```24 May 2001 - Ascension Thursday

Infinity is NOT a Number - Part 2 (Electric Bugaloo)

As on another Holy Day (in April, I believe it was), I wrote about
infinity not being a number, from the point of view of the size of
infinite sets.  The truth of the matter is, lots of time people try
talking about infinity as a number in the context of dividing by zero, and
the like.

So let me explain a few things.

First of all: you can't divide by zero.  Nuh-uh.  If you're dividing by
zero, then zero isn't what you think it is.  So perhaps this seems like
begging the question, but there's something in algebra (not the 3x + 4y =
10 algebra class you remember, but something in which you =never= see any
numbers except 0 and 1, though sometimes they are e and epsilon.  But I
nitpick (and then you say "but isn't saying 'Infinity is not a number'
nitpicking?" and I say "shut up.  read this thing and then =you= tell me
if it's nitpicking"))... where was I?  Oh yes, algebra.  Anyway, there's
something called "fields" which are a set whose elements are called
numbers and two operations on those numbers, one of which is called
addition and the other multiplication.

So what makes something a field is that addition and multiplication work
the way you expect it to meaning:

1) it doesn't matter what order you add things, and it doesn't matter what
order you multiply things
(math translation: addition and multiplication are both commutative)

2) there's a number called 0 that if you add it to anything, you get the
same thing back
(math translation: 0 is the additive identity)

3) there's something called 1 that if you multiply it by anything, you get
the same thing back.
(math translation: 1 is the multiplicative identity)

4) every number has another number such that if you add the two together
you get 0.
(math translation: every number has an additive inverse)

5) every number except 0 has another number such that if you multiply the
two together, you get 1.
(math translation: every number has a multiplicative inverse)

6) it doesn't matter how you group things together in addition, and it
doesn't matter how you group things together in multiplication
(math translation: addition and multiplication are associative)

7) a relation between multiplication and addition (a, b, and c are any
numbers): a*(b + c) = a*b + a*c
(math translation: distributive property)

Okay, how does this relate to dividing by zero?  Well, we work with the
field called real numbers, and 1, 0, addition, and multiplication are all
how you know it.  But what about subtraction and division?  Well, to
subtract a number, you simply add its additive inverse: so, because 3 +
(-3) = 0, -3 is the additive inverse of 3.  Therefore, 5 - 3 = 5 + (-3) =
2.

Likewise, to divide by a number, simply multiply by its multiplicative
inverse.  Thus division and subtraction are defined by the concept of
multiplicative and additive inverses.

Well, look at rule #5 up above.  Every non-zero number has a
multiplicative inverse.  But zero doesn't.  Thus, you cannot divide by
zero.  And actually, if you can find a number which, when multiplied by
zero, gives you 1, then you no longer have a field.  Well, you don't
have a field that has more than one element, which isn't too
interesting.  Think about it long and hard.  Let me show you something:

x * 0 = 1	  (positing an inverse for 0)
x * 3 = x*(3+0)   (additive identity)
x*(3 + 0) = x*3 + x+0  (distributive property)
x*3 + x*0 = x*3 + 1    (assumption up above)
so
x*3 = x*3 + 1   ... can you think of where this can go?
obviously, from this,
0 = 1		but this is abstract math, this isn't necessarily

but you will find that the field is only one number.

So again, you can't divide by zero.

Good, so we've gotten that out of the way.

So let's divide by zero.  No, no, not =really=.  Let's do limits.  And
let's see infinity.

So limits are a tricky sort of beast -- they let you get as close as you
want, but no touch.  We'll do limits of functions at a point, to make life
easier.

So let's look at the function f(x) = x.  What's the limit of f(x) as x
goes to 0?  Why 0, of course.  Easy.  Let's look at something harder.  Let
f(x) = x for all x not equal to zero, and f(0) = 2.  Again, what's the
limit of f(x) as x goes to 0?  Again it is zero.  Limits don't care as to
the actual value of the function at the point - only what the value gets
closer and closer to as you approach that point...  Well, let's get a
strong definition before we go any farther:

So we say the limit of f(x), as x approaches b, is L if for any epsilon >
0, there exists a delta such that if 0 < |x-b| < delta, |f(x) - L| <
epsilon.

Okay, that's the honest-to-God definition of a limit.  Let me put this out
of symbols:

Say you have a function and want to know its limit as x goes to b.  A
number L will be the limit if you can get the function value as close as
you want to L in an interval around b.

So, for example, if I tell you that lim_{x->0} f(x) = 3 (the limit of
f(x) as x goes to 0 is three), that means if I wanted a value to be within
+/- 0.1 of 3, I could find a range around 0 such that f(x) for all the x
in that range will be between 2.9 and 3.1

So that's out of the way.  But how does infinity come into it?  Well, how
about limits as x goes to infinity?  Since infinity doesn't really have an
interval about it (what would |x - infinity| < delta mean, anyway?), the
definition of limit has to change.

So we say the limit of f(x), as x goes to infinity, is L if for any
epsilon > 0, there exists an M such that if x > M, |f(x) - L| < epsilon.

In English, this means that a number L is the limit of a function at
infinity if you can get the function value as close as you want to L by
taking all numbers bigger than a certain number.

Here's a quintessential limit:  What is the limit of 1/x as x goes to
infinity?  Well, lots of people say that any finite number divided by
infinity is 0, so the answer is 0.  And they'd be =wrong=.  Because you
can't divide by infinity, because infinity is not a number.  Are we
straight on that?  For then that means infinity is the inverse of 0, and
we can't be having that now, can we?  Let's keep our fields in good
condition.

Still, the =limit= is 0, and how can we show this?  Here's the canonical
structure of a limit proof:

Let epsilon > 0  (you can prove an if-then statement by positing the if
part, and deriving the then part.)
(by the way, "let e > 0" means that epsilon can be =any=
number greater than zero, but it's a particular number)

set N = 1/epsilon.

for x > N, 1/x < 1/(1/epsilon) = epsilon (simple algebra here)

so | 1/x - 0 | < epsilon

therefore, for any positive epsilon, I have found an N such that if x> N,
|1/x - 0| < epsilon.  That means 0 is the limit as x goes to infinity.

Now they're not all that easy.  If they were, Newton would've had calculus
done in a weekend, and he could've spent more time on his studies on
Biblical fundamentalism (sad, but true.)  Deltas and Ns can be infamously
difficult to find.  Sometimes limits =don't= exist.  Then one has to show
that for some epsilon one can't find the proper N or delta.

Okay, let's look at 1/x somewhere else... how about as x approaches
0?  Ah, the old dividing by zero gag.  No, remember we are doing the
=limit=, so we never actually look at what happens when x = 0.  Still, if
one looks at a graph, it's pretty evident that (if you come from the
positive side) 1/x is getting bigger and bigger or (if you come from the
negative side) it's getting less and less.  Let's simplify matters and
consider 1/(x^2), where it seems like the function is going to infinity
from both sides...

But what does going to infinity mean in this case?  So instead of trying
to find a limit =at= infinity, we have an infinite limit at some point.

Here's your obligatory math definition:

The limit of a function f(x) as x approaches b is infinity if, for any M >
0, there exists a delta > 0 such that 0 < |x - b| < delta implies that
f(x) > M.

Basically, as you get closer and closer to the point b, the function
values can be bounded below by larger and larger numbers.

So looking at 1/x^2, for any M > 0, if I pick delta = 1/sqrt(M), f(x) > M
for all x such that -delta < x < delta.

In any case, you can see infinity means different things from different
perspectives.  It can mean boundlessness of a function in a particular way
(as in f(x) goes to infinity) , it can mean "long-term" trends (as in x
goes to infinity... this is used often in asymptotic expressions), it can
mean the size of a set.  But it is not a number.

Though it can be a point.  But that's for projective geometers to explain.

"Hey can't you see... infinity..."
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