This is great! And the best part is, this sort of "visual math" is applicable to <i>every single field</i> of mathematics; by visualizing every equation you come across, you'll find that you eventually gain a crisp (and often intuitive) understanding of the math.<p>For example: when I think of "x times y", I picture a rectangle whose sides are lengths x and y respectively; so naturally, the area of the rectangle is x times y.<p>Next time you come across an equation similar to "(x0 + x1)(y0 + y1)", you might try picturing it like this: <a href="http://content.screencast.com/users/shawnpresser/folders/Jing/media/43e220e1-2f69-49c4-a2a3-1a4801f0950d/2011-11-19_1213.png" rel="nofollow">http://content.screencast.com/users/shawnpresser/folders/Jin...</a> ... you'll discover all kinds of interesting things. E.g., Karatsuba noticed that (x0y1 + x1y0) can be computed as follows: "Find the area of the entire rectangle (x0 + x1)(y0 + y1); then subtract the area of the purple rect (x1y1); then subtract the area of the gray rect (x0y0); thus giving the answer." This was a major breakthrough in mathematics at the time, because it meant you could calculate the product of two arbitrarily large integers with N digits in less than N-squared time: <a href="http://en.wikipedia.org/wiki/Karatsuba_algorithm" rel="nofollow">http://en.wikipedia.org/wiki/Karatsuba_algorithm</a> (and <a href="http://dl.dropbox.com/u/315/books/Karatsuba%20Algorithm%20%28Fast%20Multiplication%3B%20History%29.pdf" rel="nofollow">http://dl.dropbox.com/u/315/books/Karatsuba%20Algorithm%20%2...</a> was his original paper).<p>Another random example: Have you ever played the game "pipe dream", where you have to connect the pipes together before the liquid fills them up and spills out? Well... the way I visualize "integrating a function" is: imagine the graph of the function. Now start "filling up the graph" from left to right --- just like Pipe Dream. The answer is: the total volume of the "liquid" above the zero line, minus the volume below the zero line. <a href="http://upload.wikimedia.org/wikipedia/commons/9/9f/Integral_example.svg" rel="nofollow">http://upload.wikimedia.org/wikipedia/commons/9/9f/Integral_...</a>)<p>Visualization tricks are a great way to get a "gut feeling" about "what does this equation actually <i>mean</i>? e.g., how might I relate the equation to some real-world (or imaginary-world) phenomenon?" ... you just have to be careful that your visualization is <i>exactly</i> equivalent to the mathematics, since an inaccurate visualization would throw off your intuition.
This is great. The only nitpicky thing I have is this line:<p><i>This new proof was created by a friend of mine called Stanley Tennenbaum, who has since dropped out of mathematics.</i><p>There's kind of a snobbery in mathematics that goes along the lines of Pure Mathematics > Applicable Math > Applied Math > Programming the Math. Theoretical Comp Sci falls somewhere around Applicable. Why is it that he "dropped out of mathematics" and not "decided he liked [whatever he does now]"?<p>Otherwise, a great read. Math <i>is</i> so much cooler than high school calculus drills make it seem.
There are two well known mathematicians named John Conway. A bit of Googling confirms that this paper is from John H Conway (the one well known in programming circles for the cellular automata game of "Life)), not John B Conway (the well known functional analyst and author of one of the leading textbooks on complex variables).
The first example is pretty cool...
Not to nitpick, but as he states the problem:<p><pre><code> Could there be two squares with side [sic] equal to a
whole number, n, whose total area is identical to that
of a single square with side equal to another whole
number, m?
</code></pre>
Given that he's speaking of whole numbers, the number zero comes to mind, which satisfies this.<p>So his whole proof is shot.<p>--<p>John Conway's greatest contribution to my life (as opposed to the game of life), and one I use about five times a week is The Doomsday Rule: <a href="http://en.wikipedia.org/wiki/Doomsday_rule" rel="nofollow">http://en.wikipedia.org/wiki/Doomsday_rule</a>
A bit off topic, but I was always a bit puzzled about how knots are described mathematically. This is a fantastic intro to knots in the mathematical sense.