Let's list all of the real numbers.
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Let's make this even easier on yourselves.
Instead of trying to list of all the real numbers let's just list of
the real numbers between zero and one, and what do I mean by that?
I mean I want a sequence so that every single
real number between zero and one appears somewhere in that sequence.
Well suppose that I did, suppose that this is the list.
This is the first number in my list, the second number
in my list and so on. I'm just imagining that I have a list of
all the real numbers between zero and one. Using this sequence I'll build a number x.
So here's how I'll define this number x,
I'm going to define this number x by telling
you what the nth digit after the decimal point of this number is supposed to be.
And I'm going to do that in terms of this sequence a sub n.
So let's suppose
D is the nth digit of a sub n.
Then the nth digit of x is either 1 more or 1 less than D.
Have to be 1 more than D if D is less than or equal to
7, and have to be 1 less than D if D is 8 or 9.
Well let's see how this goes.
So here is the sequence that I'm supposing is a sequence of all the real numbers.
And let's start by trying to write down this, this number x.
So x is zero point, and I don't know
what the first digit is after the decimal point.
And to do that I'll look at the first
digit of the first number, and that's an 8.
And if it's an 8, then my digit is going to be one less, which is 7.
To get the next digit, I'll look at the
second digit of the second number, and that's a 2.
And 2
is less than or equal to 7, so I'll make that a 3.
To get the next digit, I'm going to look at the third number on my list.
And the third digit of the third number is 7.
And that means my third digit of x is going to be an eight.
And I can keep on playing this game and if I keep on doing this,
I'll eventually write down a number that starts off like this, 738 and so on.
Supposedly,
I listed off every single real number between zero and one on that list.
X is also a number between zero and one.
So it must be one of the numbers on our list.
Well, here's our list, and here's x.
Is x the first number on our list? No, it's not our first number.
Because, the first digit after the decimal point of the first number is an 8.
That the first digit after
the decimal point of x is a seven.
Alright, I'm using this formula for determining digits of x.
So the first digit of the first number in an
8 and that means the digit for x is a 7.
Well maybe x is the second number on our list.
It can't be that one, because the second digit of the second number on
our list is a 2, but using this formula for the digits of x,
the second digit of x is a 3.
And because x differs from a sub 2 in the second
place after the decimal point, x can't be a sub 2.
Maybe x is a sub 3.
Well the third digit after the decimal point for a sub 3 is a
7, and the third digit after the decimal point for x is an 8, right?
X and a sub 3 differ in the third place after the decimal point.
May be x is a sub 4, well x and a sub 4 differ in the
fourth place after the decimal point.
All right, the fourth digit here is a 4, and the fourth digit of x is a 5.
Maybe x is a sub 5, well, x and a
sub 5 differ in the fifth digit after the decimal point.
The fifth digit after the decimal point is a 7.
But the fifth digit of x is an 8. So x can't be a sub 5.
Can it appear anywhere on our list? No, but why not?
Well, the number x isn't a sub n. It isn't the nth number on our list.
Because x differs from a sub n in the nth digit.
Remember how we defined the number x?
D was the nth digit of a sub n. And x was defined in such a way that the
nth digit of x was definitely not D. And because the nth digit of x isn't the
same as the nth digit of a sub n, x can't be a sub n.
X can't be any of the a's sub n's.
So x cannot appear on the list. So what does this mean?
It means there's no sequence, which mentions all
of the real numbers between 0 and 1.
If there were such a sequence, I could use that sequence to
build the number x which would have to be on the list.
Because it's a list of all real
numbers between zero and one.
And you can't be on the list because it
differs from the nth number in the nth digit.
That means that it's impossible to list off all the numbers between zero and one.
That means there's no sequence.
Which mentions every number between zero and one.
In a sense then, there's more real numbers, than there are whole numbers.
In a sense then, there's more real numbers then
there are whole numbers.
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