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[SOUND] Infinity is not a number like 
seventeen. 

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Don't treat it as a number. 
Let, let me show you something that 

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people do all of the time. 
Here it is. 

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People write down the interval from zero 
to infinity including zero, which is fine 

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but including infinity, which is not 
okay. 

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Infinity's not a number. 
What people should be writing down is 

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this. 
The interval from zero to infinity 

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including zero but not including 
infinity. 

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Because infinity's not a number. 
Why am I freaking out about this? 

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What's so bad about thinking of infinity 
as if it were some number? 

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Now, if you start trying to do arithmetic 
with infinity, you're liable to walk 

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straight into a trap. 
Let me demonstrate for you by walking 

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into one of these traps myself. 
Let's treat infinity as if it were a 

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number. 
Well, what's infinity plus one in that 

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case? 
I got infinitely many things and I add 

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one more. 
That's the same as having infinitely many 

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things. 
So, infinity plus one would equal 

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infinity. 
Now, what else would I know about 

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infinity as a number? 
Well, what would infinity minus infinity 

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be equal to? 
This would be a number minus itself, 

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and a number minus itself is zero. 
So, you're going to treat infinity as a 

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number, 
you're going to be believing these two 

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statements. 
You're going to believe infinity plus one 

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is infinity and you're going to believe 
infinity minus infinity is equal to zero. 

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This is how these numbers would work. 
This is actually very bad. 

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Let's just look at infinity plus one 
minus infinity, and let's subtract 

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infinity from both sides. 
And you said infinity minus infinity is 

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equal to zero. 
But, what's infinity plus one minus 

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infinity? 
Well, that would be infinity minus 

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infinity plus one. 
Infinity minus infinity is zero. 

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This would be zero plus one. 
That side would be one. 

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So, if you're going to treat infinity 
like a number, you're going to end up 

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telling me that one equals zero. 
That's ridiculous. 

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The upshot here is that you can't do 
arithmetic with infinity. 

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Infinity's not a number like seventeen. 
But something comes to save the day. 

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We can do limits. 
For instance, 

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consider this problem where I'm doing a 
calculation involving infinity. 

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But instead of working with infinity 
directly, I'm phrasing it as a limit 

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question. 
What's the limit of x * x - 1 as x 

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approaches infinity? 
Well, x is as large as I like. 

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By making x big enough, I can make x as 
big as I like. 

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x - 1 is also as large as I like, by 
making x big enough, I can make x - 1 as 

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big as I like. 
And what happens if I multiply together 

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two numbers which are as large as I like? 
Well, the product of two numbers that are 

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as big as I want them to be can be as big 
as I want them to be. 

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So, in that sense, the limit of x * x - 1 
as x approaches infinity is equal to 

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infinity. 
So, what about our original example? 

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What's infinity minus infinity? 
[LAUGH] Well, it depends. 

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Here's one possibility. 
Let's consider the limit of x^2 - x as x 

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approaches infinity. 
Now, this is a limit of a difference. 

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You might remember back, the limit of a 
difference is a difference of the limits 

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provided the limits exist. 
Ignoring that last part about whether the 

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limits exist, 
[LAUGH] you might just blindly start 

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writing down the limit as x approaches 
infinity of x^2 minus the limit of x as x 

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approaches infinity. 
This is no good, 

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right? 
The limit of x^2 as x approaches 

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infinity, that's equal to infinity. 
And the limit of x as x approaches 

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infinity, that's infinity, 
right? 

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I can make x^2 as large as I like if x is 
big enough, 

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and I can make x as large as I like if x 
is big enough. 

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So, what just happened? 
I'm running the limit of a difference as 

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the difference of the limits, but these 
limits don't exist, 

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right? 
They're equal to infinity and infinity is 

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not a number. 
Now, I'm left with infinity minus 

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infinity. 
I don't know what to do, right? 

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Who knows what that's equal to? 
That's exactly the sort of thing I'm not 

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permitted to think about. 
Instead, if I wanted to know the limit of 

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x^2 - x as x approaches infinity, I could 
rewrite this limit as the limit as x goes 

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to infinity of, 
what's another way of writing x^2 - x? 

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Could write it as x * x - 1. 
And we saw just a minute ago that the 

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limit of x * x - 1 as x approaches 
infinity is equal to infinity. 

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So, in this case, it seems that infinity 
minus infinity is infinity. 

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On the other hand, infinity minus 
infinity could be seventeen. 

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Let's see how. 
Here again, I've got a limit of a 

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difference of two things. 
Something minus something as x approaches 

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infinity. 
Now, if I blindly just apply the limit of 

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a difference as the difference of limits 
without remembering that that's only 

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valid if the limits exist, let's see what 
happens. 

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And then, I get the limit as x approaches 
infinity of the first thing, which is x + 

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17, minus the limit of the second thing 
which is just x. 

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Now, what's the limit of x + 17 as x 
approaches infinity. 

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Well, I can make x + 17 as large as I 
like if x is big enough, so that's 

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infinity. 
And what's the limit of x as x approaches 

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infinity? 
Well, I can make x as large as I like if 

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x is big enough, so that's also infinity. 
So I've again walked into the trap of 

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writing down infinity minus infinity, 
right? I applied the limit of a 

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difference equals the difference of the 
limits without remembering that that's 

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only valid if the limits exist. 
And in this case, the limits only exist 

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in the weak sense that I can write down 
that they equal infinity. 

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Anyway, I don't want do that. 
Now, how could I figure out what this 

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limit's equal to? 
Well, this is a limit of something and I 

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could rearrange the something, 
right? 

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This is the same as the limit as x 
approaches infinity. 

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What's another way of writing this? 
x + 17 - x, 

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I could have just written seventeen. 
Now, what the limit of seventeen is as x 

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approaches infinity? 
Well, what the limit of seventeen as x 

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approaches anything at all? 
That's just seventeen. 

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So, in this admittedly contrived case, 
infinity minus infinity ended up equaling 

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seventeen. 
[SOUND]