The Power of Mathematics - John Conway [pdf]
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thewe.net
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.
Next time you come across an equation similar to "(x0 + x1)(y0 + y1)", you might try picturing it like this: http://content.screencast.com/users/shawnpresser/folders/Jin... ... 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: http://en.wikipedia.org/wiki/Karatsuba_algorithm (and http://dl.dropbox.com/u/315/books/Karatsuba%20Algorithm%20%2... was his original paper).
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. http://upload.wikimedia.org/wikipedia/commons/9/9f/Integral_...)
Visualization tricks are a great way to get a "gut feeling" about "what does this equation actually mean? 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 exactly equivalent to the mathematics, since an inaccurate visualization would throw off your intuition.
For example we can visualize "taking the derivative of the function f(x)" by "rolling a wheel over the top of its graph": http://dl.dropbox.com/u/315/random_pics/derivative_wheel.png
The red circle is a "wheel" which happily rolls back and forth along our f(x) graph. The green line is the derivative of f(x). It's intuitively obvious what that green line is -- if you had to fit a flat piece of paper between the wheel and where it touches the graph, then you'd get that line. That's the derivative, also called the "tangent line" or "slope of the graph" at that point.
(The wheel will always touch the graph only once, because we can visualize it as small as we need it to be --- infinitely small, even.)
It's common knowledge that a curve's derivative is zero at its maximum / minimum. (In other words, "when the wheel reaches the top or bottom of a 'hill', then its derivative is zero, aka the tangent line is perfectly horizontal".) So right away, we immediately understand where the zeroes of the f'(x) graph must be located.
But the real power is, we also have a rough idea of how the f'(x) graph must look, by "watching the wheel's tangent line as we roll it along the graph", in our mind's eye.
So that's a neat trick, but why does it matter?
Well, here's one example: It's the most natural and obvious thing in the world that the derivative of cos(x) is -sin(x).
http://dl.dropbox.com/u/315/random_pics/derivative_wheel_cos...
We start at x=0; we see that the wheel is "resting on the very top of the slope", meaning the tangent line is flat, so we know right off the bat that the derivative of cos(0) is 0.
From there, the wheel "rolls downhill"; therefore the derivative "goes negative"; and it goes back to zero when the wheel reaches the bottom of the hill (which we see is at x=pi).
The wheel then begins climbing uphill, so its derivative "goes positive"; and it goes back to zero when the wheel reaches the top of the slope (at x=2pi).
So we see that the graph of the derivative of cos(x) has zeroes at 0, pi, and 2pi; and we also intuitively understand that it's negative from 0 to pi; and we see that it's positive from pi to 2pi.
Now all that's left is to think about the graph of -sin(x). It's exactly what we just described: zero at x=0, x=pi, and x=2pi; negative from 0 to pi; and positive from pi to 2pi. So that's why "it's obvious to us that the derivative of cos(x) is -sin(x)", QED.
So if you've had to worry about rote memorizing those sorts of formulas, I'd urge you to give these visual techniques a shot. You'll never again need to memorize seemingly-arbitrary equations, which is pretty sweet IMO.
Sorry for rambling on; I just get so excited about presenting math "in a visual way". It's a lot of fun, and likely wouldn't hinder the ambitious mathematician at all --- just the opposite, in fact: Feynman, for example, had an arsenal of similar techniques, which probably contributed to his intuitive understanding of the physics behind the formulas and his ability to "see past" the raw equations.
Our brains are massively parallel, visual-processing monsters... and we try to think about ideas linearly, like we're a turing machine. No -- use that GPU! :).
When I think visually, I can immediately grok the idea (it's running in hardware) and my software layer (my conscious thought process) can work on stitching together deeper ideas.
Shameless plug, but I've written about visualizing Euler's Identity (http://betterexplained.com/articles/intuitive-understanding-...):
* Radians are distance from the mover's point of view
* e is continuous growth
* i is a rotation (so e^i*anything is imaginary growth, i.e. growth in a perpendicular direction)
Combining ideas like this is just so enjoyable! It brings out the natural beauty in learning.
Not every math field uses equations, and most of those that do, use equation, where terms are very far from being ordinary (integer, real, complex) numbers, and picturing a rectangle when you have xy will only mislead you. For instance, sometimes xy is not equal to yx, sometimes x and y are not 0, but xy is 0.
Of course, there are other ways to employ visual intuition to gain better understanding of the problem -- for instance, if our multiplication comes from group structure, we can imagine an object the symmetries of which are group elements. Nevertheless, the history of mathematics teaches us that the greatest thing started to happen when we ditched our geometric and intuitive understanding of math and turned to more formal ways. I'm certainly happy to see a geometric interpretation of some definition, theorem, or proof, but I'm more interested in logical argumentation than appealing to intuition.
This new proof was created by a friend of mine called Stanley Tennenbaum, who has since dropped out of mathematics.
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]"?
Otherwise, a great read. Math is so much cooler than high school calculus drills make it seem.
I recently attended a workshop devoted entirely to Tennenbaum's Theorem, his famous result that the only recursive model of PA is the standard model.
http://www.bbk.ac.uk/philosophy/our-research/ppp/tennenbaump...
Stanley's up in heaven now.
You would have to proof that separately. I think you will be able to draw the picture that illustrates such a proof.
See the 2 & 3 proofs of Pythagorean Theorem: http://www.cut-the-knot.org/pythagoras/index.shtml
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?
So his whole proof is shot.
--
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: http://en.wikipedia.org/wiki/Doomsday_rule
Hint: it doesn't exist.
Second hint: it is trivial to demonstrate that it doesn't exist. Consider the triangle. Can you define it as a quadrilateral with one side of length zero? What is the angle at the null vertex? Proceed from there.
Finally, the problem states "another whole number". Zero is not different from zero. As with any exam, you should read the question carefully first.
A square with side n, where n=0 would be a point. Except it wouldn't, because we have to define these things differently, or geometry would be incoherent.
Incidentally, there is no need for a [sic] there because you only need one side to define a square.
Both Wikipedia [1] and Wolfram [2] indicate that there are varying interpretations regarding which integers are included in the definition of the term whole number.
Since Conway is discussing two-dimensional distance in order to determine area, his definition of the term would not include integers that are less than or equal to zero.
[1] http://en.wikipedia.org/wiki/Whole_number
"Whole number is a term with inconsistent definitions by different authors. All distinguish whole numbers from fractions and numbers with fractional parts.
Whole numbers may refer to:
natural numbers in sense (1, 2, 3, ...) - the positive integers
natural numbers in sense (0, 1, 2, 3, ...) - the non-negative integers
all integers (..., -3, -2, -1, 0, 1, 2, 3, ...)"
[2] http://mathworld.wolfram.com/WholeNumber.html
"0 is sometimes included in the list of "whole" numbers (Bourbaki 1968, Halmos 1974), but there seems to be no general agreement."
Not really. Just add the special case of zero. If you want to sound cool say "There are no two nontrivial squares with side ..."