What are some nifty and clever discoveries in mathematics?

[Mathgician]

I was reading the general discussion section and noticed that someone came with a thread about cleaver ideas. I was thinking that there may be some clever ideas, tricks, or insights that many of you experienced mathematicians may have come across. If you have anything you think it is of worth I am very interested in it. It doesn't have to be world changing discoveries, it could be anything from something minor to something major.

 

[berkeman]

Here's an earlier (short) thread about this. But it could use some expanding...

Check out the book that I mention at the end of the old thread -- it's got some great stuff in it that I use every day.

https://www.physicsforums.com/showthread.php?t=129568

 

[Mathgician]

nice... thanks, but the cover of the book don't make it look like a very respectable book. I ordered a used one through amazon for nearly the shipping price.

 

[HallsofIvy]

My greatest insight was "Gosh, I'm spending way too much time on this!"

 

[quasar987]

Check out this method of doing multiplication.



With love, Feynman.

 

[Gib Z]

That methods pretty good for 2 digit numbers but I tried 252 x 482. It wasn't fun at all, and I found it somewhat problematic when it had to carry over the tens digit, I got confused. Either way, Go Feynman! Woot!

As for discoveries, when it comes to minor things I happened to notice and prove the sum of the geometric series \sum_{n=0}^{\infty} \frac{1}{x^n} when I was 9 years old fooling around on a calculator. That was before I even knew trig so I thought i was a real genius "discovering" that, stupid me. It was a real sad shock when I studied series from a textbook, let me tell you that :D

 

[Gib Z]

This suits more the thread berkeman posted, but http://en.wikipedia.org/wiki/Mental_arithmetic reminded me of some stuff.

Theres the simple thing that when ever I multiply things that aren't too many digits and the ending digit is somewhat small, its always good to do this:

Lettings alphabetic characters represent digits - (abc)(def) = (ab0 + c)(de0 + f) then a binomial expansion. Helps more than you think.

And Newtons method works well on a lot of things, but i use it mostly for square roots. I use taylor series to the first 3 terms when doing sins, cosines and exponentials (base e). For tans, i don't actually use the taylor series for tan, but divide the sin and cos, I hate tans taylor series.

 

[mathwonk]

read "what is mathematics".

 

[Mathgician]

Check out this method of doing multiplication.



With love, Feynman.

Wow Amazing. Was it Feynman that popularized it? I'm very curious what process let that person to that discovery?

 

[quasar987]

Well, the little description of the video reads...

"An American physicist Richard Feynman developed this technique to graphically solve quantum field theory equations (actually to get approximations) that could not be solved otherwise. He got Nobel Price in physics for those "Feynman's diagrams"

 

[Mathgician]

read "what is mathematics".

Lol good one Wonk. You are quite the Zen master.

 

[Glass]

I wish there was a book of cool and clever tricks for things pivotal to other areas of math (such as solving integral tricks and algebra ones too). I always find such tricks to be quite elegant and mesmerizing. Does anyone know of such a book?

 

[Werg22]

read "what is mathematics".

Are you talking about Courant's book?

 

[Chris Hillman]

Some suggested reading for those seeking nifty arguments

I don't know why no-one has mentioned some obvious citations:

The enjoyment of mathematics; selections from mathematics for the amateur, by Hans Rademacher and Otto Toeplitz, Princeton University Press, 1957

Mathematics and logic; retrospect and prospects, by Mark Kac and Stanislaw M. Ulam, New York, Praeger, 1968 (great semipopular book despite the title)

Continued fractions, by A. Ya. Khinchin, Dover reprint, 1997.

A whole series of books called Mathematical Pearls, which I can't seem to find in our catalog,

Proofs from the book, by Martin Aigner, Günter M. Ziegler, Springer, 2004.

Modern graph theory, by Béla Bollobás, Springer, 1998.

Indra's pearls: the vision of Felix Klein, by David Mumford, Caroline Series and David Wright, Cambridge University Press, 2002.

Visual complex analysis, by Tristan Needham, Oxford University Press, 1997.

Complex analysis: the geometric viewpoint, by Steven G. Krantz, Mathematical Association of America, 2004

Fourier analysis, by T.W. Körner, Cambridge University Press, 1988.

In case you hadn't guessed, these are all really, really fun books which are full of nifty but often rather elementary mathematical reasoning. I could have added many more.

There are zillions of expository papers in the Am. Math. Monthly published over the past century which should also yield a cheap thrill or two, as close as your nearest math library, or even closer for some of you http://www.jstor.org/journals/00029890.html