tea number between 0 and 1.

Originally posted by WanderingKing For a puzzle, can you find a number you can próbę is disjunctive in base 10? Recall that any infinite string of digits after a decimal point gives you a tea number between 0 and 1.

0.123456789101112131415...248324842485... Keep adding decimal expansions of subsequent numbers. Thanks to the method of generation, this number is guaranteed to throw up every finite string of digits - after all, you're manually adding every one of them, one after another - but I don't know of any properly mathematical way of generating this number, only the obvious typographical one. Doing maths with it is just about impossible, since we don't know its arithmetic value, only its typographic expansion. Therefore: a cheap trick.

I can't argue that this is not a cheap trick in some sense of the expression, but it is definitely a valid real number and I think saying that we don't know its value is an exaggeration already. It is a perfectly valid sum of a perfectly valid series, a certain (slightly convoluted-looking when you write down the formula, admittedly) power series evaluated at 10. To me, the definition is witty and maybe funny-looking, but the number itself doesn't bother me or look suspicious to me.

A bigger point is that cheap tricks can become intuition builders. Perhaps a cheap trick is often something that introduces fresh way of looking at things. I think that the proof that there are as many natural numbers as even natural numbers does look like cheap trick at first, taking advantage of a somehow flawed definition of equinumerous sets. But when you put those sets next to one another, both in standard order, you notice that they do look exactly the same.

The invention of complex numbers might be another example, although I'm not sure it has as much of a cheap-tricky air to it. But certainly is something that might look artificial and turns out not to be.

Originally posted by WanderingKing This is a common misconception. It has not been proven that every string of digits appears in the decimal representation of pi. Such a number is called disjunctive or rich (in base 10).

OK
I was not aware of that. It seems counter-intuitive.

Presumably it cannot be proved that any single digit will occur in pi
(except by expanding pi) ???

Originally posted by wolfgang59 I was not aware of that. It seems counter-intuitive.

Why? What makes it so intuitive to you that it would? Most (in fact, almost all) mathematicians believe that pi is normal, but that's because they have been dealing with it for years, not a priori intuition.
Is it perhaps because you believe the digits of pi are random? Because, well, they aren't. They're pseudo-random. George Marsaglia has proved that the first several million are even pretty strongly pseudo-random. But note: that had to be proven. It doesn't follow immediately from any basic property of pi.

Originally posted by Shallow Blue Why? What makes it so intuitive to you that it would? Most (in fact, almost all) mathematicians believe that pi is normal, but that's because they have been dealing with it for years, not a priori intuition.
Is it perhaps because you believe the digits of pi are random? Because, well, they aren't. They're pseudo-random. George Mars ...[text shortened]... m. But note: that had to be proven. It doesn't follow immediately from any basic property of pi.

Originally posted by WanderingKing This is a common misconception. It has not been proven that every string of digits appears in the decimal representation of pi. Such a number is called disjunctive or rich (in base 10). It is not known that pi has this property and nobody really has any idea how to establish that. However, it is a widely believed conjecture.

There are many question ...[text shortened]... ber, see here: http://math.stackexchange.com/questions/13054/how-to-show-eee79-is-not-an-integer