+1090 I am curious if there could be a last digit of pi (or any irrational number).
I realize that the reason why there doesn't seem to be a last digit is because irrational numbers cannot be expressed as a ratio, and therefore go on for an infinite number of decimal places. Obviously if the last digit of any approximation is taken to be the last digit, then it can be disproved by calculating a more precise approximation with a different "last" digit. I am not talking about the last digit of an approximation, but a last digit of the value of an irrational as expressed in a base.
Regardless of whether it could be calculated or be useful, could there be a last digit of an irrational number, that would have infinitely many digits to the left? There doesn't seem to be any mathematical contradiction in this inherently that I can see (but I would love if one could be pointed out.) The best argument that I could see against a "last" digit of an irrational number is that the concept of a "last digit" in an irrational number doesn't make sense, but that doesn't necessarily mean that it doesn't exist.
+118673 Hi Mathematician,
Maybe someone else will say something interesting but I'm just going to let you ponder that one all by yourself :)
Did you know that you could compute the Nth hexadecimal digit of Pi efficiently without the previous N-1 digits. For example: the 10^50th digit of pi. The method is based on this formula:
pi = sum_(i = 0)^oo (1 /16^i) ((4/ 8i + 1) - (2/ 8i + 4) - (1/ 8i + 5) - (1/ 8i + 6))
+1090 I had heard about the spigot formula before. It is definitely an interesting formula.
+33661 "I realize that the reason why there doesn't seem to be a last digit is because irrational numbers ... go on for an infinite number of decimal places"
Hence there is no last digit!
+118673 Hi Alan,
I think that mathematician already knows this but to say that a number is irrational so therefore, by definition, the digits must go on for ever without a repetitive pattern, offers no explanation of why this is so.
Mathematician is just toying with ideas in a way that I think all mathematicians and potential mathematicians must. Only people with good analytical brains do this.
I think that many/most people just accept everything that they are told - assuming that they are told by a source that they consider credible.
+33661 "The best argument that I could see against a "last" digit of an irrational number is that the concept of a "last digit" in an irrational number doesn't make sense, but that doesn't necessarily mean that it doesn't exist."
That sentence makes grammatical sense, but is semantic nonsense! By definition, an irrational number doesn't have a last digit and hence there is no last digit!
The proof that pi is irrational was first demonstrated by Lambert in the 18th century. There are several proofs which are more or less complicated (I wouldn't call any of them simple!) - see http://www.mathpages.com/home/kmath313.htm for example.
