Talk:Normal number

"No rational number is normal to any base, since the digit sequences of rational numbers are eventually periodic."

Is this right? Obviously rationals are periodic, but what about 2/3 in base 2 (0.1010101...), or 1234567890/9999999999 in base 10 (0.12345678901234567...)?

I think it is right. Based on the definition, it isn't only single digits that have to appear equally, it's all sequences of digits. So for 2/3 to be normal to base 2, you need to show that 111010101100001101, for instance, occurs as if 2/3 were a random string of digits, when of course it does not occur at all. And similarly with your other example. Eric119 20:53, Mar 26, 2004 (UTC)

I believe there is an inconsistency with the claim at the end of the article that "not a single irrational algebraic number has ever been proven normal in any base", since we know that Copeland-Erdős constant is normal.

Oh wait, it said irrational ALGEBRAIC number - and the Copeland-Erdős constant looks pretty transcendental to me. (Michael Currie)

I think the conjecture that every irrational algebraic number is normal is not due to Bailey and Crandal as asserted but goes back to Borel: É Borel, Sur les chiffres décimaux de $\sqrt 2$ et divers problèmes de probabilités en chaîne, C. R. Acad. Sci. Paris 230 (1950) 591--593. Réédité dans : Œuvres d'É. Borel vol. 2, Éditions du CNRS, Paris, 1972, pp. 1203--1204. [J.-P. Allouche]

Layman definition of normal number and other applications to the concept

I would like to add, but am uncertain how to word it academically, a section on the way I had previously understood "normal number" to be defined, which is basically saying the same thing as what is currently in the article but phrased in a different way. Basically, I had been told that a normal number is any number in which, when treated as a string, any possible substring can be found within that string.

This has a number of implications on a number of widely held beliefs. For example, it is often believed that in an infinite universe, there must be an example of anything and everything, which is basically stating "the universe, if infinite, is normal. Another way of rephrasing this belief is "all infinite sequences are normal" which is patently false, since it is easy to come up with counterexamples of infinite sets or sequences that are NOT normal. This means that it is not necessarily a given that there is, for example, a guarantee for the existence of extraterrestrial life, which I find is usually the context for the above statement.

Any suggestions on how to insert this into the article? I think it is of importance to mention... it also has relevance in other areas, like the infinite monkey theorem (and thus relevance to things like evolution). Basically, it says that first you have to prove that a given set is normal before you can assume that it is normal, and thus assuming normality while making a proof doesn't really hold water. Fieari 18:22, 16 May 2005 (UTC)

I think the definition "a normal number is any number in which, when treated as a string, any possible substring can be found within that string" is different from the definition given in the article. For instance, take the number (the spaces are added for clarity)

0.1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 10 00 11 00 12 00 13 00 14...

which is the sequence of natural numbers (1, 2, 3, 4, ...) with zeros added between them. This number is not normal, since more than half the digits are zero, but it contains every possible string of digits.

Regarding your second, more philosophical, comment, I am not sure this article is the correct place to mention that, but I can't suggest any other place. -- Jitse Niesen 14:30, 17 May 2005 (UTC)