kiki wrote:> Hi all, > > I am an engineering student who is interested in math. I don't remember how > often I have been amazed by nice and elegant mathematical tricks that make > difficult problem suddenly very easy and trivial...I don't know if it qualifies as a "trick", but I love the elegant simplicity of Euclid's algorithm for finding the greatest common divisor of two numbers. Euclid's algorithm is especially elegant in Python, which may be why Guido uses it as an example: def gcd(a,b): while a>0: a , b = b%a , a return b I think one of the greatest tricks of all time is the FFT. Although I don't think it qualifies as easy or trivial, just computationally efficient.
collection of mathematically elegant tricks
Started by ●July 24, 2005
Reply by ●July 25, 20052005-07-25
Reply by ●July 25, 20052005-07-25
Lynn Kurtz wrote:> On Sun, 24 Jul 2005 11:14:52 -0700, "kiki" <lunaliu3@yahoo.com> wrote: > > >>Hi all, >> >>I am an engineering student who is interested in math. I don't remember how >>often I have been amazed by nice and elegant mathematical tricks that make >>difficult problem suddenly very easy and trivial... Many nice tricks >>frequently appear in these newsgroups... I am wondering if anybody has seen >>a collection of mathematical tricks ranging from high school math up to >>graduate school math? Any website, Internet resources, books that have these >>kind bags of tricks? If not, I may want to start collecting and compile one >>such resources. >> >>Recently, one very striking trick is offered by "Scott Hemphill" in >>computing the expected waiting time for a certain pattern to occur in coin >>tossing... I also remember many of the other tricks that have been >>contributed by many other authors in these newgroups... >> >>Thanks a lot >> > > > Maybe not all that clever but calculus students who have tired of > integrals of the form Int exp(ax) sin(bx) dx and its sister with > cos(bx) usually are very receptive to discovering that the hated > "integrate by parts twice and solve" can be avoided by doing the > exponential integral Int exp((a + ibx)) dx and separating real and > imaginary parts. Twice as easy and you get the sister for free.Half as difficult; agreed. But you get the sister (mostly, anyway) either way.) Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●July 25, 20052005-07-25
Robert Israel wrote:> > A trick that is used more than once is called a method. >:-) I hope you don't mind if I quote you; this applies to just about every craft I ply. Cheers - Chas
Reply by ●July 25, 20052005-07-25
In article <1122261925.361203.290020@z14g2000cwz.googlegroups.com>, <cbrown@cbrownsystems.com> wrote:>Robert Israel wrote:>> A trick that is used more than once is called a method.>:-) I hope you don't mind if I quote you; this applies to just about >every craft I ply.It's not mine. I've seen it attributed to Ron Getoor. However, I think the original version is by George Polya and Gabor Szego: An idea which can be used once is a trick. If it can be used more than once it becomes a method. (see e.g. <http://euclid.trentu.ca/math/sb/misc/quotes.html>) Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
Reply by ●July 25, 20052005-07-25
Robert Israel wrote:> In article <dbj8e15ecr7ipoieeq5sqpe59hm16hq5jh@4ax.com>, > quasi <quasi@null.set> wrote: > > >>As far as applicability to work and/or research -- yes, absolutely. >>Tricks are tricks, and once learned, they become part of your arsenal >>of creative strategies. You'd be surprised -- tricks from one field >>can often be reused in totally different fields. > > > A trick that is used more than once is called a method.And more than that, all the methods we learn in mathematics used to be just tricks.
Reply by ●July 25, 20052005-07-25
Yeah check out my website- it has lots of pretty PDE solutions that involve all kinds of elegant tricks via Fourier and Laplace. It's meant for graduate level math. www.exampleproblems.com -Todd
Reply by ●July 25, 20052005-07-25
"ToddSmith" <elliptic1@gmail.com> wrote in message news:1122267634.898662.76210@f14g2000cwb.googlegroups.com...> Yeah check out my website- it has lots of pretty PDE solutions that > involve all kinds of elegant tricks via Fourier and Laplace. It's meant > for graduate level math. > > www.exampleproblems.com > > -Todd >Wow, this is a forum that supports math equations... very nice. What kind of math equations does it support? Latex? I have been always looking for such a forum to write and communicate math more efficiently...
Reply by ●July 25, 20052005-07-25
On 25 Jul 2005 04:28:06 GMT, israel@math.ubc.ca (Robert Israel) wrote:>In article <1122261925.361203.290020@z14g2000cwz.googlegroups.com>, > <cbrown@cbrownsystems.com> wrote: > >>Robert Israel wrote: > >>> A trick that is used more than once is called a method. > >>:-) I hope you don't mind if I quote you; this applies to just about >>every craft I ply. > >It's not mine. I've seen it attributed to Ron Getoor. However, I think >the original version is by George Polya and Gabor Szego: > >An idea which can be used once is a trick. If it can be used more than >once it becomes a method. > >(see e.g. <http://euclid.trentu.ca/math/sb/misc/quotes.html>) > > >Robert Israel israel@math.ubc.ca >Department of Mathematics http://www.math.ubc.ca/~israel >University of British Columbia Vancouver, BC, CanadaOr, to paraphrase James Bond: "A trick used once, call it happenstance. The same trick twice, a coincidence. But 3 times is enemy action." quasi
Reply by ●July 25, 20052005-07-25
In article <dc1pom$o8m$1@nntp.itservices.ubc.ca>, Robert Israel <israel@math.ubc.ca> writes [...]>It's not mine. I've seen it attributed to Ron Getoor. However, I think >the original version is by George Polya and Gabor Szego: > >An idea which can be used once is a trick. If it can be used more than >once it becomes a method.I've heard this continued along these lines: `An idea which is used three times becomes a Theorem. An idea which is used four times... ah, if an idea is used four times it becomes an Axiom.' -- Graham Jones http://www.visiv.co.uk Emails to graham@visiv.co.uk may be deleted as spam Please add a j just before the @ to ensure delivery
Reply by ●July 25, 20052005-07-25
kiki wrote:> Hi all, > > I am an engineering student who is interested in math. I don't remember how > often I have been amazed by nice and elegant mathematical tricks that make > difficult problem suddenly very easy and trivial... Many nice tricks > frequently appear in these newsgroups... I am wondering if anybody has seen > a collection of mathematical tricks ranging from high school math up to > graduate school math? Any website, Internet resources, books that have these > kind bags of tricks? If not, I may want to start collecting and compile one > such resources. > > Recently, one very striking trick is offered by "Scott Hemphill" in > computing the expected waiting time for a certain pattern to occur in coin > tossing... I also remember many of the other tricks that have been > contributed by many other authors in these newgroups... > > Thanks a lotYou will probably enjoy: Proofs from the Book by Martin Aigner, Gunter M. Ziegler ISBN: 3540636986 It's a collection of the most elegant mathematical proofs* accesible to non professional mathematicians. You don't need lots of background knowledge, and the problems are easy to grasp intuitively. I certainly liked it. * in the authors opinion of course.
