[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]