[SOUND] Infinity is not a number like 
seventeen. 
Don't treat it as a number. 
Let, let me show you something that 
people do all of the time. 
Here it is. 
People write down the interval from zero 
to infinity including zero, which is fine 
but including infinity, which is not 
okay. 
Infinity's not a number. 
What people should be writing down is 
this. 
The interval from zero to infinity 
including zero but not including 
infinity. 
Because infinity's not a number. 
Why am I freaking out about this? 
What's so bad about thinking of infinity 
as if it were some number? 
Now, if you start trying to do arithmetic 
with infinity, you're liable to walk 
straight into a trap. 
Let me demonstrate for you by walking 
into one of these traps myself. 
Let's treat infinity as if it were a 
number. 
Well, what's infinity plus one in that 
case? 
I got infinitely many things and I add 
one more. 
That's the same as having infinitely many 
things. 
So, infinity plus one would equal 
infinity. 
Now, what else would I know about 
infinity as a number? 
Well, what would infinity minus infinity 
be equal to? 
This would be a number minus itself, 
and a number minus itself is zero. 
So, you're going to treat infinity as a 
number, 
you're going to be believing these two 
statements. 
You're going to believe infinity plus one 
is infinity and you're going to believe 
infinity minus infinity is equal to zero. 
This is how these numbers would work. 
This is actually very bad. 
Let's just look at infinity plus one 
minus infinity, and let's subtract 
infinity from both sides. 
And you said infinity minus infinity is 
equal to zero. 
But, what's infinity plus one minus 
infinity? 
Well, that would be infinity minus 
infinity plus one. 
Infinity minus infinity is zero. 
This would be zero plus one. 
That side would be one. 
So, if you're going to treat infinity 
like a number, you're going to end up 
telling me that one equals zero. 
That's ridiculous. 
The upshot here is that you can't do 
arithmetic with infinity. 
Infinity's not a number like seventeen. 
But something comes to save the day. 
We can do limits. 
For instance, 
consider this problem where I'm doing a 
calculation involving infinity. 
But instead of working with infinity 
directly, I'm phrasing it as a limit 
question. 
What's the limit of x * x - 1 as x 
approaches infinity? 
Well, x is as large as I like. 
By making x big enough, I can make x as 
big as I like. 
x - 1 is also as large as I like, by 
making x big enough, I can make x - 1 as 
big as I like. 
And what happens if I multiply together 
two numbers which are as large as I like? 
Well, the product of two numbers that are 
as big as I want them to be can be as big 
as I want them to be. 
So, in that sense, the limit of x * x - 1 
as x approaches infinity is equal to 
infinity. 
So, what about our original example? 
What's infinity minus infinity? 
[LAUGH] Well, it depends. 
Here's one possibility. 
Let's consider the limit of x^2 - x as x 
approaches infinity. 
Now, this is a limit of a difference. 
You might remember back, the limit of a 
difference is a difference of the limits 
provided the limits exist. 
Ignoring that last part about whether the 
limits exist, 
[LAUGH] you might just blindly start 
writing down the limit as x approaches 
infinity of x^2 minus the limit of x as x 
approaches infinity. 
This is no good, 
right? 
The limit of x^2 as x approaches 
infinity, that's equal to infinity. 
And the limit of x as x approaches 
infinity, that's infinity, 
right? 
I can make x^2 as large as I like if x is 
big enough, 
and I can make x as large as I like if x 
is big enough. 
So, what just happened? 
I'm running the limit of a difference as 
the difference of the limits, but these 
limits don't exist, 
right? 
They're equal to infinity and infinity is 
not a number. 
Now, I'm left with infinity minus 
infinity. 
I don't know what to do, right? 
Who knows what that's equal to? 
That's exactly the sort of thing I'm not 
permitted to think about. 
Instead, if I wanted to know the limit of 
x^2 - x as x approaches infinity, I could 
rewrite this limit as the limit as x goes 
to infinity of, 
what's another way of writing x^2 - x? 
Could write it as x * x - 1. 
And we saw just a minute ago that the 
limit of x * x - 1 as x approaches 
infinity is equal to infinity. 
So, in this case, it seems that infinity 
minus infinity is infinity. 
On the other hand, infinity minus 
infinity could be seventeen. 
Let's see how. 
Here again, I've got a limit of a 
difference of two things. 
Something minus something as x approaches 
infinity. 
Now, if I blindly just apply the limit of 
a difference as the difference of limits 
without remembering that that's only 
valid if the limits exist, let's see what 
happens. 
And then, I get the limit as x approaches 
infinity of the first thing, which is x + 
17, minus the limit of the second thing 
which is just x. 
Now, what's the limit of x + 17 as x 
approaches infinity. 
Well, I can make x + 17 as large as I 
like if x is big enough, so that's 
infinity. 
And what's the limit of x as x approaches 
infinity? 
Well, I can make x as large as I like if 
x is big enough, so that's also infinity. 
So I've again walked into the trap of 
writing down infinity minus infinity, 
right? I applied the limit of a 
difference equals the difference of the 
limits without remembering that that's 
only valid if the limits exist. 
And in this case, the limits only exist 
in the weak sense that I can write down 
that they equal infinity. 
Anyway, I don't want do that. 
Now, how could I figure out what this 
limit's equal to? 
Well, this is a limit of something and I 
could rearrange the something, 
right? 
This is the same as the limit as x 
approaches infinity. 
What's another way of writing this? 
x + 17 - x, 
I could have just written seventeen. 
Now, what the limit of seventeen is as x 
approaches infinity? 
Well, what the limit of seventeen as x 
approaches anything at all? 
That's just seventeen. 
So, in this admittedly contrived case, 
infinity minus infinity ended up equaling 
seventeen. 
[SOUND]