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Even nonsense can be meaningful.

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[SOUND]

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Mathematics is more than just things free
of inconsistency.

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It's more than just that which is the
case.

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Sometime we're confronted with things,
that are really not

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nonsensical or things that are just flat
out wrong.

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And when we're confronted with things like
that, we should

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really have a feeling or a need to salvage
the situation.

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We should take the nonsensical or wrong
thing and try to salvage it.

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Try to think

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of some sense in which it might make
sense.

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So what's the sum n goes from 0 to
infinity of 9 times 10 to the nth power?

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The party-pooper simply says, the series
diverges.

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And the party-pooper is right.

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This series diverges.
And why does it diverge?

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Well, it's a geometric series with

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R equals 10 and that counter ratio is

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bigger than 1.
But let's try to salvage the nonsense.

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Well, remember back how we looked at the
sum N goes from 1 to infinity

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of 9 times 10 to the negative nth power
and this was, well, it's

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equal to 1, but we could also write it as
0.9999 where the

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9s keep on going that way.

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Well, in this situation, contemplating is
really the opposite.

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I'm thinking of the sum N goes from 0 to
infinity, of 9 times 10 to the N.

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What is that?

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When I plug it in N equals 0, I get nine
time one which is 9.

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When I plug it in n equals 1, I get 9
times 10 which is 9.

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When I plug in N equals 2, I get 9 times
100 which is 900.

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And I'd keep on going.
So one way to write this might be, well at

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first 9, 9 plus 90, that's 99, 99 plus 900
that's 999.

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It's as if I could write it with 9's going
this way.

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And what if we add 1 to that.

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Well, I mean, it's not really a number,
right?

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But I can still write it down.

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So I'll write down

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[LAUGH]

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a number that's just 9s going this way.
And then I'm going to add 1 to that number

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[LAUGH].

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What do I get?
Well, 9 plus 1 is 10, carry the 1.

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9 plus 1 is 10, carry the 1.
9 plus 1 is 10, carry the 1.

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9 plus 1 is 10, carry the 1.
9 plus 1 is 10, carry the 1.

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And I'm going to keep on doing that,
right?

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So it looks like this thing, which is 9's
all the way that way, plus

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1, is 0.
So whatever this thing is, its a thing

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that if i add 1 to it, I get 0.
So what is 999 with dots all that way,

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its sort of equal and negative 1.

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because negative 1 is a thing I can add 1
to to

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get 0 and when I added 1 to this thing, I
got 0.

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Sort of, makes sense.

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So if I just ignore convergence
altogether, what would it tell me to do?

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I'm trying to evaluate the sum and goes
from 0 to infinity of 9 times 10 to the N.

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Well that would be 9 times the sum N

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goes from 0 to infinity of 10 to the N.

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This is a, is a divergent geometric
series.

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Let's just pretend the formula still
worked.

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How I calculated the value of that series.

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That'd be 9 times and the formula for this
is

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1 over 1 minus the common ratio, which is
10.

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Of course, that formula's only valid if
it, if it

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were convergent series and it's not, but
let's just pretend.

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What is this?
Is this 9 times 1 over 1 minus 10?

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That's 9 times 1 over minus 9.
Well, 9 times 1 over minus 9 is minus 1.

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Which is sort of what we're seeing here,
right?

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I mean, if we're just ignoring
convergence, it looks

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like it's telling us that the value of
this series.

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I mean, it doesn't have a value because it
diverges, but

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if the value of the series, maybe should
be minus 1.

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And that, that's wrong because the series
diverges.

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But, that's maybe the best of the wrong
answers.

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But what we can keep going with this.

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For example, let's do some more
calculations.

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So let's start with this werid number,
which is 9's all that way.

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Not really a number, but there we go and
let's multiply this by 5.

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What do I get?

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5 times 9 is 45.
So put the 5 down there and carry the 4.

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5 times 9 is 45 plus 4 is 49.
We have 4 to carry.

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5 times 9 is 45 plus 4 is 49.

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Got a 9 there and I gotta carry the 4.
5 times nine is 45 plus 4 is 49.

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You get the idea.

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I'm going to keep on getting 9's that
anyway.

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So 9, 9, 9, 9 times 5 is 5, 9, 9, 9, 9.

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What is this?
What happens if I add 5 to it?

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5 plus 5 is 0, well it's 10 but I gotta
carry the 1.

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9 plus 1 is 10, so I put down a 0 and I
carry the 1.

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9 plus 1 is 10 so I put down the 0 and
carry the 1.

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All right, so 5 with a bunch of 9's here
plus 5 is 0.

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So maybe then this number looks to be a
lot like a negative 5.

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because negative 5 is a thing I can add to
5 to get 0.

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And we already saw that 9 is this way
looked a lot like negative 1.

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So it seems like I've taken a number

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that's playing a role sort of like
negative 1

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and I've multiplied it by 5 and I've got

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a number that's playing the role of
negative 5.

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Which I could tell because when I added 5
to it,

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I got back to 0 and that worked better
than it should've.

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What if we took 999999 and multiplied it
by itself?

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So I've got

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9's all the way to the left, times itself.
9's all the way to the left.

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And what is that product?
Well, 9 times 9 is 81.

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So put the 1 down there and the 8 up there
for the carry.

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9 times 9 in 81 plus 8, which is 89, so
put a 9 there.

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I've gotta carry this 8 now.

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9 times 9 is 81, plus 8 is 89, so

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I'll write the 9 down there and I've gotta
carry

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the 8.

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Now, I keep on carrying the 8's along the
top

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and I'm going to keep on writing 9's along
the bottom.

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So it looks like working on the first
digit here 9's all the way to

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the left times 9 is just one with 9's all
the way to the left.

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But that's just working on this first
digit, now I've

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got to move to the next digit on the
bottom.

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So I'll put a 0 there, and it'll be 9
times 9 is 81, so I put the 8

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up there and the 1 down there.

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9 times 9 plus 8 is 89, so I'll put the 9
down there and then the 8 up there.

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9 times 9 is 81 plus 8, which is 89, so
I'll put the 9 down

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there and the 8 up there, and I'll keep on
going the same, exact way, all right.

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So working on that second digit, I've got
a 0, a 1 and then all 9's.

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And I gotta work on the third digit on the
bottom.

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Put down two 0's here and then it's
exactly the same pattern.

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It's going to be 1 followed by 9's going
on forever.

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And then to work on the next digit on the
bottom, it will be

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three 0's and then the same patter of 1
followed by a bunch of 9's.

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And it's going to keep on going like that.

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Well, after I do all the digits along the
bottom

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in this algorithm I'm supposed to add'
them all up.

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So now I've got 1.

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9 plus 1 is 10, so I put the 0 there and I
carry the 1.

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1 plus 9 plus 9 plus 1 is 20.

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So that's 0 here and I gotta carry a 2
there.

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2 plus 9 plus nine plus 9 plus 1, well
that's 30.

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So I put the 0 there and I'll put a 3
here.

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And I'm going to keep on going like that.

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And I'm going to get 0's all the way
across here.

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So what just happened?

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Well, it looks like we started with a
number that's playing the role of minus 1.

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And we multiplied by a number that's
playing the role of minus 1.

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And when I actually did that
multiplication,

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the answer that I ended up getting looks
to be 1.

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Which is just what I hope it would be.

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So it really is seeming like it's working.

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And when we started out with

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this nonsensical thing right, this
divergent series.

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But we're sort of taking the nonsense
seriously, we've ended up

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getting a system that works sort of like
the negative numbers.

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And in fact computers do store negative
numbers this way.

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Of course, computers usually don't

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work base 10, they work base 2 and when
you work

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base 2, you do the same kind of game
called two's complement arithmetic.

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And this sort of thing comes up in just
pure mathematics.

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This sort of thing is often called the
P-adic numbers

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and in particular when we're using powers
of ten, 10-adic numbers.

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So mathematics is so powerful, that even
nonsense can lead to a reasonable fury.

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It's one thing to win battles, just by
force alone, right?

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To be great, by virtue of simply being
correct all of the time.

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But it's something else entirely, to win
those battles, when you're weak.

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To be on the right track, even when you're
entirely wrong.

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And I think the fact that mathematics
works

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like this really shows that there's
something to it.

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It's not just symbols that we're pushing
around on a page.

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We're really out there,

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exploring something that's really there,
something really beautiful

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and we're just fortunate to get to take
part.

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[SOUND]