Let's put all the integers together.
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The goal here is to come up
with a sequence that mentions every single integer.
But before we do that, let's first try to come
up with a sequence that mentions every non negative integer.
It's the sequence a sub n equals n where the index n starts at zero.
So the terms of the sequence are zero, one, two, three, and so on.
And of course,
I could negate that sequence to get
a sequence that mentions every negative integer.
So I could look at the sequence b sub
n equals negative n, but let's start n at one.
And if I do that, then the terms of this sequence start minus one, minus two,
minus three, minus four and so on. But I want both, I want a single sequence
that includes among it's terms every
positive integer, every negative integer and zero.
Is it even possible?
Yes, I'll, I'll weave together the two sequences that we've already built.
So what do I mean?
Well let's take a look at these two sequences.
I could put them together, right. I could weave them together.
I could start with 0, then do minus 1,
then 1, then minus 2, then 2, then minus 3.
I get
a new sequence that would end up mentioning every single integer, with
start zero, minus one, one. Minus two, two,
minus three, three, minus four, four, and it would keep on going like that.
I'd like a formula for that sequence.
Well here's a formula for the sequence. C sub n will be defined
via this piecewise definition depending on the parity of n, whether n is odd or even.
If n is odd, then the nth term will be negative n plus 1 over 2.
And if n is even, then the nth term is just n over two.
And I'll start with the index zero.
And this will give me this sequence, right, the zero term, when I plug in
zero, zero is even, zero over two is zero, and that gives me the zero.
When I plug in one, one is odd, so I
get negative one plus one over two, that's negative one.
When I plug in two that's even, so it's two over two, that's this one here.
When I plug in three, three's odd, so it's negative three plus one
over two, that's negative two and it just keeps on going like that.
There's another way to think about this.
To say that I've got the same quantity of dots and squares.
Is really to say that there's some method by which I can pair off the
dots and squares so that every square gets a dot and every dot gets a square.
And once I've paired them off like this it's very
believable that there's the same quantity of dots and squares.
Well something similar is going on with non-negative integers and all integers.
If I just take a look at the non negative integers, I perhaps want
to show others the same quantity of non
negative integers as there are just all integers.
And to do that, I just have to tell you some
method for pairing off non negative integers with all the integers.
And that's exactly what this sequence does, right?
It assigns to zero the number zero.
It assigns to one, the number minus one,
it assigns to the index two, the number one,
to the index three, the sequence assigns the number negative two.
To the index four, it assigns the number two.
To the index five, it assigns the number three, to the index
six, it assigns the number three and it will keep on going.
And eventually, right, we've ended up pairing off
every single non negative integer with every single integer.
And that really means that there's the same quantity of non negative integers
as there are all integers.
That should really give you pause, that, that sounds impossible, alright?
Think about some physical object, some finite object like, coffee beans.
If I've got some coffee beans, but then I take some away, now
I've got fewer coffee beans, but the collection of all integers is different.
If I start with all the integers and just take away the
negative integers I've got the same quantity of things.
Why does that work?
Well one definition of an infinite quantity is a quantity
that needn't get smaller, even when you take something away.
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