507:(the first one) is really too much like Niven's proof. This, however, can be countered by saying that Jones' proof is much easier to understand than Niven's. It is shorter and it is arranged much differently; it argues in a different way without defining functions. I belief it is just a matter of time before those who have a vested interest in Niven and other proofs will have to give way to this new proof. The question to me is not if they will but when. I say now is a good time. I use it in my calculus course. He gave me his second proof and to me it makes for a very good application of complex calculus and motivates the transcendence of pi. Currently there is no good motivation for the technique.
2333:
publication data. It doesn't even mean anyone has read Jones' paper yet, much less that AMS is lending support to Jones' claims. If you think I'm wrong about this, then please contact MR and find out who reviewed your MONTHLY paper, and I'll be very glad and ready to discuss your fallacies with the reviewer. After all, we appeal to reason, not to authority in math. By the way, in our personal communications before, you also appealed to authority (some 'math PhD highly respected in the field'). Would you care to disclose this authority as well, so that I can have a similar discussion with him/her?
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topic. In fact, it was first posted in the 'Article' page on Dec. 3-5, 2010. You can see that before then, the page did not have the origin of
Cartwright's exam problem, and had no mention of Hermite. After Zhou's reference was posted on Dec. 3-5, 2010, many insights, connections, and motivations between Lambert's, Hermite's, Cartwright's, and Niven's proofs came along. You may see good reasons for this from a careful reading of Zhou's article.
203:
234:
2198:, we have to figure out how the ingenious integral could be sensibly discovered, instead of cheating on the masters' integral and taking it as a starting point for granted. There are of course good motivations for Hermite, Lindemann, and Hurwitz's techniques, but it takes hard work and math competency to understand. It's a typical trait of mathematical cranks (see
2749:
3819:
2299:
reviews and sometimes errors are indicated. The fact that this article was reviewed and the fact that this review is not unfavorable and the fact that it does not indicate any errors to me means that two highly reputable mathematical organizations, the MAA and the AMS, do not agree with the assessments given above.
2813:
Also, Jones' paper is not 'obviously bad'. The logical fallacies had deceived the author himself and the MAA MONTHLY referees and Editor. They become 'obvious' only after Zhou's insightful analysis. It's like a magic trick which can be quite deceptive, but obvious after pointed out. Zhou's criticisms
2798:
Zhou's article is not just a criticism of Jones's article, it also contains a very good historical discussion of the origin and motivation of
Hermite's work (which Niven polished but not referenced in 1947). Read it and you may find very interesting things and it may deepen your understandings on the
1568:
2) If Jones had just retyped Niven's proof, then it would have been much less harmful and offensive. He didn't understand what makes Niven's proof work (because Niven polished
Hermite's work and removed the clue) and introduced, in the retyping, serious red herrings and logical fallacies. His claimed
1548:
1) It's not simpler, it's not more approachable, it's not more enjoyable, and it's not easier to understand than Niven's. It's the same as Niven's. It's deliberately made to look shorter by suppressing the details Niven took care to present. The only trivial difference is whether to present the proof
4528:
You failed to prove that x and y are integers (any number with no decimal, ie 1, 42, -9, 0, etc.). A rational numbers is a number that can be expressed as x/y **where x and y are integers**, not just any x and y since if any x and y were allowed every number would be a rational number. You have only
2344:
I do agree that Niven's proof is presented too long at
Knowledge (although some parts are connections and motivations rather than proof), but it does not mean that any trivial editing of it should be claimed as someone's brand new proof. In fact, the integral proof should only belong to Hermite (see
2730:
I've contacted the MR Editor and got a reply on the MR review of Jones' paper (MR2662709): 'In actual fact, the paper was not sent out for review - the author's abstract was taken; when a correction to MONTHLY is published, it will be indexed in MathSciNet and linked to the listing of the paper.' I
2336:
Now let me inform you the situation with MAA MONTHLY. The Editor of MONTHLY has always agreed that your paper is an exposition of Niven's proof, not a new proof. After my discussions with him, he further realized that your symmetry claim is irrelevant and a red herring. The only reason that MONTHLY
914:
506:
It can be argued that Jones' proof is not sufficiently new or noteworthy to be included in the list of proofs. However, the simplicity of his proof and its geometric character will make the subject matter more approachable and enjoyable to a greater audience. It can also be argued that the proof
432:
Jones's proof has been referenced in two books: The Heart of
Calculus: Explorations and Applications (Classroom Resource Materials) by by Philip M. Anselone (Author), John W. Lee (Author), for one. Zhou's proof has not been so referenced. It is a shame that readers unable to buy such books can't
2302:
It is also apparent that the simplicity of Jones' proof will allow more people to appreciate the irrationality of pi. Presently the
Knowledge page is in my mathematical opinion appealing exclusively to professional mathematicians. Niven's original proof is half a page; his proof as given in the
2298:
I've just checked the review of Jones' proof as given by the
American Mathematical Society via its MathSciNet service. This review, just completed in the last week or so, does not indicate any errors in Jones' proof. Not all proofs are reviewed and sometimes those that are are given unfavorable
2703:
I have found a proof of the irrationaity of pi that the reviewers indicated had an error. They further said that the error could not be fixed. Not all articles are reviewed. To me if the article was as bad as indicated above -- in other people's opinion -- it would have been indicated in the
2683:
Jones doesn't have much right to criticize others' presentation of Niven's proof either. Niven's paper is half-page long. Jones stretches it to three pages in his MAA MONTHLY paper, with two and half of those three pages filled with confusions, superstitions (about the magic power of graphical
1539:
If only mathematicians read these pages, then there is no need for me to waste time to debunk Jones at all, because they can see easily the faulty logic in Jones' claims. However, Jones is trying to peddle his own misunderstandings in math and his faulty logic to mathematically unsophisticated
2332:
Jones not only lacks basic logic and math competency, but also lacks simple knowledge about the math profession. All articles and notes from Amer. Math. MONTHLY are automatically included in the MR (Math Review at MathSciNet) database. Most of these reviews are just the author's abstract and
2341:). There are backlogs with many math journals, and a criticism of your paper may need to wait for some months before getting published. So be patient. Meanwhile, it only makes you look more like a charlatan to claim that you have AMS and MAA's supports to rob Niven (ultimately Hermite).
1552:
459:
Jones's proof has been cited in two books: More
Calculus of a Single Variable (Undergraduate Texts in Mathematics) 2014th Edition by Mercer and The Heart of Calculus: Explorations and Applications (Classroom Resource Materials) by Philip M. Anselone (Author), John W. Lee (Author).
2202:
for definition) to believe that if a problem is easy to understand, then its solution is easy to find (think of all the trisectors and circle-squarers); and that if a proof looks easy (like Niven's), then its motivation is also easy. This is what led Jones to use his unexplainable
1071:
2278:
It's of course forgivable for a mathematician to make honest mistakes and has logical lapses, so long as he acknowledges them once pointed out. But it's very much unacceptable and unethical to knowingly perpetuate falsehood. So 'I say now is a good time' to stop his nonsense.
1549:
with integration-by-parts or the equivalent product-rule. Even if this trivial difference really counts for anything, then Jones does not have any right to claim for credit, because partial integration has been used by many mathematicians ever since
Hermite. For example, see
2199:
2684:
similarity), red herrings, and logical fallacies. 'Many amateur and student mathematicians' will be seriously misled and deceived to believe that the nonsensical 'visual concept that the product of two symmetric functions is symmetric' is what makes Niven's proof work!
2122:
at all. Let me give the reader an analogy. Imagine a school kid who is taught the quadratic formula. He then plugs in three concrete numbers a=2, b=5, c=-4 into the formula and gets an answer. Can he claim that he came up with a 'new' and 'his very own' way of solving
3544:
4131:
481:
It has been several years since all of these comments were made. In that time the
Mathematical Monthly has not retracted, corrected, or commented on Jones' article. Zhou's article "On Discovering and Proving the Irrationality of Pi" has not been published.
2337:
hasn't published a correction yet is format and space. The Editor wants to limit my criticisms to one page for his Editor's Endnotes. I believe a paper so flawed (a rare blunder in the history of MONTHLY) requires a more detailed and careful refutation (see
2543:
3617:
3943:
2303:
Knowledge article is much longer and much more complicated; it delves into the esoterica. Knowledge will be the main entry point for many amateur and student mathematicians and I believe Knowledge by its very nature can and should serve both audiences.
153:
1800:
1512:
I've reverted the last couple of edits, because 999ers inserted his comments in the middle of the IP's. This is against Knowledge guidelines, so please just leave comments at the end of the thread, or elsewhere as described by
2814:
really poke Niven's proof more in depth (than ever before, with new variations of integrals and examples) to reveal what's really crucial and what's irrelevant. The reader can benefit much from these examples and comparisons.
2704:
review. What is the purpose of the review other than to inform the mathematical community about the article's value. However I will leave it to some other of the math community to post the proof, if they should see fit.
789:
2951:
1187:
925:
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3831:
3622:
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930:
794:
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Ok, you justified it last time, and nobody said anything. If nobody argues this time, I won't remove it again, as long as you make it clear that it's not related to the Jones in the infobox. --
3376:
352:
3981:
2271:
as the motivation for Niven's proof. He has neither the competency nor the patience to understand continued-fractions, recurrences, and Hermite's works to discover the real motivation (see
3814:{\displaystyle {\begin{aligned}\int _{0}^{\pi i}{\frac {d}{dz}}\left(e^{-z}F(z)\right)={\Bigr |}_{0}^{\pi i}\left(e^{-z}F(z)\right)=\int _{0}^{\pi i}e^{-z}f(z){\text{ dz}}.\end{aligned}}}
1292:
147:
2013:
1565:. These other mathematicians have the good sense and math competency not to claim anything new, which shows that the size of one's ego is inversely proportional to one's math talent.
2355:
1634:
750:
683:
4432:
1540:'greater audience' and 'calculus students' (more to satisfy his ego than to enlighten the students), even after I have explained these mistakes to him in great depths and details.
3826:
2771:
Zhou's article referring to Jones' paper should also be removed. I don't really see why it was ever put in in the first place. It is just referring to an obviously bad paper.
2604:
44:
4462:
A rational number is one that can be expressed as a ratio of two integers. Pi cannot be thus expressed i.e. your x and y cannot both be integers. This is a mathematical fallacy.
573:
4388:
is constant, regardless of the circle's size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio
3610:
2731:
believe now that it's also appropriate to remove Jones' MONTHLY paper from 'Further reading', considering the potential deception done to a casual browser of the article page.
4222:
4175:
3275:
2172:
2082:
after Lindemann, Hurwitz, and Hilbert. Jones went to this standard proof and retyped it for the simplest case of linear equation (instead of a degree-n equation satisfied by
3369:
1842:. It is very important to notice the irrelevance of the graphical similarity, because the students need to see how to generalize later to harder proofs (transcendence of
194:
3228:
2865:
2678:
545:
2243:
1840:
611:
4532:
Because x and y come from geometric lengths we can know that they are non-negative real numbers but we do not know if they are able to be expressed as integers or not
4257:
3095:
2269:
2048:
1952:
1906:
1660:
4625:
4320:
4300:
3974:
3324:
3173:
3124:
3031:
3002:
2196:
2120:
2100:
2080:
1880:
1321:
1216:
781:
4589:
3196:
1467:
1444:
4277:
3295:
3144:
2973:
2644:
2624:
1926:
1860:
1421:
1401:
1381:
1361:
1341:
643:
3059:
190:
919:
where the upper and lower bounds follow from the symmetry of the curve. Evaluating this integral using integration by parts (or tabular integration) gives
909:{\displaystyle {\begin{aligned}0<\int _{0}^{\frac {p}{q}}f(x)\sin x{\text{ dx }}\leq {\frac {p}{q}}\left({\frac {p^{2}}{4q}}\right)^{n},\end{aligned}}}
4655:
342:
79:
1665:
418:
I feel that this article should be included in the wikipedia Category `Irrational numbers', I don't know how to do this however. Can anyone add it?
318:
2174:? Can he claim that his plugging-in has motivated the formula? Of course not! To motivate the formula, he has to see how the formula could be
433:
get the information from Knowledge. They are prejudiced against Americans I suspect and like bombast and stories by jealous mathematicians.
168:
4650:
4516:
85:
135:
4329:
2711:
2349:), and no one else has much left to claim (Cartwright never claimed anything for herself). The simplest way to present Hermite's proof is:
2310:
1487:
492:
385:
2872:
467:
440:
4628:
4484:
1066:{\displaystyle {\begin{aligned}\int _{0}^{\frac {p}{q}}f(x)\sin x{\text{ dx }}=\sum _{k=0}^{n}(-1)^{k}f^{(2k)}(\pi ,0),\end{aligned}}}
4453:
309:
270:
129:
1079:
2787:
99:
30:
104:
20:
125:
74:
24:
3539:{\displaystyle {\begin{aligned}{\frac {d}{dz}}\left(e^{-z}F(z)\right)=-e^{-z}F(z)+e^{-z}F'(z)=-e^{-z}f(z),\end{aligned}}}
4126:{\displaystyle {\begin{aligned}0=F(0)(e^{\pi i}+1)=F(0)+F(\pi i)+\int _{0}^{\pi i}e^{-z}f(z){\text{ dz}},\end{aligned}}}
245:
1536:
Thanks to SarekOfVulcan for removing T. W. Jones' self-claimed proofs from the list. It's a most appropriate decision.
175:
404:
65:
202:
185:
485:
Meantime Jones' second proof has been published by the Journal of College Mathematics. Does this mean anything?
2538:{\displaystyle u_{n}(x)\sin x+v_{n}(x)\cos x={\frac {x^{2n+1}}{2^{n}n!}}\int _{0}^{1}(1-z^{2})^{n}\cos(xz)\,dz,}
213:
4333:
2715:
2314:
1483:
1221:
389:
1469:
divides the sum in the last equation. As factorial growth exceeds polynomial growth we have a contradiction.
496:
3938:{\displaystyle {\begin{aligned}e^{\pi i}F(0)=F(\pi i)+\int _{0}^{\pi i}e^{-z}f(z){\text{ dz}}.\end{aligned}}}
1957:
471:
444:
141:
4632:
2761:
1522:
1501:
400:
109:
4449:
1560:
4488:
1572:
1479:
688:
648:
4404:
251:
295:
4445:
2551:
1662:
has nothing to do with the proof. The lower and upper bounds of the integral are easy consequences of
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4325:
2823:
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2791:
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2740:
2719:
2707:
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2318:
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2288:
1526:
1505:
1475:
550:
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488:
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463:
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381:
3551:
233:
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4537:
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419:
161:
55:
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4138:
3233:
2833:
This proof uses Euler's identity and the ideas of Hermite, Lindemann, Hurwitz, and Niven. Assume
2126:
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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2800:
2757:
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1518:
1497:
423:
301:
218:
70:
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3332:
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448:
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393:
51:
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2649:
516:
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2819:
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2736:
2689:
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2210:
1805:
578:
215:
4227:
3065:
2248:
2018:
1931:
1885:
1639:
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And no, ((-1/2)!)² is not rational either. Please tell me where you got that information
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755:
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1514:
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1911:
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1795:{\displaystyle 0<f(x),0<\sin x\leq 1,x^{n}<(p/q)^{n},(p-qx)^{n}<p^{n}}
314:
1569:'geometric character' is nothing but faulty logic. The shared symmetry between
2102:) and claimed it as 'his own' again. It doesn't motivate the transcendence of
291:
2054:
for a detailed discussion of this and other red herrings in Jones' claims.
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2346:
2338:
2272:
2051:
510:
Here is the proof again -- parking it here for purposes of discussion.
4357:
2946:{\displaystyle {\begin{aligned}f(z)=z^{p-1}(nz-mi)^{p}\end{aligned}}}
2178:
discovered. Therefore, to motivate the proof of the transcendence of
752:
will share these properties. The maximum of this function occurs at
4353:
399:
This link doesn't work. Does anyone have more info on that proof?
613:
will have the same shape, roots, and symmetry in the interval
227:
219:
15:
1182:{\displaystyle f^{(2k)}(\pi ,0)=f^{(2k)}(\pi )+f^{(2k)}(0)}
2975:
is a prime; the function is a complex polynomial. Define
377:
http://mathforum.org/kb/plaintext.jspa?messageID=1614449
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3181:
3152:
3132:
3103:
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3010:
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2632:
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2358:
2251:
2213:
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2129:
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1960:
1934:
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1868:
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1808:
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1409:
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1195:
1082:
928:
792:
758:
691:
651:
619:
581:
553:
519:
2680:
completes the proof. The rest are all trivialities.
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There is a well-known proof of the transcendence of
313:, a collaborative effort to improve the coverage of
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2007:
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1335:
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908:
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605:
567:
539:
4515:harvnb error: no target: CITEREFArndtHaenel2006 (
3699:
372:A shorter proof than all of the ones on the page
33:for general discussion of the article's subject.
2294:The AMS does not agree that Jones' proof is bad
4279:with this division, we have a contradiction.
1446:in all their coefficients. We conclude that
174:
8:
3823:With some algebraic manipulation this gives
2626:of integer coefficients and degrees at most
4510:
4477:But ((-1/2)!)^2 is rational, which is pi.
3033:. Using Leibniz' formula we find that if
1287:{\displaystyle f^{(2k)}(0)=f^{(2k)}(\pi )}
486:
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434:
379:
259:
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4414:
4406:
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4264:
4229:
4182:
4177:, Euler's identity. The complex part of
4146:
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4111:
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4014:
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3923:
3902:
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3778:
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3334:
3302:
3282:
3235:
3230:. We have then that the complex part of
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2933:
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2427:
2421:
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2212:
2183:
2137:
2128:
2107:
2087:
2067:
2020:
1990:
1965:
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1933:
1913:
1887:
1867:
1847:
1821:
1807:
1786:
1773:
1745:
1733:
1718:
1667:
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1620:
1595:
1574:
1451:
1428:
1408:
1388:
1368:
1348:
1328:
1299:
1260:
1229:
1223:
1194:
1155:
1124:
1087:
1081:
1026:
1016:
997:
986:
974:
942:
937:
929:
927:
893:
873:
867:
852:
844:
812:
807:
793:
791:
762:
757:
736:
711:
690:
670:
650:
618:
580:
560:
552:
529:
518:
4282:This proof generalizes to all powers of
3277:is not zero. If the sum were zero then
2008:{\displaystyle e^{x},\cos x,e^{x}\cos x}
1563:, Nieuw Archief voor Wiskunde, Dec(2000)
4503:
2524:
261:
231:
1423:. After this all derivatives have an
4302:and also yields the transcendence of
1323:is a polynomial whose least power of
7:
4224:remains a non zero upon division by
3146:does not divide the complex part of
2699:The AMS reviews are not as described
1629:{\displaystyle f(x)=x^{n}(p-qx)^{n}}
1532:Rebuttal of Jones' fallacious claims
745:{\displaystyle f(x)=x^{n}(p-qx)^{n}}
678:{\displaystyle f(x)\sin {\text{ x}}}
307:This article is within the scope of
4427:{\displaystyle \pi ={\frac {x}{y}}}
1954:can be replaced by non-symmetrical
250:It is of interest to the following
23:for discussing improvements to the
14:
4656:Mid-priority mathematics articles
3004:as the sum of the derivatives of
2599:{\displaystyle u_{n}(x),v_{n}(x)}
327:Knowledge:WikiProject Mathematics
3097:does divide the complex part of
2747:
2058:Jones' self-claimed second proof
568:{\displaystyle \sin {\text{ x}}}
330:Template:WikiProject Mathematics
294:
284:
263:
232:
201:
45:Click here to start a new topic.
3605:{\displaystyle F(z)-F'(z)=f(z)}
2050:can have different powers. See
347:This article has been rated as
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3420:
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3313:
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3264:
3255:
3246:
3240:
3217:
3208:
3162:
3156:
3113:
3107:
3081:
3069:
3020:
3014:
2991:
2985:
2930:
2911:
2889:
2883:
2593:
2587:
2571:
2565:
2521:
2512:
2497:
2477:
2406:
2400:
2375:
2369:
2232:
2217:
1829:
1809:
1770:
1754:
1742:
1727:
1684:
1678:
1617:
1601:
1585:
1579:
1310:
1304:
1281:
1275:
1270:
1261:
1250:
1244:
1239:
1230:
1205:
1199:
1176:
1170:
1165:
1156:
1145:
1139:
1134:
1125:
1114:
1102:
1097:
1088:
1053:
1041:
1036:
1027:
1013:
1003:
962:
956:
832:
826:
733:
717:
701:
695:
661:
655:
632:
620:
600:
585:
1:
4217:{\displaystyle F(0)+F(\pi i)}
4170:{\displaystyle e^{\pi i}+1=0}
3612:. Integrating this, we have
3270:{\displaystyle F(0)+F(\pi i)}
2207:about the similarity between
2167:{\displaystyle 2x^{2}+5x-4=0}
1544:Jones' self-claimed 1st proof
501:19:02, 27 November 2014 (UTC)
321:and see a list of open tasks.
42:Put new text under old text.
4651:B-Class mathematics articles
4556:22:25, 5 December 2022 (UTC)
4542:22:24, 5 December 2022 (UTC)
3198:divides the complex part of
2824:15:32, 6 February 2011 (UTC)
2809:03:47, 6 February 2011 (UTC)
2792:18:06, 5 February 2011 (UTC)
2766:21:13, 4 February 2011 (UTC)
2741:20:52, 4 February 2011 (UTC)
2720:22:46, 1 February 2011 (UTC)
2694:03:49, 31 January 2011 (UTC)
2325:Jones' new logical fallacy:
2319:13:09, 30 January 2011 (UTC)
2289:18:56, 28 January 2011 (UTC)
1527:14:38, 25 January 2011 (UTC)
1506:19:17, 15 January 2011 (UTC)
409:19:14, 29 October 2020 (UTC)
4398:. This proves that pi=x/y.
4352:is commonly defined as the
4259:. As the integral goes to
50:New to Knowledge? Welcome!
4672:
4529:shown that pi is a number
3364:{\displaystyle e^{-z}F(z)}
428:10:56, 24 April 2010 (UTC)
394:20:46, 14 March 2015 (UTC)
25:Proof that π is irrational
4637:17:33, 5 March 2020 (UTC)
4493:08:51, 15 June 2014 (UTC)
4437:Base pi: 3.14₁₀=10₃.₁₄₁₆
4338:13:23, 1 March 2011 (UTC)
1561:A rational approach to pi
1189:. Using the symmetry of
476:00:59, 30 June 2016 (UTC)
449:00:45, 30 June 2016 (UTC)
346:
279:
258:
80:Be welcoming to newcomers
4472:11:42, 18 May 2014 (UTC)
3223:{\displaystyle F(\pi i)}
2860:{\displaystyle \pi =m/n}
2673:{\displaystyle x=\pi /2}
1490:) 14:22, 11 January 2011
1383:derivatives evaluate to
540:{\displaystyle \pi =p/q}
353:project's priority scale
4511:Arndt & Haenel 2006
3976:to both sides, we have
2238:{\displaystyle x(p-qx)}
2200:Underwood Dudley's book
1835:{\displaystyle (0,p/q)}
606:{\displaystyle x(p-qx)}
310:WikiProject Mathematics
4621:
4585:
4428:
4348:Pi is not irrational.
4316:
4296:
4273:
4253:
4252:{\displaystyle (p-1)!}
4218:
4171:
4127:
3970:
3939:
3815:
3606:
3540:
3365:
3320:
3291:
3271:
3224:
3192:
3169:
3140:
3120:
3091:
3090:{\displaystyle (p-1)!}
3055:
3027:
2998:
2969:
2947:
2861:
2674:
2640:
2620:
2600:
2539:
2265:
2264:{\displaystyle \sin x}
2239:
2192:
2168:
2116:
2096:
2076:
2044:
2043:{\displaystyle x,p-qx}
2009:
1948:
1947:{\displaystyle \sin x}
1922:
1902:
1901:{\displaystyle \cos r}
1876:
1856:
1836:
1796:
1656:
1655:{\displaystyle \sin x}
1630:
1555:, MAA, 2001, pp. 6--7
1463:
1440:
1417:
1397:
1377:
1357:
1337:
1317:
1288:
1212:
1183:
1067:
1002:
910:
777:
746:
679:
639:
607:
569:
541:
240:This article is rated
75:avoid personal attacks
4622:
4620:{\displaystyle (n!)!}
4586:
4429:
4317:
4297:
4274:
4254:
4219:
4172:
4128:
3971:
3940:
3816:
3607:
3541:
3366:
3321:
3297:would have to divide
3292:
3272:
3225:
3193:
3175:. We also find that
3170:
3141:
3121:
3092:
3056:
3053:{\displaystyle p: -->
3028:
2999:
2970:
2948:
2862:
2675:
2641:
2621:
2601:
2540:
2266:
2240:
2205:graphical sixth sense
2193:
2169:
2117:
2097:
2077:
2045:
2010:
1949:
1923:
1908:for rational nonzero
1903:
1877:
1857:
1837:
1797:
1657:
1631:
1464:
1441:
1418:
1398:
1378:
1358:
1338:
1318:
1289:
1213:
1184:
1068:
982:
911:
778:
747:
680:
640:
608:
570:
542:
195:Auto-archiving period
100:Neutral point of view
4599:
4569:
4405:
4344:Pi is not irrational
4315:{\displaystyle \pi }
4306:
4295:{\displaystyle \pi }
4286:
4263:
4228:
4181:
4139:
3982:
3969:{\displaystyle F(0)}
3951:
3827:
3618:
3552:
3377:
3333:
3329:We use the function
3319:{\displaystyle F(0)}
3301:
3281:
3234:
3202:
3179:
3168:{\displaystyle F(0)}
3150:
3130:
3119:{\displaystyle F(0)}
3101:
3066:
3038:
3026:{\displaystyle f(z)}
3008:
2997:{\displaystyle F(z)}
2979:
2959:
2873:
2837:
2650:
2630:
2610:
2552:
2356:
2249:
2211:
2191:{\displaystyle \pi }
2182:
2127:
2115:{\displaystyle \pi }
2106:
2095:{\displaystyle \pi }
2086:
2075:{\displaystyle \pi }
2066:
2019:
1958:
1932:
1912:
1886:
1875:{\displaystyle \pi }
1866:
1846:
1806:
1666:
1640:
1573:
1553:Conjecture and proof
1450:
1427:
1407:
1387:
1367:
1347:
1327:
1316:{\displaystyle f(x)}
1298:
1222:
1211:{\displaystyle f(x)}
1193:
1080:
926:
790:
776:{\displaystyle p/2q}
756:
689:
649:
617:
579:
551:
517:
454:
333:mathematics articles
105:No original research
4584:{\displaystyle n!!}
4135:where we have used
4085:
3897:
3773:
3717:
3642:
2829:Jones' second proof
2606:are polynomials in
2476:
2327:Appeal_to_authority
1882:, irrationality of
1551:Miklós Laczkovich,
952:
822:
4617:
4581:
4424:
4312:
4292:
4269:
4249:
4214:
4167:
4123:
4121:
4068:
3966:
3935:
3933:
3880:
3811:
3809:
3756:
3696:
3625:
3602:
3536:
3534:
3361:
3316:
3287:
3267:
3220:
3191:{\displaystyle p!}
3188:
3165:
3136:
3116:
3087:
3050:
3023:
2994:
2965:
2943:
2941:
2857:
2670:
2636:
2616:
2596:
2535:
2525:
2462:
2261:
2235:
2188:
2164:
2112:
2092:
2072:
2040:
2005:
1944:
1918:
1898:
1872:
1852:
1832:
1792:
1652:
1626:
1462:{\displaystyle n!}
1459:
1439:{\displaystyle n!}
1436:
1413:
1393:
1373:
1353:
1333:
1313:
1284:
1208:
1179:
1063:
1061:
933:
906:
904:
803:
773:
742:
675:
635:
603:
575:and the quadratic
565:
537:
302:Mathematics portal
246:content assessment
86:dispute resolution
47:
4483:comment added by
4458:
4444:comment added by
4422:
4328:comment added by
4272:{\displaystyle 0}
4114:
3926:
3802:
3656:
3397:
3290:{\displaystyle p}
3139:{\displaystyle p}
2968:{\displaystyle p}
2795:
2778:comment added by
2710:comment added by
2639:{\displaystyle n}
2619:{\displaystyle x}
2460:
2309:comment added by
1921:{\displaystyle r}
1855:{\displaystyle e}
1492:
1478:comment added by
1416:{\displaystyle 0}
1396:{\displaystyle 0}
1376:{\displaystyle n}
1356:{\displaystyle n}
1336:{\displaystyle x}
977:
950:
887:
860:
847:
820:
673:
563:
503:
491:comment added by
478:
466:comment added by
451:
439:comment added by
401:George Albert Lee
396:
384:comment added by
367:
366:
363:
362:
359:
358:
226:
225:
66:Assume good faith
43:
4663:
4626:
4624:
4623:
4618:
4593:double factorial
4590:
4588:
4587:
4582:
4521:
4520:
4508:
4495:
4457:
4438:
4433:
4431:
4430:
4425:
4423:
4415:
4397:
4387:
4377:
4368:
4351:
4340:
4321:
4319:
4318:
4313:
4301:
4299:
4298:
4293:
4278:
4276:
4275:
4270:
4258:
4256:
4255:
4250:
4223:
4221:
4220:
4215:
4176:
4174:
4173:
4168:
4154:
4153:
4132:
4130:
4129:
4124:
4122:
4115:
4112:
4098:
4097:
4084:
4076:
4022:
4021:
3975:
3973:
3972:
3967:
3944:
3942:
3941:
3936:
3934:
3927:
3924:
3910:
3909:
3896:
3888:
3846:
3845:
3820:
3818:
3817:
3812:
3810:
3803:
3800:
3786:
3785:
3772:
3764:
3752:
3748:
3735:
3734:
3716:
3708:
3703:
3702:
3692:
3688:
3675:
3674:
3657:
3655:
3644:
3641:
3633:
3611:
3609:
3608:
3603:
3577:
3545:
3543:
3542:
3537:
3535:
3516:
3515:
3488:
3480:
3479:
3452:
3451:
3433:
3429:
3416:
3415:
3398:
3396:
3385:
3370:
3368:
3367:
3362:
3348:
3347:
3325:
3323:
3322:
3317:
3296:
3294:
3293:
3288:
3276:
3274:
3273:
3268:
3229:
3227:
3226:
3221:
3197:
3195:
3194:
3189:
3174:
3172:
3171:
3166:
3145:
3143:
3142:
3137:
3125:
3123:
3122:
3117:
3096:
3094:
3093:
3088:
3061:
3058:
3057:
3051:
3032:
3030:
3029:
3024:
3003:
3001:
3000:
2995:
2974:
2972:
2971:
2966:
2952:
2950:
2949:
2944:
2942:
2938:
2937:
2910:
2909:
2866:
2864:
2863:
2858:
2853:
2794:
2772:
2755:
2751:
2750:
2726:MathSciNet reply
2722:
2679:
2677:
2676:
2671:
2666:
2645:
2643:
2642:
2637:
2625:
2623:
2622:
2617:
2605:
2603:
2602:
2597:
2586:
2585:
2564:
2563:
2544:
2542:
2541:
2536:
2505:
2504:
2495:
2494:
2475:
2470:
2461:
2459:
2452:
2451:
2441:
2440:
2422:
2399:
2398:
2368:
2367:
2321:
2270:
2268:
2267:
2262:
2244:
2242:
2241:
2236:
2197:
2195:
2194:
2189:
2173:
2171:
2170:
2165:
2142:
2141:
2121:
2119:
2118:
2113:
2101:
2099:
2098:
2093:
2081:
2079:
2078:
2073:
2049:
2047:
2046:
2041:
2014:
2012:
2011:
2006:
1995:
1994:
1970:
1969:
1953:
1951:
1950:
1945:
1927:
1925:
1924:
1919:
1907:
1905:
1904:
1899:
1881:
1879:
1878:
1873:
1861:
1859:
1858:
1853:
1841:
1839:
1838:
1833:
1825:
1802:on the interval
1801:
1799:
1798:
1793:
1791:
1790:
1778:
1777:
1750:
1749:
1737:
1723:
1722:
1661:
1659:
1658:
1653:
1635:
1633:
1632:
1627:
1625:
1624:
1600:
1599:
1491:
1472:
1468:
1466:
1465:
1460:
1445:
1443:
1442:
1437:
1422:
1420:
1419:
1414:
1402:
1400:
1399:
1394:
1382:
1380:
1379:
1374:
1362:
1360:
1359:
1354:
1342:
1340:
1339:
1334:
1322:
1320:
1319:
1314:
1293:
1291:
1290:
1285:
1274:
1273:
1243:
1242:
1217:
1215:
1214:
1209:
1188:
1186:
1185:
1180:
1169:
1168:
1138:
1137:
1101:
1100:
1072:
1070:
1069:
1064:
1062:
1040:
1039:
1021:
1020:
1001:
996:
978:
975:
951:
943:
941:
915:
913:
912:
907:
905:
898:
897:
892:
888:
886:
878:
877:
868:
861:
853:
848:
845:
821:
813:
811:
782:
780:
779:
774:
766:
751:
749:
748:
743:
741:
740:
716:
715:
684:
682:
681:
676:
674:
671:
645:. The function
644:
642:
641:
638:{\displaystyle }
636:
612:
610:
609:
604:
574:
572:
571:
566:
564:
561:
546:
544:
543:
538:
533:
335:
334:
331:
328:
325:
304:
299:
298:
288:
281:
280:
275:
267:
260:
243:
237:
236:
228:
220:
206:
205:
196:
179:
178:
164:
95:Article policies
16:
4671:
4670:
4666:
4665:
4664:
4662:
4661:
4660:
4641:
4640:
4597:
4596:
4567:
4566:
4563:
4526:
4525:
4524:
4514:
4509:
4505:
4478:
4439:
4403:
4402:
4389:
4379:
4373:
4364:
4349:
4346:
4323:
4304:
4303:
4284:
4283:
4261:
4260:
4226:
4225:
4179:
4178:
4142:
4137:
4136:
4120:
4119:
4086:
4010:
3980:
3979:
3949:
3948:
3932:
3931:
3898:
3834:
3825:
3824:
3808:
3807:
3774:
3723:
3722:
3718:
3663:
3662:
3658:
3648:
3616:
3615:
3570:
3550:
3549:
3533:
3532:
3504:
3481:
3468:
3440:
3404:
3403:
3399:
3389:
3375:
3374:
3371:next. We have
3336:
3331:
3330:
3299:
3298:
3279:
3278:
3232:
3231:
3200:
3199:
3177:
3176:
3148:
3147:
3128:
3127:
3099:
3098:
3064:
3063:
3035:
3034:
3006:
3005:
2977:
2976:
2957:
2956:
2940:
2939:
2929:
2895:
2871:
2870:
2835:
2834:
2773:
2748:
2746:
2705:
2648:
2647:
2628:
2627:
2608:
2607:
2577:
2555:
2550:
2549:
2496:
2486:
2443:
2442:
2423:
2390:
2359:
2354:
2353:
2304:
2247:
2246:
2209:
2208:
2180:
2179:
2133:
2125:
2124:
2104:
2103:
2084:
2083:
2064:
2063:
2017:
2016:
1986:
1961:
1956:
1955:
1930:
1929:
1928:, etc.), where
1910:
1909:
1884:
1883:
1864:
1863:
1844:
1843:
1804:
1803:
1782:
1769:
1741:
1714:
1664:
1663:
1638:
1637:
1616:
1591:
1571:
1570:
1559:Frits Beukers,
1473:
1448:
1447:
1425:
1424:
1405:
1404:
1385:
1384:
1365:
1364:
1345:
1344:
1325:
1324:
1296:
1295:
1256:
1225:
1220:
1219:
1191:
1190:
1151:
1120:
1083:
1078:
1077:
1060:
1059:
1022:
1012:
924:
923:
903:
902:
879:
869:
863:
862:
788:
787:
754:
753:
732:
707:
687:
686:
647:
646:
615:
614:
577:
576:
549:
548:
515:
514:
457:
416:
411:
374:
332:
329:
326:
323:
322:
300:
293:
273:
244:on Knowledge's
241:
222:
221:
216:
193:
121:
116:
115:
114:
91:
61:
12:
11:
5:
4669:
4667:
4659:
4658:
4653:
4643:
4642:
4616:
4613:
4610:
4607:
4604:
4580:
4577:
4574:
4562:
4559:
4523:
4522:
4502:
4501:
4497:
4475:
4474:
4435:
4434:
4421:
4418:
4413:
4410:
4345:
4342:
4330:76.101.178.158
4311:
4291:
4268:
4248:
4245:
4242:
4239:
4236:
4233:
4213:
4210:
4207:
4204:
4201:
4198:
4195:
4192:
4189:
4186:
4166:
4163:
4160:
4157:
4152:
4149:
4145:
4118:
4110:
4107:
4104:
4101:
4096:
4093:
4089:
4083:
4080:
4075:
4071:
4067:
4064:
4061:
4058:
4055:
4052:
4049:
4046:
4043:
4040:
4037:
4034:
4031:
4028:
4025:
4020:
4017:
4013:
4009:
4006:
4003:
4000:
3997:
3994:
3991:
3988:
3987:
3965:
3962:
3959:
3956:
3930:
3922:
3919:
3916:
3913:
3908:
3905:
3901:
3895:
3892:
3887:
3883:
3879:
3876:
3873:
3870:
3867:
3864:
3861:
3858:
3855:
3852:
3849:
3844:
3841:
3837:
3833:
3832:
3806:
3798:
3795:
3792:
3789:
3784:
3781:
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3207:
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3135:
3115:
3112:
3109:
3106:
3086:
3083:
3080:
3077:
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3049:
3046:
3043:
3022:
3019:
3016:
3013:
2993:
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2936:
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2712:169.139.115.67
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2347:Zhou's article
2339:Zhou's article
2311:76.101.178.158
2273:Zhou's article
2260:
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2111:
2091:
2071:
2052:Zhou's article
2039:
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2030:
2027:
2024:
2004:
2001:
1998:
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1989:
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1979:
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1917:
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1480:76.101.178.158
1458:
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1392:
1372:
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1312:
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1038:
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1032:
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1019:
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1008:
1005:
1000:
995:
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989:
985:
981:
973:
970:
967:
964:
961:
958:
955:
949:
946:
940:
936:
932:
931:
917:
916:
901:
896:
891:
885:
882:
876:
872:
866:
859:
856:
851:
843:
840:
837:
834:
831:
828:
825:
819:
816:
810:
806:
802:
799:
796:
795:
772:
769:
765:
761:
739:
735:
731:
728:
725:
722:
719:
714:
710:
706:
703:
700:
697:
694:
669:
666:
663:
660:
657:
654:
634:
631:
628:
625:
622:
602:
599:
596:
593:
590:
587:
584:
559:
556:
536:
532:
528:
525:
522:
493:98.208.142.242
456:
453:
415:
412:
398:
386:98.155.236.135
373:
370:
365:
364:
361:
360:
357:
356:
345:
339:
338:
336:
319:the discussion
306:
305:
289:
277:
276:
268:
256:
255:
249:
238:
224:
223:
214:
212:
211:
208:
207:
181:
180:
118:
117:
113:
112:
107:
102:
93:
92:
90:
89:
82:
77:
68:
62:
60:
59:
48:
39:
38:
35:
34:
28:
13:
10:
9:
6:
4:
3:
2:
4668:
4657:
4654:
4652:
4649:
4648:
4646:
4639:
4638:
4634:
4630:
4614:
4608:
4605:
4594:
4578:
4575:
4572:
4560:
4558:
4557:
4553:
4549:
4544:
4543:
4539:
4535:
4530:
4518:
4512:
4507:
4504:
4500:
4496:
4494:
4490:
4486:
4482:
4473:
4469:
4465:
4461:
4460:
4459:
4455:
4451:
4447:
4443:
4419:
4416:
4411:
4408:
4401:
4400:
4399:
4396:
4392:
4386:
4382:
4376:
4372:
4367:
4363:
4362:circumference
4359:
4355:
4343:
4341:
4339:
4335:
4331:
4327:
4309:
4289:
4280:
4266:
4246:
4240:
4237:
4234:
4208:
4205:
4199:
4196:
4190:
4184:
4164:
4161:
4158:
4155:
4150:
4147:
4143:
4133:
4116:
4105:
4099:
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4073:
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4047:
4041:
4035:
4032:
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4023:
4018:
4015:
4011:
4001:
3995:
3992:
3989:
3977:
3960:
3954:
3945:
3928:
3917:
3911:
3906:
3903:
3899:
3893:
3890:
3885:
3881:
3877:
3871:
3868:
3862:
3859:
3853:
3847:
3842:
3839:
3835:
3821:
3804:
3793:
3787:
3782:
3779:
3775:
3769:
3766:
3761:
3757:
3753:
3749:
3742:
3736:
3731:
3728:
3724:
3719:
3713:
3710:
3705:
3693:
3689:
3682:
3676:
3671:
3668:
3664:
3659:
3652:
3649:
3645:
3638:
3635:
3630:
3626:
3613:
3596:
3590:
3587:
3581:
3574:
3571:
3567:
3561:
3555:
3546:
3529:
3523:
3517:
3512:
3509:
3505:
3501:
3498:
3492:
3485:
3482:
3476:
3473:
3469:
3465:
3459:
3453:
3448:
3445:
3441:
3437:
3434:
3430:
3423:
3417:
3412:
3409:
3405:
3400:
3393:
3390:
3386:
3372:
3355:
3349:
3344:
3341:
3337:
3327:
3310:
3304:
3284:
3261:
3258:
3252:
3249:
3243:
3237:
3214:
3211:
3205:
3185:
3182:
3159:
3153:
3133:
3110:
3104:
3084:
3078:
3075:
3072:
3047:
3044:
3041:
3017:
3011:
2988:
2982:
2962:
2953:
2934:
2926:
2923:
2920:
2917:
2914:
2906:
2903:
2900:
2896:
2892:
2886:
2880:
2868:
2854:
2850:
2846:
2843:
2840:
2831:
2830:
2826:
2825:
2821:
2817:
2811:
2810:
2806:
2802:
2796:
2793:
2789:
2785:
2781:
2777:
2767:
2763:
2759:
2758:SarekOfVulcan
2754:
2745:
2744:
2743:
2742:
2738:
2734:
2728:
2727:
2723:
2721:
2717:
2713:
2709:
2701:
2700:
2696:
2695:
2691:
2687:
2681:
2667:
2663:
2659:
2656:
2653:
2633:
2613:
2590:
2582:
2578:
2574:
2568:
2560:
2556:
2532:
2529:
2526:
2518:
2515:
2509:
2506:
2501:
2491:
2487:
2483:
2480:
2472:
2467:
2463:
2456:
2453:
2448:
2444:
2437:
2434:
2431:
2428:
2424:
2418:
2415:
2412:
2409:
2403:
2395:
2391:
2387:
2384:
2381:
2378:
2372:
2364:
2360:
2352:
2351:
2350:
2348:
2342:
2340:
2334:
2330:
2329:
2328:
2322:
2320:
2316:
2312:
2308:
2300:
2296:
2295:
2291:
2290:
2286:
2282:
2276:
2274:
2258:
2255:
2252:
2229:
2226:
2223:
2220:
2214:
2206:
2201:
2185:
2177:
2161:
2158:
2155:
2152:
2149:
2146:
2143:
2138:
2134:
2130:
2109:
2089:
2069:
2060:
2059:
2055:
2053:
2037:
2034:
2031:
2028:
2025:
2022:
2002:
1999:
1996:
1991:
1987:
1983:
1980:
1977:
1974:
1971:
1966:
1962:
1941:
1938:
1935:
1915:
1895:
1892:
1889:
1869:
1849:
1826:
1822:
1818:
1815:
1812:
1787:
1783:
1779:
1774:
1766:
1763:
1760:
1757:
1751:
1746:
1738:
1734:
1730:
1724:
1719:
1715:
1711:
1708:
1705:
1702:
1699:
1696:
1693:
1690:
1687:
1681:
1675:
1672:
1669:
1649:
1646:
1643:
1621:
1613:
1610:
1607:
1604:
1596:
1592:
1588:
1582:
1576:
1566:
1564:
1562:
1556:
1554:
1546:
1545:
1541:
1537:
1534:
1533:
1529:
1528:
1524:
1520:
1519:SarekOfVulcan
1516:
1507:
1503:
1499:
1498:SarekOfVulcan
1495:
1494:
1493:
1489:
1485:
1481:
1477:
1470:
1456:
1453:
1433:
1430:
1410:
1390:
1370:
1350:
1330:
1307:
1301:
1278:
1267:
1264:
1257:
1253:
1247:
1236:
1233:
1226:
1202:
1196:
1173:
1162:
1159:
1152:
1148:
1142:
1131:
1128:
1121:
1117:
1111:
1108:
1105:
1094:
1091:
1084:
1056:
1050:
1047:
1044:
1033:
1030:
1023:
1017:
1009:
1006:
998:
993:
990:
987:
983:
979:
971:
968:
965:
959:
953:
947:
944:
938:
934:
922:
921:
920:
899:
894:
889:
883:
880:
874:
870:
864:
857:
854:
849:
841:
838:
835:
829:
823:
817:
814:
808:
804:
800:
797:
786:
785:
784:
770:
767:
763:
759:
737:
729:
726:
723:
720:
712:
708:
704:
698:
692:
667:
664:
658:
652:
629:
626:
623:
597:
594:
591:
588:
582:
557:
554:
547:we note that
534:
530:
526:
523:
520:
511:
508:
504:
502:
498:
494:
490:
483:
479:
477:
473:
469:
468:98.208.142.38
465:
452:
450:
446:
442:
441:98.208.142.38
438:
430:
429:
425:
421:
413:
410:
406:
402:
397:
395:
391:
387:
383:
378:
371:
369:
354:
350:
344:
341:
340:
337:
320:
316:
312:
311:
303:
297:
292:
290:
287:
283:
282:
278:
272:
269:
266:
262:
257:
253:
247:
239:
235:
230:
229:
210:
209:
204:
200:
192:
189:
187:
183:
182:
177:
173:
170:
167:
163:
159:
155:
152:
149:
146:
143:
140:
137:
134:
131:
127:
124:
123:Find sources:
120:
119:
111:
110:Verifiability
108:
106:
103:
101:
98:
97:
96:
87:
83:
81:
78:
76:
72:
69:
67:
64:
63:
57:
53:
52:Learn to edit
49:
46:
41:
40:
37:
36:
32:
26:
22:
18:
17:
4629:95.49.72.119
4564:
4545:
4531:
4527:
4506:
4498:
4485:83.24.97.120
4479:— Preceding
4476:
4440:— Preceding
4436:
4394:
4390:
4384:
4380:
4378:: The ratio
4374:
4365:
4347:
4281:
4134:
3978:
3946:
3822:
3614:
3547:
3373:
3328:
2954:
2869:
2832:
2828:
2827:
2812:
2797:
2774:— Preceding
2770:
2752:
2729:
2725:
2724:
2702:
2698:
2697:
2682:
2547:
2343:
2335:
2331:
2324:
2323:
2301:
2297:
2293:
2292:
2277:
2204:
2175:
2061:
2057:
2056:
2015:, etc., and
1567:
1558:
1550:
1547:
1543:
1542:
1538:
1535:
1531:
1530:
1510:
1474:— Preceding
1471:
1363:, the first
1075:
918:
512:
509:
505:
487:— Preceding
484:
480:
462:— Preceding
458:
455:Jones' proof
435:— Preceding
431:
417:
380:— Preceding
375:
368:
349:Mid-priority
348:
308:
274:Mid‑priority
252:WikiProjects
198:
184:
171:
165:
157:
150:
144:
138:
132:
122:
94:
19:This is the
4446:83.28.62.68
4324:—Preceding
2706:—Preceding
2305:—Preceding
1517:. Thanks.--
783:. We have
324:Mathematics
315:mathematics
271:Mathematics
148:free images
31:not a forum
4645:Categories
4499:References
4464:BethNaught
3060:m}" /: -->
2867:. Define
2646:. Letting
4591:mean the
4548:JGHFunRun
4534:JGHFunRun
2780:Theonanda
513:Assuming
88:if needed
71:Be polite
21:talk page
4561:Notation
4481:unsigned
4454:contribs
4442:unsigned
4371:diameter
4326:unsigned
3037:m}": -->
2788:contribs
2776:unsigned
2708:unsigned
2307:unsigned
2176:sensibly
1488:contribs
1476:unsigned
1218:we have
489:unsigned
464:unsigned
437:unsigned
420:Octonion
414:Category
382:unsigned
186:Archives
56:get help
29:This is
27:article.
4369:to its
3947:Adding
3062:, then
1515:WP:TALK
351:on the
242:B-class
199:30 days
154:WP refs
142:scholar
4513:, p. 8
4358:circle
3126:, but
2955:where
2816:999ers
2801:999ers
2733:999ers
2686:999ers
2548:where
2281:999ers
1557:, or
1294:. As
1076:where
248:scale.
126:Google
4565:Does
4356:of a
4354:ratio
3045:: -->
685:with
169:JSTOR
130:books
84:Seek
4633:talk
4552:talk
4538:talk
4517:help
4489:talk
4468:talk
4450:talk
4334:talk
2820:talk
2805:talk
2784:talk
2762:talk
2753:Done
2737:talk
2716:talk
2690:talk
2315:talk
2285:talk
2245:and
1862:and
1780:<
1725:<
1694:<
1673:<
1636:and
1523:talk
1502:talk
1484:talk
801:<
497:talk
472:talk
445:talk
424:talk
405:talk
390:talk
162:FENS
136:news
73:and
4595:or
4360:'s
4322:.
3548:as
2507:cos
2410:cos
2379:sin
2275:).
2253:sin
1997:cos
1975:cos
1936:sin
1890:cos
1697:sin
1644:sin
1403:at
1343:is
976:dx
966:sin
846:dx
836:sin
665:sin
555:sin
343:Mid
176:TWL
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