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PROFESSOR: There's other kinds
of bogus proofs that come up.

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Let's just run through
this one quickly.

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Here's a fact that you
about, roots of polynomials.

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Every polynomial has
two roots, at least

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over the complex
numbers, over c.

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And how do you prove that?

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Well, you just write down
the formulas for the roots.

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You know the quadratic formula.

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One root is a minus b
plus this square root

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of this quantity of 2a.

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And the other root is minus
b minus this square root

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of b squared minus 4ac over 2a.

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And that's the
end for the proof.

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You can just plug-in
this formula for r1

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for x into this
polynomial and it

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would simplify to
be equal to 0, which

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shows that this is a root.

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You could plug that one
into this formula for x

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and simplify algebraically
and discover it was 0,

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proving that r2 is a root.

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We've just proved that every
polynomial has two roots.

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Well, that's not true.

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We haven't proved it.

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This is a proof by
calculation that has problems.

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What's the problem? well let's
look at a counter example.

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What about the polynomial
0x squared plus 0x plus 1?

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It doesn't have any roots.

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It's just a constant 1 which
never crosses the origin.

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So it's got no roots.

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What about 0x squared
plus 1x plus 1?

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Well that's 45 degree
line, the y equals x line,

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and it only crosses
the origin once.

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It has only one root.

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What happened to the
two formulas, r1 and r2?

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And the answer
was, in this case,

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we had to divide by 0 error.

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If you look at that
formula, there's a quotient,

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there's a denominator of 2a.

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Divide by 0 and these
formulas don't work right.

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They aren't the roots.

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And so implicitly, in
order to have two roots,

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we need to assume that the
denominator, a, the leading

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coefficient of the
polynomial is not 0.

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So that fixes those two bugs.

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Is that all?

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Well, no, because look
at this polynomial.

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1x squared plus 0x
plus 0 has one root.

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The only possible
root of this is 0.

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Because if you look
at this, the only way

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to get 1 times something
plus 0 to equals 0

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is if the something is 0.

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So there's only one root.

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And what's going on here?

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Well, what's happened is that
in this case, the two formulas,

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r1 and r2, which were
different formulas, evaluate

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to the same thing when b
is 0 and c is 0 and a is 1.

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And that's why they look
like different formulas

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but they evaluate to the same
thing so there's only one root.

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The fix to that is to
require the quantity

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by which the two
root formulas, r1

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and r2 differ to be non-zero.

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And that's the quantity that we
were taking the square root of,

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the discriminant it's called.
b squared minus 4ac needs

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to be 0 and then r1
and r2 will differ

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and we will get the two roots.

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Now, there's still
a complication.

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It sounds like
we've now verified

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that indeed our
proof by calculation

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is correct now that we've
put in these qualifiers,

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that a is positive
and d is non-zero.

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But what happens if d is
non-zero but negative?

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It's an interesting
complication.

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And let's look at the
formula, x squared plus 1,

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where b squared minus 4ac
is going to be minus 3.

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And that will turn out
to have two roots, namely

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i and minus i.

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And it's not possible
to tell which

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is which. r1 is taking the
square root of minus 1,

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and r2 is taking the
square root of minus 1.

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One of them is adding the
square root of minus 1.

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The other one's subtracting
the square root of minus 1.

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But which square
root of minus 1?

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There's no way to tell
the difference between i

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and minus i, abstractly.

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They both behave the same way.

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And so we have an ambiguity
about which one is r1

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and which one is r2.

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It doesn't hurt at
all for the theorem

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that r1 and r2 are different.

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And so there are two roots.

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But ambiguity can
be problematic.

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And let me give you an
illustration of that.

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When there's ambiguity,
I can do things

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like proving easily that
1 is equal to minus 1.

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Here's the proof.

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And I will let you contemplate
that and try to figure out just

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where in this reasoning
that step by step

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seems pretty reasonable,
but nevertheless I've

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concluded that 1 is equal
to minus 1, which is absurd.

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It's taking
advantage of the fact

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that you don't know whether
the square root of minus 1

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means i or minus i.

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So the moral of all of
this is that, first of all,

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be sure that you are
applying the rules properly.

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There's an assumption of
an algebraic rule in there

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that isn't true.

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And again, that kind
of mindless calculation

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is risky when you don't really
understand what you're doing,

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you don't have a clear memory
of what the exact rules are.

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So it's understanding
that bails you out

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of this kind of blunder.

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Let's look at 1 equals
minus 1 a little

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because it lets us wrap
up with an amusing remark.

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What's terrible about
1 equals minus 1?

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Well, it's false, and you
don't want to ever conclude

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something that's false.

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That's worrisome.

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It's disastrous
when you conclude

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that something is false.

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Let me give you an
illustration of why.

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Because let's suppose the 1
is equal to minus 1 and let's

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reason in a correct
from that assumption

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that we have falsely proved.

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But let's assume that we
start off with the assumption

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that 1 is minus 1.

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Well, if I multiply both
sides of an equation

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by the same thing, it's equal.

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So I can multiply
both sides by 1/2,

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and I get 1/2 is
equal to minus 1/2.

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Now I can also add the
same thing to both sides.

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That's a perfectly sound move
for reasoning about equalities.

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If I add 3/2 to both sides, I've
turn 1 equals minus 1 into 2

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is equal to 1.

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Now I'm in great shape to
prove all kinds of things.

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Here's a famous one.

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"Since I am the
Pope are clearly 2,

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we conclude that I
and the Pope are one.

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That is, I am the Pope."

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And I've just proved to
you this absurd fact.

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This is a joke
that's attributed--

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a witty remark that's
attributed to Bertrand

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Russell, the famous philosopher,
logician, pacifist, Nobel Prize

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winner, who supposedly was
approached by some socialite

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at a party who had heard
that mathematicians thought

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that if 1 is equal to minus 1
then you could prove anything.

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And so she challenged him,
prove that you're the Pope.

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And supposedly Russell, who
was a notoriously quick wit,

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came up with this example.

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Who knows whether it's true.

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It's a good story.

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There's a picture
of the great man.

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And you might care
to learn more about

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this remarkable contributor
to logic, and philosophy,

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and politics.