# recipe for infinitely many irrational numbers - or is it?

What if we write 0. and then throw a coin and depending on the result continue the number with 1 or 0 and continue this process indefinitely. It seems like a recipe for producing irrational numbers.

Are these numbers really irrational? Are they transcendental? Are they normal? Maybe I should also ask: is anything well-defined by this recipe? :-)

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this is related: math.stackexchange.com/questions/41150/… –  Adam Nov 26 '13 at 11:11
Your "recipe" is equivalent to "a uniformly random real number in [0,1]", which obviously is irrational with probability 1. However, you can't say that it defines an irrational. It doesn't, just like the uniform random variable doesn't. –  user21820 Dec 26 '13 at 9:08

UPDATE if you are interested in a probabilistic / philosophical answer (i.e., a supernatural being which performs a supertask tossing the coin at times $t$, $t+\frac{1}{2}$, $t+\frac{3}{4}$, $t+\frac{7}{8} \ldots$, at time $t+1$ she will with near certainty ($p=1$) obtain a normal number. But this is not a "recipe", in the sense she cannot be sure it is. (Ok, if she is a supernatural being maybe she has some other way to know it :-) )