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Irrational numbers like $pi$, $e$ and $sqrt2$ have a unique and non-repeating sequence after the decimal point. If we extract the $n$-th digit from such numbers (where $n$ is the number of times the method is called) and make a number with the digits as it is, should we not get a perfect random number generator? For example, if we're using $sqrt2$, $e$ and $pi$, the first number is 123, second one is 471, the next one is 184 and so on.

It is cryptographically useless because an adversary can predict every single digit. It is also very time consuming.

For any reasonable definition of perfect, the mechanism you describe is not a perfect random number generator.

• Non-repeating isn't enough. The decimal number $0.10100100010001dots$ is non-repeating but it's a terrible generator of random digits, since the answer is "always" zero, occasionally one, and never anything else.




• We don't actually know if every digit occurs equally often in the decimal expansion of $pi$ or $mathrme$ (though we suspect they do).




• In many situations, we require random numbers to be unpredictable (indeed, if you asked a random person what "random" means, they'd probably say something about unpredictability). The digits of well-known constants are totally predictable.




• We usually want to generate random numbers reasonably quickly, but generating successive digits of mathematical constants tends to be quite expensive.




• It is, however, true that the digits of $pi$ and $mathrme$ look statistically random, in the sense that every possible sequence of digits seems to occur about as often as it should. So, for example, each digit does occur very close to one time in ten; each two-digit sequence very close to one in a hundred, and so on.

Duplicate, but nice question. It was already answered https://math.stackexchange.com/questions/51829/distribution-of-the-digits-of-pi:

...it is believed that $pi$ is a normal number (~uniform distribution of every digits sequence).

For digit distribution data, see e.g. http://www.eveandersson.com/pi/precalculated-frequencies or https://thestarman.pcministry.com/math/pi/RandPI.html (first 1000 digits):

At mathoverflow, there are also nice answers at:

• What is the state of our ignorance about the normality of pi?


• Does pi contain 1000 consecutive zeroes?

In general, this approach does not work: "randomness" does not mean that you get a lot of different digits, but there are other aspects as well. For example, a classic test is to see if all two-digit, or three-digit etc. combinations occur with the same frequency. This would be a very simple test, which can rule out obvious non-random results, but is still by far too simplistic to check for really random behaviour.

See the Wikipedia page about Randomness Tests as a collection of links to primary sources regarding this. They do mention a good amount of quite complicated-sounding concepts; it is it not so important to go into deep detail about this - but it is clear that it is not intuitively possible to declare a specific number to be a good source for such digits.

On a positive note: For a specific irrational number, you are of course free to just try it; i.e., calculate the number to a sufficiantly large degree of digits, and run it through all known tests (there are tools for that, see above link). If the measure is good enough for your use case, and if you are aware that this is obviously useless for cryptographical applications, and always get the same numbers if you should start over, and that the quality might degrade if you get past the n you picked for testing the randomness, you could use those numbers. But it will be far better to use a dedicated (pseudo-)random number generator; and nothing beats a good physical source of randomness.