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 lotI've collected some over the years (math tricks and some useful device designs) and have thought about expanding the group and publishing them, but it is a low priority project. Although it probably isn't exactly what you're thinking of, IMO there are already excellent references, called math handbooks. :-) Over the years I've found a good knowledge of trig identities, series that sum to useful quantities, and other such things one finds in such books to be some of the most useful things I've learned. One doesn't have to remember all the details (although that's helpful), but just recall that one might have seen something involving the functions involved in the problem and look up various possibilities. Cheers, Russell
collection of mathematically elegant tricks
Started by ●July 24, 2005
Reply by ●July 24, 20052005-07-24
Reply by ●July 24, 20052005-07-24
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...Something I have described as magic to my friends is evaluating integrals using residues. You start with an integral that looks, not just difficult, but impossible. Then you sketch a clever curve in the complex plane, shake the Residue Theorem over it, and the answer pops out. I'm not sure I've got the terminology right, but this is what I mean: http://mathworld.wolfram.com/ComplexResidue.html http://mathworld.wolfram.com/ResidueTheorem.html http://mathworld.wolfram.com/ContourIntegration.html Jim Burns
Reply by ●July 24, 20052005-07-24
In article <42E4253A.C2598E82@osu.edu>, Jim Burns <burns.87@osu.edu> wrote:>Something I have described as magic to my friends is evaluating >integrals using residues.>You start with an integral that looks, not just difficult, but >impossible. Then you sketch a clever curve in the complex plane, >shake the Residue Theorem over it, and the answer pops out. > >I'm not sure I've got the terminology right, but this is >what I mean:Oh, I think that's an excellent way of putting it. 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 24, 20052005-07-24
On Sun, 24 Jul 2005 15:35:01 -0700, "kiki" <lunaliu3@yahoo.com> wrote:> >"quasi" <quasi@null.set> wrote in message >news:oq78e15ss5hk4pro0q4qanj32resstk7td@4ax.com... >> On Sun, 24 Jul 2005 11:14:52 -0700, "kiki" <lunaliu3@yahoo.com> wrote: >> >>>... 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. >> >> Try some competition problem books. The math competitions often have >> problems where the solution is simple and elegant after applying some >> inspired tricks. >> >> 2 very good ones that I know of: >> >> The USSR Olympiad Problem Book >> >> 500 Mathematical Challenges >> >> quasi > >Hi Quasi, > >Are these problems real problems that will appear in study and research? >These mathematical Olympiad always gives me an impression that they are >unreal and heavenly; that working out them does not improve one's study, >work, and research... am I correct?Well, in some cases, yes, so ignore those problems where the statement or solution doesn't seem worthwhile. Choose problems which intrigue you. 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. quasi
Reply by ●July 24, 20052005-07-24
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. 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 24, 20052005-07-24
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. > > Robert Israel israel@math.ubc.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, BC, CanadaFrobenius comes to mind. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●July 24, 20052005-07-24
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...You can restate all the trig identities half- and double-angle theorems and the rest as complex algebra, and cover a semester's worth of trigonometry ground in under two weeks. I won't argue that it's a trick, but it beats looking for that table of formulas. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●July 24, 20052005-07-24
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 lotOne trick that I particularly like is the proof that an irreducible polynomial of the complex numbers cannot have double roots. (The proof is to consider the derivative of the polynomial.) This is actually rather important in Galois Theory in showing things like the quintic cannot be solved by radicals. (I have never studied Galois Theory in fields of characteristic p, but I presume that this must be quite a problem.) Stephen
Reply by ●July 24, 20052005-07-24
David Kastrup wrote:> "kiki" <lunaliu3@yahoo.com> writes: > > >>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. > > > Ah, one thing that I remember: prove that the Eigenvalues of an > Antihermitian/Hermitian matrix are purely imaginary/real. > > The proof for that is just pretty. Something like > > A x = lambda x > > (for lambda an Eigenvalue, and x the corresponding Eigenvector) > > <Ax, x> = <lambda x, x> (scalar product) > > <x, A^{T*} x> = lambda <x,x> (factor gets transposed and conjugated > when switching sides) > <x, +/- A x> = lambda <x,x> (Hermitian/Antihermitian property) > <x, +/- lambda x> = lambda <x,x> (Eigenvector equation) > +/- lambda^* <x,x> = lambda <x,x> > +/- lambda^* = lambda (<x,x> is nonzero) > > Im/Re lambda = 0 >And then go onto show that the eigenvectors are orthogonal. I really like this trick when you apply it to differential operators. For example, showing that the terms of the Bessel series are orthogonal. This uses integration by parts in a nice way.
Reply by ●July 24, 20052005-07-24
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. --Lynn
