Pi Problem
A friend of mine is a crammer and a bragger. He claims he remembers the value of pi up to 100 places by heart. The trouble is I do not. And I am marooned with him on an island with no books, computers, mobile or calculator. The question is, how can I verify that the 100 digits he is rattling off is not gibberish?
posted by Rajesh Lal at
7:44 AM
18 Comments:
You could always write it in the sand as recites it and get him to repeat it again and again.
although this could lead to miss trust and thats not a good thing when alone on an island. I would just humor him until rescue.
is there something to write with? If so, you can just use long division to figure it out as far as you want to. Unfortunately, I can't remember the proper fraction to use.
If you are trying to remember the proper fraction, you might be thinking of 22/7, but this is only accurate to 4-5 decimal places.
Writing it down and making him repeat it only ensures that he can remember 100, non-repeating, numbers. If you had a perfect circle and very accurate measuring devices you could use the pi*r(squared) equation. But it would be difficult to be accurate to 100 decimals.
Gregory series:
pi/4 = 1 - 1/3 + 1/5 - 1/7 . . .
You need 100 terms to be accurate to 100 decimals.
I seem to recall that there's a variation of this somewhere that gets you there a lot quicker . . .
Hi Wiz, I can't let you get away with that ;) To calculate pi correctly to a mere 10 places requires ove 10 000 000 000 operations. You die long before you got there. However there are some strange sweet spots in the calculations where you get many of the digits right, but with a few erroneous ones. See http://en.wikipedia.org/wiki/Leibniz_formula_for_pi
I think the solution is more along the lines of the statistical one that Adam rejected. I guess you could learn a few things like how many times does each digit actually appears and the lengths of runs without a particular digit appearing. But you'd have to have the foresight to do this before being marooned. better still, using precognition, tattoo the first 100 digits of pi on your body ready for the duel ;)
I think anything that involves accuratee calculations or measurements are quite impractical. 100 dp is hopelessly beyond current measurement technology.
Ramanujan developed algorithms that can knock out at least 8 extra decimal places per iteration. But you'd still need a computer to use his formula.
See http://mathworld.wolfram.com/PiFormulas.html for more.
Hi Chris,
The very same Leibniz that you quoted above also stated that in an alternating series of terms decreasing to zero the error in the sum is less than the last term that you stop at.
On that basis, 100 terms should be sufficient in the Gregory series.
How much paper and time are allowed on this island?
Hi Wiz. Your Liebniz statement can't be right. At e.g. the nth DP, we are dealing with terms of absolute value near 1/n and would probably make the nth DP oscillate for quite a while. I jumped in because I've known that the Gregory-Liebniz series is quite hopeless for calculating pi for many decades.
I think Liebniz was talking about a test for convergence of an alternating power series, it doesn't attempt to give any info about the rate of convergence. See http://en.wikipedia.org/wiki/Alternating_series_test for more info. But meanwhile, see http://www.math.ucla.edu/~mvanvalk/31Bleib.pdf for information about the error bound.
Check those links for more info on good pi algorithms. At least one of Ramanujan's (if you don't know about him, look him up, he was an astonishing mathematician) formulae is what is used in the most modern high precision calculations of pi. Pi is known to about 10^12 digits. Sorry,lots of mmaths for you to read, but you might enjoy it as it's only for interest. Greetz.
This post has been removed by the author.
Really writing this to kick the post through. Most of my deletes are because I'm kicking my posts through.
I still think that the problem is to come up with easier ways than actually calculating pi. Assuming I was unprepared for this essential activity whilst being marooned on an island, probably without paper and pen, I can't think of anything better than getting your mate to write it in the sand. Get him to do it again, somewhere else, then compare the results. It would seem very unlikely that he would be able to accurately reproduce a random number to hundred DP, rather than a famous one. That's why I can't agree with adam's rejection of such a feat.
There are a few things you could do. One, you could put up with him until you get to land, have him copy the digits down, then google.com would do wonders.
If you are impatient and want proof on the island, things would get trickier.
You could have him say the numbers, write them on the beach or memorize them, then have him recite them, day after day, without looking at your writing. you would then have proof that he has memorized 100 digits-although not necessarily pi.
Another thing you could do is have a circle yourself that you know the circumference of and the radius....although, as mentioned, no resources on the island, so that might not be an option.
You could also throw coconutts at him until he admits hes wrong, as pi clearly does not equal 42. That is, if you live with "Hitchhikers Guide to the Galaxy" as your bible.
You could also worry about more important things, such as, you know, surviving on an island where you are marooned.
If you are completely desperate to find out what pi equals so as to prove him wrong, good luck. It has taken an extremely long time for a computer to find out admittedly millions of digits of pi, but still...a COMPUTER. There is probably some formula to find out 100 digits of pi without a calculator, but whatever.
Enjoy my boredom-induced super long, pointless answer ^.^
Inspired by the last Anonymous, I just calculated Pi on my computer to 1 000 000 digits. It took less than 3 seconds. To 10 000 000 digits took about 50 seconds.
I won't publish the results, so think yourselves lucky.
For the rates of convergence of various formulae, see: http://documents.wolfram.com/mathematica/Demos/Notebooks/CalculatingPi.html
I think I'd opt for anonymous's sugestion number 5 =)
This is from http://en.wikipedia.org/wiki/Pi:
"In 1706 John Machin was the first to compute 100 decimals of π, using the formula
π/4 = 4 arctan 1/5 - arctan 1/239
with
arctan x = [...] = x - (x^3)/3 + (x^5)/5 - (x^7)/7 + ...
Formulas of this type, now known as Machin-like formulas, were used to set several successive records and remained the best known method for calculating π well into the age of computers. A remarkable record was set by the calculating prodigy Zacharias Dase, who in 1844 employed a Machin-like formula to calculate 200 decimals of π in his head at the behest of Gauss. The best value at the end of the 19th century was due to William Shanks, who took 15 years to calculate π with 707 digits, although due to a mistake only the first 527 were correct."
So it's possible, but I highly doubt that that is the answer Rajesh expected us to come up with.
Is the island perfectly circular? If so, measure it's circumferance and diameter (using steps) and divide one into the other writing in the sand..
355/113 will get you the first six decimals of pi. After that you'll just have to take him on trust, especially if he can repeat the 100 digits over again.
Measurement is out as far as 100 SF is concerned. I can't quickly find a confirmation, but the very best measurement of size of an island sized, smooth, rigid body, is probably in the neighbourhood of 7 SF. No measurement of any physical quantity of any type has ever been made to remotely near 100 SF.
Mo, thanks, especially for that bit on Zacharias Dase. I think that the Machin method is probably the optimum approach if you really had to work out the accurate answer by hand.
Wiz, I'm going along with your last answer.
I though I'd estimate how well EBs method would work. A total error of one tenth of a step of 0.5 m over a distance of 1000 m would be apparent in the about the fourth decimal place. I'm been unrealistic in allowing exactly equal lengths, but don't fancy doing the statistics to put bounds on the accumulated error of step length with standard deviation of 1% or so over a 1000 m distance. If the friend was like me, he'd fool you as he'd know 3.141593 which is as far as I bother with. I remember e as 2.7 1828 1828 though.
THATS AN EASY ANSWER. YOU DON'T NEED TO KNOW THE VALUE OF PI TO SOLVE THIS DILLEMA.
HAVE THEM RECITE THE VALUE OF PI TO THE 100TH DIGIT. JUST MEMORIZE THE LAST 10-20 DIGITS HE SAID, AND HAVE HIM RECITE THE VALUE OF PI AGAIN. IF THE LAST DIGITS HE SAID CHANGED, THEN HE WAS FULL OF S#!%.
Answer, I like your answer, but DON'T SHOUT please.
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