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PROFESSOR: In 6.042, we're
going to be pretty concerned

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with proofs.

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We're going to try
to help you learn

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how to do rudimentary proofs
and not be afraid of them.

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The most important
skill, in some ways,

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is the ability to distinguish
a very plausible argument that

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might not be totally right from
a proof which is totally right.

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That's an important skill.

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And it's a basic
understanding of what math is.

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It's that distinction
between knowing

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when a thing is
mathematically, absolutely

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unarguable and inevitable
as opposed to something

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that's just very likely.

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It's interesting.

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Physicists by and
large do a lot of math,

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and they tend not to worry
so much about proofs.

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But all the theoreticians
and the mathematicians

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are in agreement
that you don't really

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understand the
subject until you know

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how to prove the basic facts.

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Pragmatically, the
value of proofs

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is that there's an awful lot
of content in this subject

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and in many other
mathematical subjects.

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And if your only way to figure
out what the exact details are

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is memorization, you're
going to get lost.

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Most of these rules and
theorems that we prove,

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I can never remember
them exactly.

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But I know how to prove them,
so I can debug them and get them

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exactly right.

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So let's begin by looking
at just examples of proofs

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before we start to try to get
abstract about what they are.

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And we'll look at
a famous theorem

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that you've all seen from
early on in high school,

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the Pythagorean theorem.

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It says that if I have a right
triangle with sides a and b

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and hypotenuse c, then there's
a relationship between a, b

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and c-- namely that a
squared plus b squared

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equals c squared.

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Now, this is, as I said,
completely familiar.

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But is it obvious?

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Well, every once in a while,
students say it's obvious,

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but what I think they really
mean is that it's familiar.

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It's not obvious.

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Part of the argument for the
fact that it's not obvious

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is that for thousands
of years, people

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have kept feeling
the need to prove it

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in order to be sure that it's
true and explain why it's true.

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There's a citation in the
notes of a website devoted

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to collecting Pythagorean
theorem proofs.

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There's over a hundred
of them, including

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one by a former president
of the United States.

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So let's look at one
of my favorite proofs

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of the Pythagorean theorem.

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And it goes this way.

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There are four
triangles that are

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all the same size, four
copies of this abc triangle,

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which we've put in different
colors to distinguish them,

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and a square, which for the
moment, is of unknown size.

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And the proof of the
Pythagorean theorem

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is going to consist of
taking these four shapes

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and reassembling them so that
they form a c by c square

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first, and then finding
a second arrangement

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so that they form two squares--
an a by a square and a b

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by b square.

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Then by the theorem
of conservation

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of paper or
conservation of area,

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it has to be that
the c by c area is

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the same as the a by
a plus b by b area.

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And so a squared plus b
squared is equal to c squared.

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Well, let's look at
those rearrangements.

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And probably, you
should take a moment

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to try it yourself before
I pop the solution up.

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But there's the solution
to the first one.

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It's the easier of the two.

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This is the c by c arrangement.

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The hint is that if
it's going to be c by c,

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you don't have a
lot of choice except

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to put the c [? long ?]
hypotenuses on the outside.

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And then it's a matter of
just fiddling the triangles

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around so they fit together.

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And you discover there's
a square in the middle.

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And that's just where that
extra square that is provided

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will fit.

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Also, this enables
you to figure out

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what the dimensions
of the square are.

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Because if you look at
it, this is a b side.

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We're letting b be the longer of
the two sides of the triangle.

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And this is the a
side, the shorter side

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of another triangle.

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So what's left here
has to be b minus a.

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So now we know that
it's a b minus a

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by b minus a square
from this arrangement.

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And that's what
we've indicated here.

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Now, the next arrangement
is the following.

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We're going to take
two of the triangles

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and form a rectangle,
another two triangles

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to from a rectangle,
line them up in this way,

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and fit the b minus a by
b minus a square there.

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Now, where are the two squares?

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Well, I didn't say that
the a by a and a and b by b

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square needed to be separate.

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In fact, they're not.

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They're attached.

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But where are they?

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Well, let's look at this line.

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How long is it?

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Well, it's a plus
b minus a long,

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which means that it's b long.

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And suddenly, there is
a b and there's a b,

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and I've got a b by b
rectangle right there.

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But wait a second.

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Here's a b minus a, and it's
lined up against a b side.

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So if I look at what's left,
it's b minus b minus a.

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It tells me that that
little piece is a.

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And so sure enough, when
I add this hidden line--

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conceptual line to
separate the two squares,

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this part's a by a,
and that part's b by b.

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And we've proved the
Pythagorean theorem.

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So what about this process?

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It's really very elegant,
and it's absolutely correct.

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And I hope it's
kind of convincing.

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And so this is a wonderful
case of a proof by picture

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that really works in this case.

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But unfortunately, proofs by
pictures worry mathematicians,

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and they're illegitimately
worrisome because there's

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lots of hidden assumptions.

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An exercise that
you can go through

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is to go back and
think about all

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of the geometric
information that's

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kind of being taken for
granted in this picture.

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Like over here, how
did we know that that

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was a right angle, that
this thing was a rectangle?

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We needed that to be a right
angle because we were claiming

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that this was a square.

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Well, how did we know
that that was a rectangle?

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Well, the answers are obvious.

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We're using the fact that
the complementary angles

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of a right triangle
sum to 90 degrees

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because the angles of a triangle
in general sum to 180 degrees.

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We're using that in a
bunch of other places.

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We're also using the fact that
this is a straight line, which

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may or may not be obvious.

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But it's true, and
that's why it's

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safe to add those distances
to figure out what it was.

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My point is that
there are really

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a whole lot of hidden
assumptions in the diagram

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that it's easy to
overlook and be fooled by.

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So let me show you an example
of getting fooled by a proof

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by diagram.

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And here is how to
get infinitely rich.

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Let's imagine that I have a
10 by 11 piece of gold foil.

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Actually, they could
be slabs of gold,

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but let's think of this as
a rectangular shape that's

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made out of gold.

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And it's going to be rectangles.

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Those are right angles there.

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And what I'm going to do
is mark off the corners.

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I'm going to mark off
a length down of 1,

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and then I'm going to
mark off a length of 1

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and shift it so that it
touches the diagonal,

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and do the same thing
in this lower corner.

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And now, let's just shift
these shapes, the top one going

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southwest and the second
one going northeast.

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And what I wind up
with is this picture so

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that I've now got those little
red triangles protruding

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above the shape.

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OK, cool.

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Well, what do we know?

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This is now side 10 because I've
subtracted 1 from its length

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here .

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And this is side 11
because that used to be 10,

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and I've added 1 to
its length there.

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So that's cool because
now, what I can do

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is take those protruding
triangles out,

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and they'll form a
little 1 by 1 square.

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And suddenly, I have this
little bit of gold that's extra.

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But look what's here.

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It's a 10 by 11 rectangular
shape of gold foil again.

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So I just rotate
this by 90 degrees,

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and I start all over again.

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I can keep generating
these little 1

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by 1 shapes of
gold foil forever.

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I could get infinitely rich.

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OK, well there's
something wrong with that.

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It's violating all kinds
of conservation principles,

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not to mention that it would
undermine the price of gold.

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So what's the bug?

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Well, you probably
can spot this,

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but maybe you've been fooled.

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What's going on is there's
an implicit assumption

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that those little triangles that
I cut off were right triangles

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and that this line that I
claimed was of length 11

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was a straight
line, and it's not.

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Those triangles have two
sides that are of length 1.

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They're isosceles
triangles, but they're

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lying up against a diagonal
that's not 45 degrees.

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And so they're not
right triangles.

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And that line isn't straight.

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And 10 and 11 were
close enough that it

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wasn't visually obvious.

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So this is a way to
simply put one over

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on you with a proof by picture.

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And if I had been
asked to justify

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how do I know it's
a straight line,

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that bug would have emerged.

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But you're not likely to notice
that if it isn't visually

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obvious, which is why we worry
about that some of these proofs

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[? by picture. ?]