Looking for reading material on: Numbers, in whose internal decimal places appear all natural numbers as a sequence of digits

Question

For example one can have the number 0.12 and can look at the sequence of the digits of it's internal decimal places and see, that 0.12 contains the numbers 1,2 and 12. It is also easy to construct a number, that contain all natural numbers by just numerating the natural numbers and write them one after the other behind the 0. . However I was wondering, if it were possible to find all natural numbers in beautiful numbers like Pi or e or $ \sqrt2 $ etc. , also if this is some sort of property only a few numbers share.

Can anyone recommend some sort of reading material on that topic or knows more about it? As always: Many thanks in advance.

Answer 1

You should look for articles on Normal Numbers which are slightly different from what you specify - they have each natural number an infinite number of times (not just once) and occurring in the right proportions.

It can be shown that most real numbers are normal numbers. There is a chapter in Hardy and Wright's "An Introduction to the Theory of Numbers" which contains the proof and other information.

It is a very hard thing to prove that any particular number is Normal, unless it has been particularly constructed with a proof in mind. I believe the cases of $e$ and $\pi$ and $\sqrt 2$ are open.

Answer 2

This has been asked before on MathOverflow and as far as I know based on the responses there, there is no known "typical" irrational number like $\sqrt{2}$ or $\pi$ which is known to contain all natural numbers as some sequence of consecutive digits. Note there are uncountably many irrational numbers that do NOT have this property, e.g. all the irrational numbers that do not contain a certain digit in their expansion.