Are irrational numbers completely random?
As far as I know the decimal numbers in any irrational appear randomly showing no pattern. Hence it may not be possible to predict the $n^{th}$ decimal point without any calculations. So I was wondering if there is a chance that somewhere down the line in the infinite list of decimal numbers in an irrational could reveal something like our date of birth in order (eg: 19901225) or a even a paragraph in binary that would reveal something meaningful.
Since this a infinite sequence of random numbers ;
- Is it possible to calculate the probability of a birthday (say 19901225) appearing in order inside the sequence?
- Does the probability approach to 1 since this is an infinite series.
Any discussions and debate will be welcomed.
Here are two examples of irrational numbers that are not 'completely random':
$$.101001000100001000001\ldots\\.123456789101112131415\ldots$$
Notice the string $19901225$ does not appear in the first number, and appears infinitely many times in the second.
Now, as to your question of probability, let's consider the interval $[0,1]$. Using a modified version of the argument in this question, it can be shown that given any finite string of digits, the set of all numbers containing the string in their decimal expansion is measurable, and has measure $1$.
So, as you suspect, if we choose a number at random between $0$ and $1$, the probability that it has the string $19901225$ in its decimal expansion is indeed $1$. Also, more surprisingly, and perhaps a bit creepy, the probability that we choose a number that contains the story of your life in binary is also $1$.
Not necessarily. Consider $$0.1010010001000010000010000001....$$
Some irrational numbers do have the property you're looking for. They are called "disjunctive numbers."
There is a notion on normal number see http://en.wikipedia.org/wiki/Normal_number which basically says that all sequence of digits of a given length are equiprobable. It is known that the set of normal number has full measure and therefore is dense. Note that normality depend on the base. So there is a notion of absolute normality which says that a number is normal in every base. $\pi$, $\sqrt(2)$, $e$ are all believed to be normal but, as far as I know, there is no proof of that. By the way, I'm pretty sure that your birth date already appeared in the known decimal of $\pi$.