most recent 30 from http://mathoverflow.net 2013-05-18T08:13:39Z http://mathoverflow.net/feeds/user/17457 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/74841/an-example-of-a-beautiful-proof-that-would-be-accessible-at-the-high-school-level Manya 2011-09-08T08:10:47Z 2013-04-19T22:02:28Z

<p>The background of my question comes from an observation that what we teach in schools does not always reflect what we practice. Beauty is part of what drives mathematicians, but we rarely talk about beauty in the teaching of school mathematics. </p> <p>I'm trying to collect examples of good, accessible proofs that could be used in middle school or high school. Here are two that I have come across thus far:</p> <p>(1) Pick's Theorem: The area, A, of a lattice polygon, with boundary points B and interior points I is A = I + B/2 - 1.</p> <p>I'm actually not so interested in verifying the theorem (sometimes given as a middle school task) but in actually proving it. There are a few nice proofs floating around, like one given in "Proofs from the Book" which uses a clever application of Euler's formula. A very different, but also clever proof, which Bjorn Poonen was kind enough to show to me, uses a double counting of angle measures, around each vertex and also around the boundary. Both of these proofs involve math that doesn't go much beyond the high school level, and they feel like real mathematics.</p> <p>(2) Menelaus Theorem: If a line meets the sides BC, CA, and AB of a triangle in the points D, E, and F then (AE/EC) (CD/DB) (BF/FA) = 1. (converse also true) See: <a href="http://www.cut-the-knot.org/Generalization/Menelaus.shtml" rel="nofollow">http://www.cut-the-knot.org/Generalization/Menelaus.shtml</a>, also for the related Ceva's Theorem.</p> <p>Again, I'm not interested in the proof for verification purposes, but for a beautiful, enlightening proof. I came across such a proof by Grunbaum and Shepard in Mathematics Magazine. They use what they call the Area Principle, which compares the areas of triangles that share the same base (I would like to insert a figure here, but I do not know how. -- given triangles ABC and DBC and the point P that lies at the intersection of AD and BC, AP/PD = Area (ABC)/Area(DBC).) This principle is great-- with it, you can knock out Menelaus, Ceva's, and a similar theorem involving pentagons. And it is not hard-- I think that an average high school student could follow it; and a clever student might be able to discover this principle themselves.</p> <p>Anyway, I'd be grateful for any more examples like these. I'd also be interested in people's judgements about what makes these proofs beautiful (if indeed they are-- is there a difference between a beautiful proof and a clever one?) but I don't know if that kind of discussion is appropriate for this forum.</p> <p><strong>Edit</strong>: I just want to be clear that in my question I'm really asking about proofs you'd consider to be beautiful, not just ones that are neat or accessible at the high school level. (not that the distinction is always so easy to make...)</p>

http://mathoverflow.net/questions/107153/first-known-proof-of-sqrt-2-is-irrational-with-prime-factorization Manya 2012-09-14T08:18:18Z 2012-09-15T18:33:33Z

<p>Do any of you happen to know the history of the standard prime factorization proof of $\sqrt 2$ is irrational? I know this theorem was known to Aristotle, and that the Fundamental Theorem of Arithmetic, on which the proof rests, is found already in Euclid, but I've not been able to track down the origin of this particular proof. </p> <p>These sites I know about: <a href="http://www.cut-the-knot.org/proofs/sq_root.shtml" rel="nofollow">http://www.cut-the-knot.org/proofs/sq_root.shtml</a>, <a href="http://www.math.ufl.edu/~rcrew/texts/pythagoras.html" rel="nofollow">http://www.math.ufl.edu/~rcrew/texts/pythagoras.html</a>, and of course <a href="http://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic" rel="nofollow">http://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic</a></p> <p>But any other references, online or in paper form, would be greatly appreciated!</p>

http://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time Manya 2012-05-03T10:07:24Z 2012-07-09T07:27:58Z

<p>I am looking for examples of theorems that may have originally had a clunky, or rather technical, or in some way non-illuminating proof, but that eventually came to have a proof that people consider to be particularly nice. In other words, I'm looking for examples of theorems for which have some early proof for which you'd say "ok that works but I'm sure this could be improved", and then some later proof for which you'd say "YES! That is exactly how you should do it!"</p> <p>Thanks in advance.</p>

http://mathoverflow.net/questions/80451/casual-tours-around-proofs/80460#80460 Manya 2011-11-09T08:42:28Z 2011-11-09T08:42:28Z

<p>Polya's "Mathematics and Plausible Reasoning" is a good source. In chapter XVI he compares how many textbooks present proofs (which he calls a "deus ex machina" approach) and how it could be done showing the discovery process. The example he uses to illustrate is: If the terms of the sequence a1, a2, a3... are non-negative real numbers not all equal 0, then sum(1, infinity) (a_1, a_2, ...a_n)^1/2 &lt; e sum(1, infinity)a_n.</p>

http://mathoverflow.net/questions/75922/is-beauty-at-the-high-school-level-even-possible Manya 2011-09-20T08:21:01Z 2011-09-20T11:33:23Z

<p>This question is a follow up to <a href="http://mathoverflow.net/questions/74841/an-example-of-a-beautiful-proof-that-would-be-accessible-at-the-high-school-level" rel="nofollow">74841</a>, and follows from a suggestion by <a href="http://mitpress.mit.edu/books/chapters/0262550636chap1.pdf" rel="nofollow">Gian-Carlo Rota</a> that beauty as judged by the educated public differs from that experienced by mathematicians (he gives Euclidean geometry as an example of a theory that non-mathematicians would find beautiful, but mathematicians would hesitate to classify as so.) Rota also cautions against creating a list of beautiful theorems, even though it might be a best-seller, because one cannot really experience beauty when taken out of context. </p> <p>He explains, "The fact is that the beauty of a mathematical theorem is best observed when the theorem is presented as the crown jewel within a context of results of a theory. When instead mathematical theorems from disparate areas of mathematics are strung together and presented as “pearls”, then they lose their relevance, and are likely to be appreciated only by those who are already familiar with them."</p> <p><em><strong>This brings me to my question</strong>: Is the task of trying to introduce beauty in high school mathematics in vain?</em> (This question resonates with one comment on my original question regarding whether high school students are capable of really appreciating proof.)</p> <p>It seems to be there are two obvious answers, one of which is yes, just forget it. The other is, no, but don't expect students to get a full appreciation of beauty by the time they leave high school. The third, bolder, answer is that one can achieve beauty early if exposed to the right mathematics in the right way.</p> <p><em><strong>So:</strong> If answer three is correct, does anyone have any existence proofs (either from experience or the literature)?</em></p> <p>(Also, just to be careful, in my original question, I was specifically asking for examples of beautiful proofs, while Rota is discussing theories and theorems as well. And I want to emphasize that I ask this question only because it is something that genuinely worries me, not that I fail to appreciate the thoughtful answers given to my original question.)</p>

http://mathoverflow.net/questions/74707/mathematical-habits-of-thought-and-action-which-would-be-of-use-to-non-mathematic/74713#74713 Manya 2011-09-07T05:31:50Z 2011-09-08T07:22:57Z

<p>Edit: I have taken away my comments comparing my experience in mathematics and math ed departments. I still think there is something true about the kind of character that emerges from working on hard problems which demand a certain kind of rigor, and I also think there is something special (and not known to the general public) about the strong social bonds that exist in the mathematical community, but I don't think that one needs to contrast math and math ed groups to make that point.</p> <p>My short answer (to which Gil alludes below) was: Persistence and humility.</p> <p>Though I'm not sure this applies only to mathematicians, but in general to people who work on hard, and perhaps in some ways technical, problems.</p>

http://mathoverflow.net/questions/48771/proofs-that-require-fundamentally-new-ways-of-thinking/73954#73954 Manya 2011-08-29T09:18:17Z 2011-08-29T09:18:17Z

<p>How about Bolzano's 1817 proof of the intermediate value theorem?</p> <p>In English here: Russ, S. B. "A Translation of Bolzano's Paper on the Intermediate Value Theorem." Hist. Math. 7, 156-185, 1980. </p> <p>Or in the original here: Bernard Bolzano (1817). Purely analytic proof of the theorem that between any two values which give results of opposite sign, there lies at least one real root of the equation. In Abhandlungen der koniglichen bohmischen GeseUschaft der Wissenschaften Vol. V, pp.225-48.</p> <p>Not fully rigorous, according to today's standards, but perhaps his method of proof could be considered a breakthrough nonetheless.</p>

http://mathoverflow.net/questions/74841/an-example-of-a-beautiful-proof-that-would-be-accessible-at-the-high-school-level/128134#128134 Manya 2013-04-23T07:40:01Z 2013-04-23T07:40:01Z

Ok thanks for the comment.

http://mathoverflow.net/questions/74841/an-example-of-a-beautiful-proof-that-would-be-accessible-at-the-high-school-level/128137#128137 Manya 2013-04-23T07:37:57Z 2013-04-23T07:37:57Z

Thanks for the example.

http://mathoverflow.net/questions/74841/an-example-of-a-beautiful-proof-that-would-be-accessible-at-the-high-school-level/83957#83957 Manya 2013-03-18T10:09:02Z 2013-03-18T10:09:02Z

Thanks, that is nice.

http://mathoverflow.net/questions/74841/an-example-of-a-beautiful-proof-that-would-be-accessible-at-the-high-school-level/111552#111552 Manya 2012-11-06T08:18:37Z 2012-11-06T08:18:37Z

Thanks, Neil. Nice examples.

http://mathoverflow.net/questions/74841/an-example-of-a-beautiful-proof-that-would-be-accessible-at-the-high-school-level/108766#108766 Manya 2012-10-04T06:33:40Z 2012-10-04T06:33:40Z

Ok, thanks for sharing these.

http://mathoverflow.net/questions/107153/first-known-proof-of-sqrt-2-is-irrational-with-prime-factorization/107170#107170 Manya 2012-09-18T19:08:03Z 2012-09-18T19:08:03Z

Ok, thanks, that seems like an interesting book.

http://mathoverflow.net/questions/107153/first-known-proof-of-sqrt-2-is-irrational-with-prime-factorization/107220#107220 Manya 2012-09-18T11:32:28Z 2012-09-18T11:32:28Z

Yeah, I agree, this thread is off-topic, but there are of course other cool proofs out there. You can find a nice collection at the link I gave above (<a href="http://www.cut-the-knot.org/proofs/sq_root.shtml" rel="nofollow">cut-the-knot.org/proofs/sq_root.shtml</a>). If you haven't seen 8''' before, that one is quite nice.

http://mathoverflow.net/questions/107153/first-known-proof-of-sqrt-2-is-irrational-with-prime-factorization/107158#107158 Manya 2012-09-18T11:22:20Z 2012-09-18T11:22:20Z

Ok, I accept that :-)

http://mathoverflow.net/questions/107153/first-known-proof-of-sqrt-2-is-irrational-with-prime-factorization/107158#107158 Manya 2012-09-18T08:47:53Z 2012-09-18T08:47:53Z

Thanks for the Mazur paper. It is very good! But is it so clear that Gauss applied the fundamental theorem to the $\sqrt 2$. Or am I missing something?

http://mathoverflow.net/questions/107153/first-known-proof-of-sqrt-2-is-irrational-with-prime-factorization Manya 2012-09-14T15:39:07Z 2012-09-14T15:39:07Z

No, thanks for the reference.

http://mathoverflow.net/questions/107153/first-known-proof-of-sqrt-2-is-irrational-with-prime-factorization/107158#107158 Manya 2012-09-14T09:55:28Z 2012-09-14T09:55:28Z

Thanks for the information and the links. I wonder if Gauss himself could have proven it...?

http://mathoverflow.net/questions/107153/first-known-proof-of-sqrt-2-is-irrational-with-prime-factorization Manya 2012-09-14T08:19:35Z 2012-09-14T08:19:35Z

I'm not very good with tags, so please feel free to edit if there are better choices. Thanks.

http://mathoverflow.net/questions/74841/an-example-of-a-beautiful-proof-that-would-be-accessible-at-the-high-school-level/87744#87744 Manya 2012-06-18T19:44:26Z 2012-06-18T19:44:26Z

Cool. That one came up in a seminar I attended recently (on the topic of beauty in mathematics...)

http://mathoverflow.net/questions/74841/an-example-of-a-beautiful-proof-that-would-be-accessible-at-the-high-school-level/99731#99731 Manya 2012-06-18T19:43:19Z 2012-06-18T19:43:19Z

Thanks. I don't think I saw that one in high school!

http://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time/95856#95856 Manya 2012-05-04T08:20:25Z 2012-05-04T08:20:25Z

@Liviu: Thanks. Though some of the proofs in that book (as the authors themselves admit) are not necessarily the nicest or cleanest versions. Are there any proofs in there that you think are particularly good?