I was reading today that someone found $\pi$ to the ten trillionth digit. Whenever I read that $\pi$ has been calculated to more digits, I ask myself whether this is useful. I know that there are conjectures out there about distributions of numbers in $\pi$ and such. So, I supposed knowing more digits helps us test conjectures. But, are there more reasons that we would want to know the digits? Anything really cool I'm ignoring or forgetting?
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Not for any real-life calculations according to wikipedia
It is however useful to test supercomputers for accuracy
Knowing more and more digits of $\pi$ has absolutely no value to anybody. Digits of $\pi$, beyond the twentieth or so, are completely worthless. The only value in this enterprise, if any, lies in the process by which the digits are generated, not the digits themselves.
A less mathematical reason for calculating more and more decimal places is because we know they are there. Man is inherently curious and always wants to see what's over the next hill, round the next bend etc.
Another reason is for its properties of randomness.
This can be used to test software for analyzing random sequences. It can also be used as a teaching aide. Example: if the digits of pi exhibit statistical randomness (it is believed they do), then at some point in pi's expansion there will be a sequence of one million consecutive 0's.
Whenever we calculate the first $n$ digits of an irrational number, we simply estimate it and there is a small error in our approximation.
If we use that for practical applications, we should always be aware of the error and check if it is resonable or not for the practical application. If not we need more digits, so theoretically there is always a need for extra digits.
The above answers covered well why 1 billion digits are probably more than enough for $\pi$, anyhow in general it is hard to say how many digits we need to know to cover any possible application. A resonable number of digits shouldn't suffice, we always need an unresonable number.
Not for $\pi$, but here is a known example where 26 digits (well not exactly, they were 26 binary digits) were not enough for a practical application, and unfortunatelly some people died because the error in that application was too big:
I find this example interesting, because most people would think that 5-6 digits should suffice in all cases, and it is easy to understand why in this case the estimation wasn't good (of course people should had thought about it before it happend)...
protected by Zev Chonoles Sep 27 '15 at 5:44
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