I know that, to date, there has never been found an end to the numerical value of Pi, and recall from my high school and college days the notion that there wasn’t a final digit.
I was wondering what the implications would be, if any, if someone found that Pi did in fact have a definable mathmatical end. And could Pi have a final digit that we just haven’t yet calculated?
Nope, pi has been proved to be irrational and transcendental, so by definition it can never have a last digit.
This was proved in the 19th century, BTW, so you should have been taught in both high school and college that this was no notion.
The current record for calculating Pi is 1.24 trillion decimal places and it hasn’t repeated yet.
Pi is an irrational number so it won’t repeat. Further it is a transcendental number* (all transcendental numbers are also irrational). Pi was proven to be a transcedental number in 1882 by the German mathematician Ferdinand Lindemann. I won’t pretend to understand any of it but I trust the results since no one seems to have disputed it in the last 120+ years.
More Piey goodness here: Are the digits of pi random, or not?
I might well have been taught that (I did recall that there was some notion that there couldn’t be an end to Pi, but couldn’t recall if it was proved.) but it just hasn’t been something I’ve had to think about for about 15 years…
One of the first proto-computer viruses was in the Star Trek TOS ep in which the captain or Spock instructs the computer to compute the last digit of pi. They figured that would tie it up for a while. Does nothing to further the OP, but I had never considered in that light (as a sort of virus).
I heard on NPR yesterday that there is a company that makes fake ATM slips, with a bogus balance printed on it, for those people really desperate to make an impression. The balance that they say is $314,159.26.
Well, I’ve memorized 350, and boy, it sure would be nice if it repeated!
The last digit of pi is 6. So there.
No, it’s Inf. So saith IEEE.
Aha, but what you (and they) don’t realize is that the next 1.24 trillion are a repeat of the first 1.24 trillion. 
When come back, calculate Pi.
That’s only if you round off to the nearest quadrillionth.
I’ve tried to understand this on my own, but failed, so if anyone has the time and inclination, could you explain how you prove that a number is irrational and transcendental?
transcendental?
A number is irrational if it cannot be represented as a fraction of any whole numbers. i.e. 22/7 used to be used for pi.
The (IMO) most elegant proof that Pi (or the square root of 2) are irrational is the Reverse Proof where you start off assuming that, yes - as sure as eggs is eggs, you can represent pi as a numerator divided by a denominator. As sure as I am Priceguy this will be proved.
The beauty of a reverse-proof is you end up with something like 1=2, hence the term “Good enough for government work”. :b
Since you are trying to prove that something is not the case, you use proof by contradiction: you assume that the number is rational ( or algebraic) and show that this leads to two contradictory conclusions.
The classic example is the proof that [symbol]Ö[/symbol]2 is irrational. Suppose that it not. That means we can write it as a fraction, say
[symbol]Ö[/symbol]2 = m/n
Now it may be that m/n is not in its lowest terms, but if this is the case then we cancel it down until it is. Then
[symbol]Ö[/symbol]2 = a/b
and there is no number greater than 1 which goes into both a and b ( or we could cancel it down further).
Now square both sides of the equation and multiply up by b[sup]2[/sup]. We now have
a[sup]2[/sup] = 2b[sup]2[/sup] (*)
The number 2b[sup]2[/sup] is clearly even. Therefore a[sup]2[/sup] is even. What does this tell us about a? Well, the square of any odd number is odd [(2m+1)[sup]2[/sup] = 4m[sup]2[/sup] + 4m + 1 = 2(2m[sup]2[/sup] + 2m) +1]. Since the square of a is even, it follows that a must be even as well ( if it were odd, its square would be as well).
By definition of even, this means we can write a = 2c, say. Substituting this into (*) we have
2b[sup]2[/sup] = a[sup]2[/sup] = (2c)[sup]2[/sup] = 4c[sup]2[/sup]
Cancelling a factor of 2,
b[sup]2[/sup] = 2c[sup]2[/sup]
This equation shows that b[sup]2[/sup] is even. By exactly the same reasoning as above, it follows that b is even. But now we are done, for a/b was in its lowest terms and we have shown that the numerator and denominator both have a factor of 2. These are the two contadictory conclusions mentioned above.
Note that it would be quite wrong to argue “Well, why not just cancel the factor of 2, and then the fraction might be in its lowest terms.” This would be wrong on two counts:
(i) The above argument shows that after we had cancelled the factor of 2 from the fraction, there would still be a factor of 2 remaining.
(ii) More importantly, the precise nature of the contradiction is unimportant. What matters is that the assumption that [symbol]Ö[/symbol]2 is rational leads inevitably to a contradiction. Therefore, that assumption is false.
Since being transcendental is a much stronger property than being irrational, proofs are naturally more difficult. We can spot some common themes, though.
We assume that the number in question is algebraic, i.e. it is the root of a polynomial equation. By the exercise of great ingenuity, we deduce the existence of a number m having three properties:
(a) m is an integer
(b) m > 0
© m < 1
These three statements clearly produce a contradiction.
Using various forms of this strategy we can prove
e is transcendental
[symbol]p[/symbol] is irrational
[symbol]p[/symbol] is transcendental
The Gelfand-Schneider Theorem: Suppos that [symbol]a[/symbol] and [symbol]b[/symbol] are algebraic, that [symbol]a[/symbol] is not 0 or 1, and that [symbol]b[/symbol] is irrational. Then [symbol]a[/symbol][sup][symbol]b[/symbol][/sup] is transcendental.
Well, Jabba, thanks… I guess. I think I get it now. I’ll read through that a couple more times though.
G. H. Hardy said that the proof above is one which any intelligent reader should be able to assimilate in an hour or two. You should never expect to grasp a proof fully in one go, so don’t worry if you haven’t.
Recent advances allow us to calculate the Nth digit of pi. I do not believe there are any restrictions on N. In other words, if 1.24 Trillion isn’t enough for you, go ahead and calculate the google’th digit. No? How about the googleplex’th digit? Graham’s numbers?
Jabba, I wish to congratulate and thank you for your explanation. Excellent, excellent job. Clear and concise. And fascinating.
I am obviously not a mathematician (though I had to take a lot of math to obtain my Comp Sci degree). Following your post was the most enjoyment I’ve gotten out of math in many a year. I recall a special thrill when helping my young teenage daughter as she struggled with some mathematical concept (I forget precisely what it was), and when the “light went on” and her face lit up with a breakthrough understanding, it was a special moment.
I can’t really say that your post was that rewarding of course, but it was enjoyable nonetheless.