When I worked at the University of Hertfordshire I had access to computers that ran Mathematica but I think I only made use of it once, to invert a 4x4 matrix. I am not up to date with these things and I don't know how Mathematica compares to the competition but if I needed to do a lot of math I would be looking into it. The thing that prompted me to write this entry is something I read on the Wolfram blog about The Incredible Convenience of Mathematica Image Processing - very cool indeed.
The Bridges Organisation deals with mathematical connections in Art, Music and Science.
For a formal introduction to number theory you might like to try A Primer for Logic and Proof by Holly P. Hirst and Jeffry L. Hirst.
An international team of mathematicians has succeeded in mapping the 248-dimensional Lie group called E8.
In a 1999 article in the Times Higher Educational Supplement Going to war over prime numbers, Duncan Campbell explained how revelations from the secret world of spying raise academic questions for both history and mathematics.
Paul Bourke's website has an excellent section on fractals.
I always remember that pi is approximately 22/7 but have trouble remembering the far closer approximation 355/113.
I once came up with this useful idea (~ means approximately equal to):
If sqrt(n) ~ a/b then sqrt(n) ~ nb/a
The geometric mean of these two approximations is exactly sqrt(n)
The arithmetic mean must be a better approximation, hence:
sqrt(n) ~ (a^2 + nb^2)/2ab
Take for example n=2, a=1, b=1 then we get successive approximations:
sqrt(2) ~ 1, 3/2, 17/12, 577/408
Squaring the last of these gives 2 + 1/332928 - not bad eh!
In order to celebrate mathematics in the new millennium, The Clay Mathematics Institute of Cambridge, Massachusetts (CMI) named seven Prize Problems with a reward of $1 million per problem for solving them. Check out the seven Millennium Problems.
What's the pattern in this sequence:
infinity, five, six, three, three, three, three,...
It is all made clear in a 1998 paper Platonic Solids in All Dimensions by John Baez.
I am not a mathematician as such, but while doing postgraduate research in space plasma physics in the late 80's I had to get my head around some quite hairy vector field theory. Now I have forgotten more mathematics than most people will ever know and am mainly interested in ways of representing mathematical knowledge on the web. There is no point trying to give any kind of overall view of mathematics because that seems to have been comprehensively taken care of in the wikipedia entry but I will make a few notes of my own.
If you wanted to produce a document (book, paper, webpage) containing significant mathematical content (equations, formulae, proofs) what sort of system would you use? As I understand it there are really only two serious contenders at the moment, LaTeX which is the defacto standard for typesetting mathematics and MathML which is generally considered the way of the future. I found this useful article on the subject by G. Donald Allen (no date but Apache says last modified 2003-06-25).
Sometime in 2001 I noticed that the excellent MathWorld had gone offline as a result of legal action by robber barons at CRC Press. The site is back, but since it has basically been owned by CRC lawyers (see Eric Weisstein's commentary) I would recommend against using it. Fortunately there are alternatives such as PlanetMath which is licensed under the GNU FDL.
Here are a few cool sites:
www.zenatode.org.uk Ian Gregory 2010