# Normal Numbers

1. Feb 24, 2005

### NateTG

This is all really hard stuff, and speculation.

Normal numbers are irrational numbers that have the property that all the digits in their decimal expression are equally distributed.
http://mathworld.wolfram.com/NormalNumber.html
An example of a normal 10-number might be:
0.123456789101112131415161718192021222324252627...

Clearly whether a number is normal can depend on the base that it is represented in, so it makes sense to refer to b-normal numbers where b is the base.

An absolutely normal number is a number that is normal in any fixed base.

I was wondering about this stuff a while ago, and the following questions came to me - insightful comments would be appreciated:
If a number is $p$-normal for all primes, is it necessarily absolutely normal?
If a number is $n^i$-normal for some whole number $n$ is it absolutely normal?
If a number is $p$-normal and $q$-normal is it also $pq$-normal? What about the converse?

What about more exotic bases - like negative bases or base factorial? I'd like to call a number !-normal if its base factorial expansion has a distribution so that all digits have zero density, in all of it's base factorial expansions. (Phrased that way to cover 0.1,2,3,4,5,6,7,8,9,10,11,12,13...=1.0,0,0... (base factorial) and it's cousins.) Base factorial seems particularly interesting because it should dovetail well with the Taylor-series-type stuff we have for caculating constants.