# Two questions about Pi **URL:** \ **Category:** Factual Questions\ **Created:** [March 14, 2015, 11:58pm UTC](https://boards.straightdope.com/t/two-questions-about-pi/715016 "2015-03-14T23:58:28Z")\ **Posts on this page:** 14\ **Page:** 3
**Author:** ![UncleRojelio](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/unclerojelio/32/3160_2.png) [@UncleRojelio](https://boards.straightdope.com/u/UncleRojelio)\ **Post date:** [March 16, 2015, 7:28pm UTC](https://boards.straightdope.com/t/two-questions-about-pi/715016/41 "2015-03-16T19:28:19Z")
[Donald Knuth on some Pi associations](https://www.youtube.com/watch?v=cI6tt9QfRdo). ---
**Author:** ![Suburban\_Plankton](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/suburban_plankton/32/219_2.png) [@Suburban\_Plankton](https://boards.straightdope.com/u/Suburban_Plankton)\ **Post date:** [March 16, 2015, 7:38pm UTC](https://boards.straightdope.com/t/two-questions-about-pi/715016/42 "2015-03-16T19:38:33Z")
> [@Derleth](#): > > Technically, it’s equally correct to say that _all_ rational numbers repeat, either in an infinity of zeroes or, if written another way, in an infinity of nines. So 1/1 is rational, and it can be written 1.0… or 0.9… and both forms end in an endless repetition. So you’re saying that 0.99999… = 1? I think this assertion calls for further discussion 😃 ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [March 16, 2015, 7:43pm UTC](https://boards.straightdope.com/t/two-questions-about-pi/715016/43 "2015-03-16T19:43:56Z")
> [@Suburban\_Plankton](#): > > So you’re saying that 0.99999… = 1? Only for small values of 1. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [March 16, 2015, 7:55pm UTC](https://boards.straightdope.com/t/two-questions-about-pi/715016/44 "2015-03-16T19:55:03Z")
> [@Indistinguishable](#): > > Thus, the n-fold donvolution of x with itself is the result of integrating **its derivative** n times, x[sup]n[/sup]/n! [that this is the n-fold integral of **x[sup]0[/sup]** corresponds to the basic power rule that x[sup]n[/sup] integrates to x[sup]n + 1[/sup]/(n + 1)]. I could swear I had already made these corrections, but at any rate, here they are now, in bold. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [March 16, 2015, 8:03pm UTC](https://boards.straightdope.com/t/two-questions-about-pi/715016/45 "2015-03-16T20:03:32Z")
Shit, another typo: > [@Indistinguishable](#): > > Which means x[sup]n[/sup]/n! donvolved with x[sup]m[/sup]/m! must be x[sup]n + m[/sup]/**(n + m)!**. ---
**Author:** ![Senegoid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/senegoid/32/6606_2.png) [@Senegoid](https://boards.straightdope.com/u/Senegoid)\ **Post date:** [March 16, 2015, 8:06pm UTC](https://boards.straightdope.com/t/two-questions-about-pi/715016/46 "2015-03-16T20:06:35Z")
When is _e_ day? ---
**Author:** ![74westy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/74westy/32/3950_2.png) [@74westy](https://boards.straightdope.com/u/74westy)\ **Post date:** [March 16, 2015, 8:11pm UTC](https://boards.straightdope.com/t/two-questions-about-pi/715016/47 "2015-03-16T20:11:54Z")
> [@Senegoid](#): > > When is _e_ day? Feb. 72nd ---
**Author:** ![Coriolanus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/coriolanus/32/18769_2.png) [@Coriolanus](https://boards.straightdope.com/u/Coriolanus)\ **Post date:** [March 16, 2015, 8:14pm UTC](https://boards.straightdope.com/t/two-questions-about-pi/715016/48 "2015-03-16T20:14:46Z")
> [@Senegoid](#): > > When is _e_ day? [It actually is a thing](http://www.timeanddate.com/holidays/world/e-day) a day each for MDY and DMY types. ---
**Author:** ![Knowed\_Out](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/knowed_out/32/7380_2.png) [@Knowed\_Out](https://boards.straightdope.com/u/Knowed_Out)\ **Post date:** [March 16, 2015, 8:20pm UTC](https://boards.straightdope.com/t/two-questions-about-pi/715016/49 "2015-03-16T20:20:28Z")
Here’s a [presentation](https://docs.google.com/presentation/d/15RQlAHZOBj4xvWhifLuUoVNXlYjuCuA8TtylbqDNQaw/edit?usp=sharing)I did that shows how Pi could be calculated using polygons. You could therefore say the area of a circle is the sum of the areas of an infinite amount of polygons. Each polygon would have a rational number for its area, but since there’s an infinite amount of them, all those rational numbers add up to an irrational number. ---
**Author:** ![Coriolanus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/coriolanus/32/18769_2.png) [@Coriolanus](https://boards.straightdope.com/u/Coriolanus)\ **Post date:** [March 16, 2015, 8:27pm UTC](https://boards.straightdope.com/t/two-questions-about-pi/715016/50 "2015-03-16T20:27:56Z")
> [@Knowed\_Out](#): > > Here’s a [presentation](https://docs.google.com/presentation/d/15RQlAHZOBj4xvWhifLuUoVNXlYjuCuA8TtylbqDNQaw/edit?usp=sharing)I did that shows how Pi could be calculated using polygons. Nice. I like your “Near Infinity” - It reminds me of the Classic Trek (“Wolf in the Fold”) where Spock, in order to keep the computer busy, orders it to calculate π to the last digit. ---
**Author:** ![Lumpy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lumpy/32/446_2.png) [@Lumpy](https://boards.straightdope.com/u/Lumpy)\ **Post date:** [March 16, 2015, 8:47pm UTC](https://boards.straightdope.com/t/two-questions-about-pi/715016/51 "2015-03-16T20:47:23Z")
> [@Cabin\_Fever](#): > > ETA: I want to be a Quantum Mechanic one day. [https://html1-f.scribdassets.com/3jchbwyh6oa2awc/images/81-26cb6891af.jpg](https://html1-f.scribdassets.com/3jchbwyh6oa2awc/images/81-26cb6891af.jpg) ---
**Author:** ![Cabin\_Fever\_1](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cabin_fever_1/32/2908_2.png) [@Cabin\_Fever\_1](https://boards.straightdope.com/u/Cabin_Fever_1)\ **Post date:** [March 17, 2015, 12:59am UTC](https://boards.straightdope.com/t/two-questions-about-pi/715016/52 "2015-03-17T00:59:03Z")
> [@Lumpy](#): > > [https://html1-f.scribdassets.com/3jchbwyh6oa2awc/images/81-26cb6891af.jpg](https://html1-f.scribdassets.com/3jchbwyh6oa2awc/images/81-26cb6891af.jpg) OMG. 😛 You owe me a new keyboard… ---
**Author:** ![Hector\_St\_Clare](https://avatars.discourse-cdn.com/v4/letter/h/b5ac83/32.png) [@Hector\_St\_Clare](https://boards.straightdope.com/u/Hector_St_Clare)\ **Post date:** [March 18, 2015, 4:29pm UTC](https://boards.straightdope.com/t/two-questions-about-pi/715016/53 "2015-03-18T16:29:13Z")
> [@septimus](#): > > For anyone who’s not yet seen it, [Buffon’s needle “trick” for estimating _pi_ might be interesting.](http://www.numberphile.com/pi/pi_matches.html) Another way to visualize pi that doesn’t rely on circles or polygons: pi is 4 times the area under the curve _y_ = 1 / (1 + _x_[sup]2[/sup]) between _x_ = 0 and _x_ = 1. The hidden connexion to circles here is that the integral of that curve is the arctangent. You can use that integral to calculate pi, although it converges very slowly so no one ever does. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [March 18, 2015, 5:47pm UTC](https://boards.straightdope.com/t/two-questions-about-pi/715016/54 "2015-03-18T17:47:32Z")
> [@Hector\_St\_Clare](#): > > Another way to visualize pi that doesn’t rely on circles or polygons: pi is 4 times the area under the curve _y_ = 1 / (1 + _x_[sup]2[/sup]) between _x_ = 0 and _x_ = 1. > > The hidden connexion to circles here is that the integral of that curve is the arctangent. You can use that integral to calculate pi, although it converges very slowly so no one ever does. Au contraire. One of the fastest way people compute π is by integrating 1/(1 + _x_[sup]2[/sup]). They just do it over different ranges and combine the result. In more detail: 1/(1 + _x_[sup]2[/sup]) = 1 - x[sup]2[/sup] + x[sup]4[/sup] - x[sup]6[/sup] + … for |x| \< 1, by the standard geometric series expansion, and thus its integral from 0 to x [arctan(x)] in that range is x - x[sup]3[/sup]/3 + x[sup]5[/sup]/5 - x[sup]7[/sup]/7 + … Put another way, its integral from 0 to 1/x [arccot(x)] is 1/x - 1/(3x[sup]3[/sup]) + 1/(5x[sup]5[/sup]) - 1/(7x[sup]7[/sup]) + … This gives a very efficient means of computing arccot(x) for x \> 1 [note that the terms in the series alternate between positive and negative but get smaller in magnitude by a factor of at least x[sup]2[/sup] each time; thus, the partial sums alternate between a sequence of lower upper bounds and a sequence of increasing lower bounds, with the distance between these shrinking in accordance with the size of the terms]. [We can also compute arccot(1) directly with this series (which, for x = 1, would be 1 - 1/3 + 1/5 - 1/7 + …), but very slowly (now the shrinking by at least a factor of x[sup]2[/sup] is of no use; here, instead, convergence comes from the fact that 1/n goes to 0, but much more slowly).] So instead of computing π/4 as arccot(1), we can use representations of arccot(1) as combinations of arccot(n) for larger n [“Machin-like formulae”]. The simplest of these is arccot(1) = arccot(2) + arccot(3) [evidenced by the observation that (2 + i) \* (3 + i) = 5 \* (1 + i)], and this gives a method of computing π which is already very efficient. World record computations are often done by the same technique, just using slightly more honed decompositions [e.g., arccot(1) = 12 arccot(49) + 32 arccot(57) - 5 arccot(239) + arccot(110443), again verifiable by simple complex number calculation as before]. [Previous page](https://boards.straightdope.com/t/two-questions-about-pi/715016.md?page=2)