What is the deep idea in the Pythagorean theorem?
- Hi Pamela, Kiyavilo and all! I'm getting great response to a letter
that I wrote to the Math Future group. I share my second letter.
Perhaps we have thoughts? Andrius
I'm replying to the thread "Doubling the Area and Discovery" at
http://groups.google.com/group/mathfuture/, and yet starting a new
thread, which might be interesting to authors Eli Maor and Mark Levi,
and so I CC them.
Linda, going back to your original question about "fourths" of a
rectangle. I found it interesting, as you said, that if you cut up a
rectangle by its diagonals, then all of the fourths have the same area.
I think a straightforward proof is to cut up the rectangle into
"eighths". For example, you can cut up the rectangles into four smaller
rectangles, then slash those in half along the diagonals. Then you can
see that there are eight right triangles, all of the same size. They
make up the fourths.
Alexander, what a fantastic site you have! I appreciate your page of
proofs of the Pythagorean theorem:
I just finished looking through "The Pythagorean theorem: a 4,000-year
history" by Eli Maor, which I found quite inspiring.
I'll take the opportunity to explain the kind of book I'd like to write
in the Public Domain for adult self-learners of basic math. I'm trying
to explain teach math in terms of the deepest principles, and thus the
fewest principles, but also classic problems that illustrate them.
Here's a classic problem: Your usual sandwich at your Usual Cafe costs
$9. Across the street, at the Unusual Cafe, the same sandwich costs
one-third more. But today you have a coupon good at the Unusual Cafe
with which everything is one-third off. Where is it cheaper to buy the
I've had stock traders and professors tell me "the price is the same".
But actually, one-third more than $9 is $12, and one-third off of $12 is
$8. So today it's cheaper across the street. The lesson is a deep
idea: "Algebra is thinking step-by-step". You can't solve this problem
by doing it in one step. That's because the meaning of "one-third"
changes. This may be the simplest problem to illustrate this point.
And it's a real life problem that builds math intuition, not a contrived
problem that destroys it. I think that several dozen such "classic
problems" could illustrate all of the deep ideas in basic math.
Some more deep ideas: "Every answer is an amount and a unit".
(Otherwise, we don't know how to interpret it, we don't have the
context.) "If the units are the same, then addition means combine."
"If the units are different, then addition means list." In other words,
2 feet + 3 feet is 5 feet, but 2 feet + 3 inches isn't 5 anything. And
similarly, we can add 2 / 7 + 3 /7, or 2 % + 3 %, or 2 X + 3 X, or 2
million + 3 million (noting that a number can serve as a unit). We use
a single unit to calculate, but we use multiple units to make final
answers understandable, as in running a marathon in 2 hours + 9 minutes
+ 36 seconds. This is the spirit that I want to teach in. I wrote a
set of notes for precalculus where I was able to cover all of it in 10
weeks at a rapid clip by just focusing on the deep ideas:
I want to write a short, punchy paperback (something like Darrell Huff's
"How to Lie With Statistics" which is about 150 pages, or George Polya's
"How to Solve It") that covers counting, arithmetic, algebra, geometry,
precalculus. Here are my notes:
For example, I teach trigonometry from the deep idea "A Right Triangle
is Half of a Rectangle". I think that's much more easy and meaningful
to memorize than, say, the formula for the area of a triangle, which
follows directly from this deep idea, yet means nothing of itself. Now,
this deep idea lets us think of a shape in two ways (it's a map, a
morphism) so that (in the world of right triangles) the shape is given
by one angle, whereas (in the world of rectangles) the shape is given by
the ratio. So immediately we see that there is a one-to-one
correspondence between an angle and a ratio. And we see that no formula
is involved! And that we can calculate, that is, approximate the angle
and the ratio by constructions. So much learned, so quickly, with
nothing to memorize. (Except that "A Right Triangle is Half of a
Rectangle" is a very deep idea! which we're never taught.)
So I ask myself, what's the main point of the Pythagorean theorem a**2 +
b**2 = c**2. I have a Ph.D. in math, but I didn't learn how to prove
the Pythagorean theorem until a couple of years ago. At that time I saw
the picture of the Chinese proof (Alexander's proof #4) and I thought it
through as a principle "Four times a right triangle is the difference of
two squares", which, if you see the picture, makes sense to a small
child, and gives a straightforward algebraic derivation: 4 x 1/2 ab =
(a + b)**2 - c**2. The diagram also suggests the relationship between
two different coordinate systems.
Yet it didn't seem central enough of an idea.
So I found Eli Maor's book from the Chicago Public Library. It has a
wonderful variety of proofs and insights. What struck me? And what
The deepest idea that I found was "A right triangle is two copies of
itself". Actually, that's a consequence of another deep idea, which is
that "A triangle is two right triangles" which we see by drawing the
altitude, and which yields the formula for the area of a triangle.
So if your initial triangle happens to be a right triangle, then you
draw the altitude and see that it's two right triangles. Then note that
they all have a right angle. And each piece shares an acute angle with
the whole. But then, for each piece, the third angle must be the same
as the third angle of the whole. So the pieces and the whole must have
the same shape.
Then another deep idea is "Any area with a dimension of length X is
proportional to a square of area X**2".
Then we see, as in Eli Maor's own noteworthy "instantaneous" proof of
the Pythagorean theorem, that the whole right triangle's area equals the
two pieces' area, and so their areas A + B = C relate also as k * a**2 +
k * b**2 = k * c**2 for some proportion k. That means that a**2 + b**2
= c**2, but much more importantly, that k * a**2 + k * b**2 = k * c**2
and A + B = C for ANY three similar shapes with some dimension a, b, c!
At this point, I start to see, Why are we learning a**2 + b**2 = c**2 ?
Why are we fixating on that? It's just as meaningless as a*b/2.
Whereas the idea "A right triangle is two copies of itself" is a
beautiful, fractal, self similar idea with great implications. (And I
never knew this idea!) It's saying that a right triangle works as a
mechanism that can split anything into two copies of itself and yet
preserve the total area. That's the point of Euclid's classic theorem
(Alexander's proof #1) except that Euclid, following the Greek
aesthetic, wants to construct it for a square, and so the square gets
split along the altitude into two pieces, two rectangles, then moving
along (as halves, which is to say, two right triangles), preserving
their areas, because their widths don't change, until they are shown to
be halves of the desired squares.
"A right triangle is two copies of itself" is, I suppose, at the heart
of "Four times a right triangle is the difference of two squares"
because if you rotate a right triangle, then after four rotations you
get back to where you started.
But how to truly think about this? There's something going on here by
which, thanks to the self-similarity of the right triangle, two
dimensions become independent, and yet, I suppose, comparable. What is
it all about!
Here's another proof that's so simple:
which says, look at a tesellation, an infinite tiling of squares of
lengths a and b, and see on the diagonal the squares of length c they
form, and note that there is a one-to-one correspondence between the
squares of lengths a, b and c, and you can mark out a region as large as
you like, and it will be more and more clear that the a**2 and the b**2
must end up covering the same ground as the c**2.
Mark Levi has several proofs of the Pythagorean theorem in his book "The
Mathematical Mechanic" which I have yet to fathom. They are very simple
but require some physics insights. He's playing slick with physical
units, setting them equal to 1 as needed, as physicists do. And then
he's able to show, for example, that the Pythagorean theorem follows
immediately from the fact that a water-filled right triangular fish
tank, free to rotate, won't. Or that it follows from the areas swept
out by a, b, c if they are rotated around a circle. Or he proves it
with springs balancing forces.
I think you see how I'm thinking, how I'm trying to find and share the
gist, which I was never taught. We're given piles of textbooks with
thousands of pages when perhaps fifty problems could communicate all of
I hope to write such a book. I think that's one way I should try to
make a living. I think it would be honest work. I'm impressed that Tom
Henderson was so successful raising money at http://www.kickstarter.com
write a book "Punk Math". He certainly presents his math viscerally.
So I thought that "Gospel Math" would represent my aesthetic.
I appreciate thoughts, how do we like such an approach? and what do we
think is at the heart of the Pythagorean theorem? And also, how might I
try to make a living from creating such books or learning materials in
the Public Domain? There seems to be a real need and there must be a
way to earn at least a few months pay.
Wow! I really appreciate the great responses to my post from yesterday.
I hope to reply tomorrow.
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