Even nonsense can be meaningful.
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Mathematics is more than just things free
of inconsistency.
It's more than just that which is the
case.
Sometime we're confronted with things,
that are really not
nonsensical or things that are just flat
out wrong.
And when we're confronted with things like
that, we should
really have a feeling or a need to salvage
the situation.
We should take the nonsensical or wrong
thing and try to salvage it.
Try to think
of some sense in which it might make
sense.
So what's the sum n goes from 0 to
infinity of 9 times 10 to the nth power?
The party-pooper simply says, the series
diverges.
And the party-pooper is right.
This series diverges.
And why does it diverge?
Well, it's a geometric series with
R equals 10 and that counter ratio is
bigger than 1.
But let's try to salvage the nonsense.
Well, remember back how we looked at the
sum N goes from 1 to infinity
of 9 times 10 to the negative nth power
and this was, well, it's
equal to 1, but we could also write it as
0.9999 where the
9s keep on going that way.
Well, in this situation, contemplating is
really the opposite.
I'm thinking of the sum N goes from 0 to
infinity, of 9 times 10 to the N.
What is that?
When I plug it in N equals 0, I get nine
time one which is 9.
When I plug it in n equals 1, I get 9
times 10 which is 9.
When I plug in N equals 2, I get 9 times
100 which is 900.
And I'd keep on going.
So one way to write this might be, well at
first 9, 9 plus 90, that's 99, 99 plus 900
that's 999.
It's as if I could write it with 9's going
this way.
And what if we add 1 to that.
Well, I mean, it's not really a number,
right?
But I can still write it down.
So I'll write down
[LAUGH]
a number that's just 9s going this way.
And then I'm going to add 1 to that number
[LAUGH].
What do I get?
Well, 9 plus 1 is 10, carry the 1.
9 plus 1 is 10, carry the 1.
9 plus 1 is 10, carry the 1.
9 plus 1 is 10, carry the 1.
9 plus 1 is 10, carry the 1.
And I'm going to keep on doing that,
right?
So it looks like this thing, which is 9's
all the way that way, plus
1, is 0.
So whatever this thing is, its a thing
that if i add 1 to it, I get 0.
So what is 999 with dots all that way,
its sort of equal and negative 1.
because negative 1 is a thing I can add 1
to to
get 0 and when I added 1 to this thing, I
got 0.
Sort of, makes sense.
So if I just ignore convergence
altogether, what would it tell me to do?
I'm trying to evaluate the sum and goes
from 0 to infinity of 9 times 10 to the N.
Well that would be 9 times the sum N
goes from 0 to infinity of 10 to the N.
This is a, is a divergent geometric
series.
Let's just pretend the formula still
worked.
How I calculated the value of that series.
That'd be 9 times and the formula for this
is
1 over 1 minus the common ratio, which is
10.
Of course, that formula's only valid if
it, if it
were convergent series and it's not, but
let's just pretend.
What is this?
Is this 9 times 1 over 1 minus 10?
That's 9 times 1 over minus 9.
Well, 9 times 1 over minus 9 is minus 1.
Which is sort of what we're seeing here,
right?
I mean, if we're just ignoring
convergence, it looks
like it's telling us that the value of
this series.
I mean, it doesn't have a value because it
diverges, but
if the value of the series, maybe should
be minus 1.
And that, that's wrong because the series
diverges.
But, that's maybe the best of the wrong
answers.
But what we can keep going with this.
For example, let's do some more
calculations.
So let's start with this werid number,
which is 9's all that way.
Not really a number, but there we go and
let's multiply this by 5.
What do I get?
5 times 9 is 45.
So put the 5 down there and carry the 4.
5 times 9 is 45 plus 4 is 49.
We have 4 to carry.
5 times 9 is 45 plus 4 is 49.
Got a 9 there and I gotta carry the 4.
5 times nine is 45 plus 4 is 49.
You get the idea.
I'm going to keep on getting 9's that
anyway.
So 9, 9, 9, 9 times 5 is 5, 9, 9, 9, 9.
What is this?
What happens if I add 5 to it?
5 plus 5 is 0, well it's 10 but I gotta
carry the 1.
9 plus 1 is 10, so I put down a 0 and I
carry the 1.
9 plus 1 is 10 so I put down the 0 and
carry the 1.
All right, so 5 with a bunch of 9's here
plus 5 is 0.
So maybe then this number looks to be a
lot like a negative 5.
because negative 5 is a thing I can add to
5 to get 0.
And we already saw that 9 is this way
looked a lot like negative 1.
So it seems like I've taken a number
that's playing a role sort of like
negative 1
and I've multiplied it by 5 and I've got
a number that's playing the role of
negative 5.
Which I could tell because when I added 5
to it,
I got back to 0 and that worked better
than it should've.
What if we took 999999 and multiplied it
by itself?
So I've got
9's all the way to the left, times itself.
9's all the way to the left.
And what is that product?
Well, 9 times 9 is 81.
So put the 1 down there and the 8 up there
for the carry.
9 times 9 in 81 plus 8, which is 89, so
put a 9 there.
I've gotta carry this 8 now.
9 times 9 is 81, plus 8 is 89, so
I'll write the 9 down there and I've gotta
carry
the 8.
Now, I keep on carrying the 8's along the
top
and I'm going to keep on writing 9's along
the bottom.
So it looks like working on the first
digit here 9's all the way to
the left times 9 is just one with 9's all
the way to the left.
But that's just working on this first
digit, now I've
got to move to the next digit on the
bottom.
So I'll put a 0 there, and it'll be 9
times 9 is 81, so I put the 8
up there and the 1 down there.
9 times 9 plus 8 is 89, so I'll put the 9
down there and then the 8 up there.
9 times 9 is 81 plus 8, which is 89, so
I'll put the 9 down
there and the 8 up there, and I'll keep on
going the same, exact way, all right.
So working on that second digit, I've got
a 0, a 1 and then all 9's.
And I gotta work on the third digit on the
bottom.
Put down two 0's here and then it's
exactly the same pattern.
It's going to be 1 followed by 9's going
on forever.
And then to work on the next digit on the
bottom, it will be
three 0's and then the same patter of 1
followed by a bunch of 9's.
And it's going to keep on going like that.
Well, after I do all the digits along the
bottom
in this algorithm I'm supposed to add'
them all up.
So now I've got 1.
9 plus 1 is 10, so I put the 0 there and I
carry the 1.
1 plus 9 plus 9 plus 1 is 20.
So that's 0 here and I gotta carry a 2
there.
2 plus 9 plus nine plus 9 plus 1, well
that's 30.
So I put the 0 there and I'll put a 3
here.
And I'm going to keep on going like that.
And I'm going to get 0's all the way
across here.
So what just happened?
Well, it looks like we started with a
number that's playing the role of minus 1.
And we multiplied by a number that's
playing the role of minus 1.
And when I actually did that
multiplication,
the answer that I ended up getting looks
to be 1.
Which is just what I hope it would be.
So it really is seeming like it's working.
And when we started out with
this nonsensical thing right, this
divergent series.
But we're sort of taking the nonsense
seriously, we've ended up
getting a system that works sort of like
the negative numbers.
And in fact computers do store negative
numbers this way.
Of course, computers usually don't
work base 10, they work base 2 and when
you work
base 2, you do the same kind of game
called two's complement arithmetic.
And this sort of thing comes up in just
pure mathematics.
This sort of thing is often called the
P-adic numbers
and in particular when we're using powers
of ten, 10-adic numbers.
So mathematics is so powerful, that even
nonsense can lead to a reasonable fury.
It's one thing to win battles, just by
force alone, right?
To be great, by virtue of simply being
correct all of the time.
But it's something else entirely, to win
those battles, when you're weak.
To be on the right track, even when you're
entirely wrong.
And I think the fact that mathematics
works
like this really shows that there's
something to it.
It's not just symbols that we're pushing
around on a page.
We're really out there,
exploring something that's really there,
something really beautiful
and we're just fortunate to get to take
part.
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