# The End of Pi? **URL:** \ **Category:** Factual Questions\ **Created:** [June 11, 2003, 11:37pm UTC](https://boards.straightdope.com/t/the-end-of-pi/181227 "2003-06-11T23:37:15Z")\ **Posts on this page:** 9\ **Page:** 2
**Author:** ![kabbes](https://avatars.discourse-cdn.com/v4/letter/k/e99b99/32.png) [@kabbes](https://boards.straightdope.com/u/kabbes)\ **Post date:** [June 12, 2003, 12:42pm UTC](https://boards.straightdope.com/t/the-end-of-pi/181227/21 "2003-06-12T12:42:06Z")
Just to reinforce: showing that _e_ and pi are transcendental is _hard_. People had been essentially trying to find out whether pi could be represented algebreically or whether it _transcended_ algebra (hence the name) for 2,500 years before Lindemann came along in 1882. Hermite showed that _e_ was transcendental in 1873 after a more moderate mere 100 years of attempt. It took him many years of his life and after doing so he simply refused to attempt pi, claiming that he couldn’t face the task! However Lindemann actually used Hermite’s methods when tackling pi, the task taking a fairly modest 9 years(!) There are a few numbers that to this day have not been proved to be transcendental, though are believed to be. These include:[ul][li]Euler’s constant, gamma (the limt as n goes to infinity of sum(1/n) - ln(n))[/li][li]Catalan’s constant (the sum of -1[sup]k[/sup]/(2k+1)[sup]2[/sup] = 1 - 1/9 + 1/25 - 1/49 + …); and [/li][li]pi[sup]_e_[/sup][/ul]The fact that we still can’t prove the transcendentality of these numbers should give you an idea of just how hard the task is![/li] Other numbers that we _have_ proved to be transcendental include:[ul][li]Chaitin’s constant (the probability that a random algorithm halts);[/li][li]_e_[sup]pi[/sup]; and[/li][li]2[sup]sqrt(2)[/sup][/ul]The last is an example of the Gelfond-Schneider theorem mentioned by **Jabba** but has a special name (Hilbert’s number) because proving its transcendality was one of Hibert’s famous problems.[/li] One last thing: although proving a number is transcendental is generally hard, it can be shown that transcendental numbers have the same cardinality as the reals, which means there are more of them than there are rational numbers. How’s about that? I have a thing about transcendental numbers. Can you tell? 😃 pan ---
**Author:** ![Algernon](https://avatars.discourse-cdn.com/v4/letter/a/a183cd/32.png) [@Algernon](https://boards.straightdope.com/u/Algernon)\ **Post date:** [June 12, 2003, 12:49pm UTC](https://boards.straightdope.com/t/the-end-of-pi/181227/22 "2003-06-12T12:49:21Z")
> [@](#): > > One last thing: although proving a number is transcendental is generally hard, it can be shown that transcendental numbers have the same cardinality as the reals, which means there are more of them than there are rational numbers. How’s about that? Fascinating **kabbes** … I think I just might be having a transcendental experience. ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [June 12, 2003, 1:17pm UTC](https://boards.straightdope.com/t/the-end-of-pi/181227/23 "2003-06-12T13:17:23Z")
Further to **kabbes’s** post, it has not even been proved that Euler’s Constant [symbol]g[/symbol] is irrational, let alone transcendental. The most that has been proved in this direction is that if [symbol]g[/symbol] = a/b, then b \> 10[sup]10,000[/sup]. ---
**Author:** ![kabbes](https://avatars.discourse-cdn.com/v4/letter/k/e99b99/32.png) [@kabbes](https://boards.straightdope.com/u/kabbes)\ **Post date:** [June 12, 2003, 1:22pm UTC](https://boards.straightdope.com/t/the-end-of-pi/181227/24 "2003-06-12T13:22:32Z")
Thanks **Jabba**. Bizarrely enough, I didn’t know that! pan ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [June 12, 2003, 2:10pm UTC](https://boards.straightdope.com/t/the-end-of-pi/181227/25 "2003-06-12T14:10:51Z")
Chaitin’s constant is an interesting case. If anyone tells you they know any part of its value, you can infer that they are assuming that it has that value. We do know that it’s somewhere in [0, 1], but that’s it. For more info, read [this](http://www.cs.auckland.ac.nz/CDMTCS/chaitin/unm2.html). As an example of how difficult it is to prove that numbers are irrational, consider [symbol]p[/symbol]e and [symbol]p[/symbol] + e. We know that they’re not both rational, but that’s it. ---
**Author:** ![alimarx](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@alimarx](https://boards.straightdope.com/u/alimarx)\ **Post date:** [June 12, 2003, 2:23pm UTC](https://boards.straightdope.com/t/the-end-of-pi/181227/26 "2003-06-12T14:23:31Z")
> [@](#): > > \*Originally posted by muttrox \* > **…go ahead and calculate the google’th digit…** [nitpick] In’n’t spelled “googol”? [/nitpick] ---
**Author:** ![Colophon](https://avatars.discourse-cdn.com/v4/letter/c/f05b48/32.png) [@Colophon](https://boards.straightdope.com/u/Colophon)\ **Post date:** [June 12, 2003, 3:56pm UTC](https://boards.straightdope.com/t/the-end-of-pi/181227/27 "2003-06-12T15:56:27Z")
Yup. A googol is 10[sup]100[/sup]. A Google is currently only [3,083,324,652](http://www.google.com). ---
**Author:** ![Saxman](https://avatars.discourse-cdn.com/v4/letter/s/f6c823/32.png) [@Saxman](https://boards.straightdope.com/u/Saxman)\ **Post date:** [June 12, 2003, 7:14pm UTC](https://boards.straightdope.com/t/the-end-of-pi/181227/28 "2003-06-12T19:14:05Z")
Am I the only one for whom **Jabba** ’s proof came out, literally, in Greek? Like, the symbols are all in Greek? ---
**Author:** ![bonzer](https://avatars.discourse-cdn.com/v4/letter/b/45deac/32.png) [@bonzer](https://boards.straightdope.com/u/bonzer)\ **Post date:** [June 12, 2003, 8:16pm UTC](https://boards.straightdope.com/t/the-end-of-pi/181227/29 "2003-06-12T20:16:35Z")
They’re somewhat more complicated than the proof that sqrt(2) is irrational, but, for completeness, here are proofs for pi being [irrational](http://pi314.at/math/irrational.html) and [trancendental](http://dspace.dial.pipex.com/town/way/po28/maths/docs/pi.html). [Previous page](https://boards.straightdope.com/t/the-end-of-pi/181227.md?page=1)