P I To 31 Trillion Digits......

[ToraToraTora]

https://www.bbc.co.uk/news/technology-47524760
One thing that has always puzzled me, what is the calculation that they do to yield this number? We know it's irrational so there must be some sort of division sum they do that yields a non recurring quotient. Anyone?

[sanmac]

I really don't understand the big deal surrounding this "feat". So, she used computers to come up with a large number which is of no use to absolutely anybody for any reason whatsoever.

[jim360]

Various series expansions exist to define pi. One rather famous one is the following wonderful result:

(1 + 1/4 + 1/9 + 1/16 + 1/25 + ... + 1/n^2) = (pi^2)/(6)

ie, the sum of the reciprocals of all square numbers equals pi squared divided by six.

[jim360]

I should say that this is almost certainly *not* the series used in the present calculation, because, although it's relatively easy to set up, I think it's computationally a pain to get an accurate computation of pi quickly that way. But that is just one of thousands of formulas that can be used to compute pi numerically.

[ToraToraTora]

so they just increase the series continually? So could I code a loop of sqrt(1/(n**2))*6 - with N to infinity?

[ToraToraTora]

ok thanks Jim. The method used though, to yeild what it does must be extraordinarily accurate or it would start recurring at some point or, worse, reach an exact answer!

[Canary42]

https://en.wikipedia.org/wiki/Pi_Day

[spathiphyllum]

I'm with sanmac

[New Judge]

You only see a different result (and then only minimal) when you get beyond three decimal places - more than enough for everyday use. I understand some uses of pi may require more (but cannot imagine more than four or five places being necessary, but let's be generous and say ten may be needed). Like sanmac, I simply do not understand the enthusiasm for such a pointless exercise.

[ludwig]

I like the idea that there's 170TB of data storage somewhere containing nothing but a big useless number.

I hope someone's made a note of it in case it gets lost.

[jim360]

Yes, there have been some remarkable developments in computation techniques for pi.

I just checked quickly, and by my reckoning if you take the sum up to n=1000 you get a value pi = 3.1406, and n=10000 still only gets you to 3.1415. So yes, very slow. My pitiful laptop took 4.5 minutes to go to n = 1 million, and that got my only one more digit.

I think the one currently used is the Chudnovsky algorithm, which looks arbitrary to me to but is apparently not -- but is anyway much, much faster than my sum over inverse squares.

[ToraToraTora]

that's more like it jim!

[jim360]

Why does everything have to be "useful"? Sometimes it's just nice to do things for their own sake.

If you do need a use, then pi computations are a great way to test supercomputing power, or the efficiency of certain more general-use algorithms.

[ludwig]

Here's a thought. How do they know it's right?

[bednobs]

how would anyone know if its right or wrong anyway?

[bednobs]

great minds ...

[sanmac]

Of much more use would be the ability to prove "The Infinite Monkey Theorem".

[jim360]

There are checking algorithms, eg in the worst case you can just use a second (separate) method to obtain the same result.

[4candles]

22 ÷ 7 is near enough.

[fiction-factory]

I wonder if anyone has tried to work it out using Euler's identity
e^iπ + 1 = 0.
That's even more mind stretching than pi since it relates two important irrational numbers (Pi - roughly 3.142- and e- roughly 2.718) and an imaginary number (square root of -1)
When I retire I'll see whether I can rearrange it (maybe using the known series for pi and e) and work out the value of i. Simpler than resolving Brexit i reckon

[jim360]

you just get that i = ln(-1)/Pi so it's not that helpful.