Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

A normal number is a number where no number is favored to appear in the digits. Does this definition imply that all whole numbers appear in its digits? Because the definition involves notions from probability, I was wondering if it might happen that a certain number would not appear in the digits of a normal number without contradicting the definition.

share|cite|improve this question

Yes it does. Working in base B, a string with A digits should have the natural density $ \frac {1}{ B^A} $ by definition of a normal number. If the string doesn't appear at all, then it has natural density 0.

share|cite|improve this answer
Looking at the first $N$ numbers, let the number of times that the string appears be $N(S)$. Then, the definition of normal means that $\lim_{N \rightarrow \infty} \frac {N(S)}{S} = \frac {1}{B^A}$, which is the natural density. In fact, you can conclude that the string appears infinitely often, since otherwise the density will still be 0. – Calvin Lin Dec 29 '12 at 4:49
So is it possible to produce a non-normal number where all natural numbers appear anyway or are these notions equivalent? – user54358 Dec 29 '12 at 5:01
I believe so, but may be wrong. Take a normal number, and then insert $ 2^{n-1}$ 0's in the $2^n$ place for $n\geq 2$. For example, take 0.123456789101112... and transform it into 0.12 0 34 00 5678 0000 91011121 00000000 ... Then for given any string $S$ of length $A$ which isn't 0, $lim_{N \rightarrow \infty} \frac {N(S)} {S} = \frac {1}{2 \times 10^A}$ (which requires a short argument), while the string 0 occurs with natural density $ \frac {1}{2} $. – Calvin Lin Dec 29 '12 at 5:21
Like if you have the number 0.10203040506070809010011012... the concatenation of natural numbers with 0 inserted in between. What's the density of 0 there? Is 0 already denser or is this still normal? – user54358 Dec 29 '12 at 5:22
Hey Calvin, if you did that construction wouldn't you end up cutting some of the numbers? Like 13 for example got cut by the zeroes you inserted. – user54358 Dec 29 '12 at 5:27

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.