Sage
Sage is a free open-source software application which covers many aspects of mathematics, including algebra, combinatorics, numerical mathematics and calculus. It combines dozens of software packages.
Note: compare Sage with other free computer algebra software!
A New Puzzle Challenges Math Skills
KenKen or KEN-KEN is a style of arithmetic and logical puzzle sharing several characteristics with sudoku. The name comes from Japanese and is translated as “square wisdom” or “cleverness squared”.
Related: list of logic puzzels
Free Online Math Textbooks
Online Mathematics Textbooks. Mathematics E-Books. If you know of others, please drop a note.
How to Learn Math and Physics
How to Become a Pure Mathematician (or Statistician)
How to Become a Good Theoretical Physicist
Wikipedia portals: mathematics – physics. Not very useful to learn math and physics, but still a nice starting point to find additional references.
And once you spend several years of dedicated study, you might want to try to solve these questions…
Any additional links would be greatly appreciated.
Amusements in Mathematics
Henry Ernest Dudeney (10 April 1857–24 April 1930) was an English author and mathematician who specialised in logic puzzles and mathematical games. In 1917 he published Amusements in Mathematics, 430 puzzles with solutions. Amusing indeed!
An example:
Two trains start at the same time, one from London to Liverpool, the other from Liverpool to London. If they arrive at their destinations one hour and four hours respectively after passing one another, how much faster is one train running than the other?
Solution: One train was running just twice as fast as the other (hint: when the trains pass eachother they are running for the same time).
Indra’s Pearls
Indra’s Pearls: The Vision of Felix Klein is a geometry book written by David Mumford, Caroline Series and David Wright, and published by Cambridge University Press in 2002.
The book explores the patterns created by iterating conformal maps of the complex plane called Möbius transformations, and their connections with symmetry and self-similarity. These patterns were glimpsed by German mathematician Felix Klein, but modern computer graphics allows them to be fully visualised and explored in detail.
See also Indra’s Pearls web site
Mathematicians Reveal Secrets Of The Ancient And Universal Art Of Symmetry
Humans have used symmetrical patterns for thousands of years in both functional and decorative ways. Now, a new book (The Symmetries of Things) by three mathematicians offers both math experts and enthusiasts a new way to understand symmetry and a fresh way to see the world. More at Science Daily.
Urbain Le Verrier
Urbain Jean Joseph Le Verrier (March 11, 1811 – September 23, 1877) was a French mathematician who specialized in celestial mechanics and is best known for his part in the discovery of Neptune.
Fermat’s Last Theorem
Fermat’s Last Theorem is the name of the statement in number theory that: if an integer 
The fact that the problem’s statement is understandable by schoolchildren makes it all the more frustrating, and it has probably generated more incorrect proofs than any other problem in the history of mathematics. No correct proof was found for 357 years, when a proof was finally published by Andrew Wiles in 1995.
Flatland
How would a creature limited to two dimensions be able to grasp the possibility of a third? Edwin A. Abbott’s droll and delightful ‘romance of many dimensions’ explores this conundrum in the experiences of his protagonist, A Square, whose linear world is invaded by an emissary Sphere bringing the gospel of the third dimension on the eve of the new millennium. Part geometry lesson, part social satire, this classic work of science fiction brilliantly succeeds in enlarging all readers’ imaginations beyond the limits of our ‘respective dimensional prejudices’. In a world where class is determined by how many sides you possess, and women are straight lines, the prospects for enlightenment are boundless, and Abbott’s hypotheses about a fourth and higher dimensions seem startlingly relevant today.
Edwin Abbott’s beloved mathematical adventure novel Flatland (1884) is now being introduced to a whole new generation of readers and viewers through Flatland: The Movie, a dramatic computer-animated adaptation starring Martin Sheen, Kristen Bell, Michael York, Tony Hale, and Joe Estevez. This book is the companion to the movie–and the ultimate edition of the classic book on which it is based. A beautiful, large-format volume, Flatland: The Movie Edition includes: the full text of the original novel; the screenplay of the movie; essays on the making of the movie by the writers and filmmakers–producer Seth Caplan, director Jeffrey Travis, and director and animator Dano Johnson; color illustrations; and a new introduction by Thomas Banchoff, a Brown University mathematician and Flatland authority who served as an advisor to the filmmakers.
The Inhabitants of SPACE IN GENERAL
And H. C. IN PARTICULAR
This Work is Dedicated
By a Humble Native of Flatland
In the Hope that
Even as he was Initiated into the Mysteries
Of THREE Dimensions
Having been previously conversant
With ONLY TWO
So the Citizens of that Celestial Region
May aspire yet higher and higher
To the Secrets of FOUR FIVE OR EVEN SIX Dimensions
Thereby contributing
To the Enlargement of THE IMAGINATION
And the possible Development
Of that most rare and excellent Gift of MODESTY
Among the Superior Races
Of SOLID HUMANITY
The Evolutionary and Developmental Foundations of Mathematics
Understanding the evolutionary precursors of human mathematical ability is a highly active area of research in psychology and biology with a rich and interesting history. Are some species born with this ability or is it an environmental issue ? It seems that a combination of both is explaining mathematical abilities.
Beran MJ (2008) The Evolutionary and Developmental Foundations of Mathematics. PLoS Biol 6(2): e19
Small Infinity, Big Infinity
Infinity is bigger than any number. But saying just how much bigger is not so simple. In fact, infinity comes in infinitely many different sizes—a fact discovered by Georg Cantor in the late 1800s.
Now a mathematician has come up with a new, different proof. Based on a simple game, the proof uses a strategy that might someday shed light on one of the great unsolved questions in mathematics. Continue…
Digits and Squares
Solution to yesterday’s puzzle. The top row must be one of the four following numbers: 192, 219, 273, 327. The first was the example given.
Puzzle
It will be seen in the diagram that we have so arranged the nine digits in a square that the number in the second row is twice that in the first row, and the number in the bottom row three times that in the top row. There are three other ways of arranging the digits so as to produce the same result. Can you find them? (Answer)
– Henry Ernest Dudeney, Amusements in Mathematics, 1917
Millennium Prize Problems
The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. Currently, six of the problems remain unsolved. A correct solution to each problem results in a $1,000,000 prize (sometimes called a Millennium Prize) being awarded by the institute. Only the Poincaré conjecture has been solved, and the solver has not pursued the conditions necessary to claim the prize.
The problems are:
- P = NP problem (
- The Hodge conjecture
- The Riemann hypothesis
- Yang-Mills existence and mass gap
- Navier-Stokes existence and smoothness
- The Birch and Swinnerton-Dyer conjecture
- The Poincaré conjecture (solved)
The links in the list are the wikipedia entries as these can give you at least an idea of what these problems are about. If you want to read the official problem description, please go to the CMI Millennium Problems, select one of the problems and read the pdf files.
There are easier ways to get rich…
Update: you can also have a go at the 23 (24) Hilbert problems (Mathworld – wikipedia).
Carnegie Mellon algorithm identifies top 100 blogs for news
Using a problem-solving method called the Cascades algorithm, Carlos Guestrin, assistant professor of computer science and machine learning, and his students compiled a list of the best 100 blogs to read to find the biggest news on the Web as early as possible, blogcascades. It includes well-known blogs, such as Instapundit and Boing Boing, but also some more obscure ones like Watcher of Weasels and Don Surber. The original paper, a ppt presentation and a video presentation are available at blogcascades, but you’ll need to be familiar with optimization problems and complexity classes. Remember Pareto ? (via)
RK sez: I love it when people make a decent analysis.
Euler’s Identity
Euler’s identity is the equation:
where:
Euler’s identity is remarkable for its mathematical beauty. Three basic arithmetic functions are present exactly once: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants:
- The number 0
- The number 1
- The number
- The number
- The number
Impressed ? Well, let’s take now the natural log of both sides of the equation (after subtracting one from both sides). This actually allows us to define the natural log of the negative numbers in the complex plane as follows:
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