Well, the answer is that it has to be infinitely long, but the question is what is the most compact form of a Normal Number possible.

I was motivated to look into this from a lovely Numberphile video about all the real numbers.

Normal numbers in base 10 are those for which, in the base 10 decimal expansion, you can find every natural number.

Champernowne’s number is a very simple example of this where it is simply written as:

0.12345678910111213…etc.

I thought that it might be interesting to see if one could write a more compact Normal Number, but using a similar procedure to Champernowne. I haven’t seen this done anywhere else. For example, in the above expression, you don’t need to include the 12 explicitly as it’s already there at the beginning. You could write

0.12345678910113

So you skip the 12, and also 11 and 13 becomes 113. We will do all of this just with the list of digits, rather than the number in base 10.…