
The Power of Mathematics - John Conway [pdf] - niyazpk
http://thewe.net/math/conway.pdf
======
palish
This is great! And the best part is, this sort of "visual math" is applicable
to _every single field_ 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.

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...](http://content.screencast.com/users/shawnpresser/folders/Jing/media/43e220e1-2f69-49c4-a2a3-1a4801f0950d/2011-11-19_1213.png)
... 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...](http://dl.dropbox.com/u/315/books/Karatsuba%20Algorithm%20%28Fast%20Multiplication%3B%20History%29.pdf)
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_...](http://upload.wikimedia.org/wikipedia/commons/9/9f/Integral_example.svg))

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.

~~~
jberryman
Reminds me of how the ancient Greeks thought of mathematics in a purely
geometric way. The pythagoreans wouldn't buy a proof that you couldn't
demonstrate geometrically, with whole numbers. They couldn't conceive of
infinite or infinitesimal entities, so they explained them away, making
calculus one of those ideas that "the Greeks could have had but didn't".

~~~
palish
Mm, fair point. Although, "geometric thinking" wasn't a hindrance; check this
out, it's pretty interesting:

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...](http://dl.dropbox.com/u/315/random_pics/derivative_wheel_cos.png)

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.

~~~
kalid
Awesome example about the derivative! I think the best part of geometric
thinking is running ideas on your "native hardware".

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-...](http://betterexplained.com/articles/intuitive-
understanding-of-eulers-formula/)):

* 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.

------
kevinalexbrown
This is great. The only nitpicky thing I have is this line:

 _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.

~~~
hyperbovine
That's ironic because John Conway does a lot of applied math. I think he might
have just been explaining why the audience did not know the name. Mathematics
is still a relatively small field.

~~~
kevinalexbrown
Oh yeah, Game of Life and all that. Oops, thanks for pointing that out.

------
tzs
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).

------
abecedarius
I've only read as far as the first example, and wow, it's beautiful -- you
don't even need to read the proof, the diagram says it all.

~~~
Someone
If you read it really carefully, you will notice that it does not say it all.
The picture can only be drawn if m < 2n.

You would have to proof that separately. I think you will be able to draw the
picture that illustrates such a proof.

------
alf
Visual proofs are uniquely beautiful.

See the 2 & 3 proofs of Pythagorean Theorem: <http://www.cut-the-
knot.org/pythagoras/index.shtml>

------
jerfelix
The first example is pretty cool... Not to nitpick, but as he states the
problem:

    
    
        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?
    

Given that he's speaking of whole numbers, the number zero comes to mind,
which satisfies this.

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>

~~~
Arjuna
_"Given that he's speaking of whole numbers, the number zero comes to mind,
which satisfies this. So his whole proof is shot."_

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."

~~~
slowpoke
Where I study, it's a general consensus that in Math, N does not include 0
unless explicitly included as N index 0. In CompSci on the other hand, 0 is
assumed to be included for practical and technical reasons.

------
anrope
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.

------
38leinad
always a good idea to already insult your reader in the second sentence...

~~~
ejenkinsiii
Ha...I agree but I do appreciate this thread I currently had to review
geometry while studying C++ and most of what's explained on the document is
what I discovered while reviewing so it's nice to have it documented, now it's
time to get the logic down math is everything I'm discovering when it comes to
C.S./Programming, it defines how you build Data Structures/Algorithms well
apologies for the rant it's just good to see a thread which helps in an area
of my current study, and it's true using geometry makes programming and math
easier to comprehend

