Hi, one of the excellent math book availabe in net. http://www.physics.miami.edu/nearing/mathmethods/ Regards, Ajith timm@amsta.leeds.ac.uk wrote:> 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 lot > > You 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.
collection of mathematically elegant tricks
Started by ●July 24, 2005
Reply by ●July 25, 20052005-07-25
Reply by ●July 25, 20052005-07-25
Two tricks that you see fairly often are "adding zero" or "multiply by one." example: show why 0^0 is undefined... rewrite 0^(1-1) ** trick of adding zero you show 0^0 is equivalent to 0/0 ** my apologies to the group and anyone who teaches undergrad mathematics. simply writing 0/0 gives me a headache. Reading it, moreso. but suffice to say, there are a lot of proofs/shows that utilize one of these two. ajith_pc@yahoo.com wrote:> Hi, > > one of the excellent math book availabe in net. > http://www.physics.miami.edu/nearing/mathmethods/ > > Regards, > Ajith > > > timm@amsta.leeds.ac.uk wrote: > > 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 lot > > > > You 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.
Reply by ●July 25, 20052005-07-25
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...The proof of sin(a + b) = (sin a) * (cos b) + (sin b) * (cos a) according to E. Schmidt runs as follows: f(x) := sin(a + b - x)*cos(x) + cos(a + b - x)*sin(x) is constant on R (as well on C), because f' = 0. From f(0) = f(b) follows the desired theorem. Michael
Reply by ●July 25, 20052005-07-25
On 25 Jul 2005 09:44:37 -0700, mark_pfannenstiel@yahoo.com wrote:> Two tricks that you see fairly often are "adding zero" or "multiply by > one."> example: > show why 0^0 is undefined... > rewrite 0^(1-1) ** trick of adding zero > you show 0^0 is equivalent to 0/0Bad example, because 0^0 *is* defined. Rewriting 0^0 as 0^(1-1) doesn't demonstrate anything except that the trick doesn't work in this case. It doesn't show that 0^0 is undefined. A better trick is to write 0^0 = (1-1)^0 and expand using the binomial theorem, obtaining 0^0 = 1. See the sci.math FAQ for more about why 0^0 makes perfectly good sense. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. <http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>
Reply by ●July 25, 20052005-07-25
mark_pfannenstiel@yahoo.com writes:> Two tricks that you see fairly often are "adding zero" or "multiply by > one." > > example: > show why 0^0 is undefined... > rewrite 0^(1-1) ** trick of adding zero > you show 0^0 is equivalent to 0/0Nonsense. According to your reasoning: Show why 0^1 is undefined... rewrite 0^(2-1) You show 0^1 is equivalent to 0/0 -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
Reply by ●July 25, 20052005-07-25
On Mon, 25 Jul 2005 19:18:45 +0200, mh@michael-hoppe.de (Michael Hoppe) wrote:>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... > >The proof of > > sin(a + b) = (sin a) * (cos b) + (sin b) * (cos a) > >according to E. Schmidt runs as follows: > > f(x) := sin(a + b - x)*cos(x) + cos(a + b - x)*sin(x) > >is constant on R (as well on C), because f' = 0. From > > f(0) = f(b) > >follows the desired theorem. > >MichaelThe above proof looks great except for one thing -- the reasoning is circular, since the usual proof that the derivative of sin(x) is cos(x) makes use of the formula for sin(a+b). quasi
Reply by ●July 25, 20052005-07-25
"quasi" <quasi@null.set> wrote in message news:3fpae1tfchhmbmcajgl69p2j3hdfmuq4u2@4ax.com...> On Mon, 25 Jul 2005 19:18:45 +0200, mh@michael-hoppe.de (Michael > Hoppe) wrote: > >>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... >> >>The proof of >> >> sin(a + b) = (sin a) * (cos b) + (sin b) * (cos a) >> >>according to E. Schmidt runs as follows: >> >> f(x) := sin(a + b - x)*cos(x) + cos(a + b - x)*sin(x) >> >>is constant on R (as well on C), because f' = 0. From >> >> f(0) = f(b) >> >>follows the desired theorem. >> >>Michael > > The above proof looks great except for one thing -- the reasoning is > circular, since the usual proof that the derivative of sin(x) is > cos(x) makes use of the formula for sin(a+b). > > quasisuch a proof always seemed silly to me. how do you define sin and cos if not by the exponential function? if you define it that way, showing that the derivative of sin is cos is trivial.
Reply by ●July 25, 20052005-07-25
On Mon, 25 Jul 2005 19:33:24 GMT, "Justin Young" <x_static66@hotmail.com> wrote:> >"quasi" <quasi@null.set> wrote in message >news:3fpae1tfchhmbmcajgl69p2j3hdfmuq4u2@4ax.com... >> On Mon, 25 Jul 2005 19:18:45 +0200, mh@michael-hoppe.de (Michael >> Hoppe) wrote: >> >>>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... >>> >>>The proof of >>> >>> sin(a + b) = (sin a) * (cos b) + (sin b) * (cos a) >>> >>>according to E. Schmidt runs as follows: >>> >>> f(x) := sin(a + b - x)*cos(x) + cos(a + b - x)*sin(x) >>> >>>is constant on R (as well on C), because f' = 0. From >>> >>> f(0) = f(b) >>> >>>follows the desired theorem. >>> >>>Michael >> >> The above proof looks great except for one thing -- the reasoning is >> circular, since the usual proof that the derivative of sin(x) is >> cos(x) makes use of the formula for sin(a+b). >> >> quasi > >such a proof always seemed silly to me. >how do you define sin and cos if not by the exponential function? >if you define it that way, showing that the derivative of sin is cos is >trivial.Ok, yes, now I see what the proof intended, so I withdraw my objection. By mentioning that it's also true over C, that suggests that the author assumes a prerequisite of complex analysis, so the proof is redeemed, and it's definitely elegant. It certainly beats expanding the formal power series for sin(a+b) with a,b as unknowns and comparing it to the power series for sin(a)cos(b)+cos(a)sin(b). quasi
Reply by ●July 25, 20052005-07-25
"kiki" <lunaliu3@yahoo.com> writes:> [...] > 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...You might try LaTeX2HTML or TtH. The UK TeX FAQ, http://www.tex.ac.uk/cgi-bin/texfaq2html?label=whereFAQ has a couple more. -- % Randy Yates % "Maybe one day I'll feel her cold embrace, %% Fuquay-Varina, NC % and kiss her interface, %%% 919-577-9882 % til then, I'll leave her alone." %%%% <yates@ieee.org> % 'Yours Truly, 2095', *Time*, ELO http://home.earthlink.net/~yatescr
Reply by ●July 25, 20052005-07-25
well, I learned a new thing today. I had always believed that 0^0 was an undefined quantity due to the 2 statements 1. x^0 =1 2. 0^x = 0 x > 0. After an afternoon of considering such things as binomial theorem: (1 + 0)^1 really ought to be equal to 1 as well as some of the other arguements in the FAQ has convinced me 0^0 = 1.
