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Let's list all of the real numbers.

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[SOUND].

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Let's make this even easier on yourselves.

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Instead of trying to list of all the real
numbers let's just list of

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the real numbers between zero and one, and
what do I mean by that?

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I mean I want a sequence so that every
single

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real number between zero and one appears
somewhere in that sequence.

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Well suppose that I did, suppose that this
is the list.

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This is the first number in my list, the
second number

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in my list and so on.
I'm just imagining that I have a list of

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all the real numbers between zero and one.
Using this sequence I'll build a number x.

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So here's how I'll define this number x,

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I'm going to define this number x by
telling

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you what the nth digit after the decimal
point of this number is supposed to be.

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And I'm going to do that in terms of this
sequence a sub n.

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So let's suppose

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D is the nth digit of a sub n.

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Then the nth digit of x is either 1 more
or 1 less than D.

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Have to be 1 more than D if D is less than
or equal to

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7, and have to be 1 less than D if D is 8
or 9.

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Well let's see how this goes.

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So here is the sequence that I'm supposing
is a sequence of all the real numbers.

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And let's start by trying to write down
this, this number x.

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So x is zero point, and I don't know

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what the first digit is after the decimal
point.

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And to do that I'll look at the first

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digit of the first number, and that's an
8.

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And if it's an 8, then my digit is
going to be one less, which is 7.

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To get the next digit, I'll look at the

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second digit of the second number, and
that's a 2.

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And 2

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is less than or equal to 7, so I'll make
that a 3.

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To get the next digit, I'm going to look
at the third number on my list.

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And the third digit of the third number is
7.

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And that means my third digit of x is
going to be an eight.

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And I can keep on playing this game and if
I keep on doing this,

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I'll eventually write down a number that
starts off like this, 738 and so on.

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Supposedly,

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I listed off every single real number
between zero and one on that list.

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X is also a number between zero and one.

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So it must be one of the numbers on our
list.

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Well, here's our list, and here's x.

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Is x the first number on our list?
No, it's not our first number.

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Because, the first digit after the decimal
point of the first number is an 8.

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That the first digit after

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the decimal point of x is a seven.

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Alright, I'm using this formula for
determining digits of x.

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So the first digit of the first number in
an

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8 and that means the digit for x is a 7.

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Well maybe x is the second number on our
list.

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It can't be that one, because the second
digit of the second number on

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our list is a 2, but using this formula
for the digits of x,

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the second digit of x is a 3.

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And because x differs from a sub 2 in the
second

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place after the decimal point, x can't be
a sub 2.

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Maybe x is a sub 3.

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Well the third digit after the decimal
point for a sub 3 is a

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7, and the third digit after the decimal
point for x is an 8, right?

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X and a sub 3 differ in the third place
after the decimal point.

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May be x is a sub 4, well x and a sub 4
differ in the

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fourth place after the decimal point.

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All right, the fourth digit here is a 4,
and the fourth digit of x is a 5.

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Maybe x is a sub 5, well, x and a

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sub 5 differ in the fifth digit after the
decimal point.

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The fifth digit after the decimal point is
a 7.

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But the fifth digit of x is an 8.
So x can't be a sub 5.

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Can it appear anywhere on our list?
No, but why not?

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Well, the number x isn't a sub n.
It isn't the nth number on our list.

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Because x differs from a sub n in the nth
digit.

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Remember how we defined the number x?

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D was the nth digit of a sub n.
And x was defined in such a way that the

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nth digit of x was definitely not D.
And because the nth digit of x isn't the

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same as the nth digit of a sub n, x can't
be a sub n.

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X can't be any of the a's sub n's.

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So x cannot appear on the list.
So what does this mean?

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It means there's no sequence, which
mentions all

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of the real numbers between 0 and 1.

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If there were such a sequence, I could use
that sequence to

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build the number x which would have to be
on the list.

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Because it's a list of all real

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numbers between zero and one.

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And you can't be on the list because it

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differs from the nth number in the nth
digit.

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That means that it's impossible to list
off all the numbers between zero and one.

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That means there's no sequence.

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Which mentions every number between zero
and one.

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In a sense then, there's more real
numbers, than there are whole numbers.

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In a sense then, there's more real numbers
then

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there are whole numbers.

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[SOUND].