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Infinity, Zero


Date: 4 Jan 1995 08:18:50 -0500
From: Ellen Charny
Subject: Infinity

Hi!  

My name is Ellen Charny and I'm a high school student.  I was  
wondering a few things about infinity, and also zero.  First of all,  
I've come to understand that you can't divide by zero, but no one can  
actually prove WHY.  I tried to...but my reasons had no actual basis.   
I was inventing theorems, i think.  If you're interested, I'll send you 
the proof.  But I wanted to see a real proof first.

Second, We learned in trig that you can't raise zero to the zero-th  
power because zero would equal one, obviously.  But I was wondering  
if you knew anything of further interest about this.

Lastly, (I was asking my old geometry teacher about this and he  
thought it made sense) infinity.  I realize this is not so much a  
number as an endless amount.  But, if there are an infinite number of  
numbers between 1 & 2, and an infinite number of numbers between 1 &  
50, wouldn't the second infinity be bigger than the first?

Thanks for your time-- maybe you can set my mind at ease.

-Ellen


Date: 5 Jan 1995 03:05:43 -0500
From: Dr. Sydney
Subject: Re: Infinity

Dear Ellen,

Hello!  My name is Sydney, and I am one of the "math doctors" here
at Swarthmore.  We are glad you wrote with your questions.  You asked 
some GREAT questions that explore part of the wonderful depth of math.  
One of the other math doctors has a good answer for your first question, 
so I'll let him mail you a copy of it, and I'll say something about the second 
two questions.

You say that in trig you learned you can't raise 0 to the 0th power,
right?  Well, did you also learn that you can't determine what 0 times
infinity is (or to put it another way, you can't determine what 0/0 is or
infinity /infinity is)?  I don't know what you are studying now, but in
calculus you will learn that these are called INDETERMINATE FORMS.  
That means we can't say what these expressions are unless we are given a
specific problem.  For instance, say we were given a problem like finding
the limit as x goes to -1 of (3x^2 + 2x - 1)/(x^2 + x).  

If we plug a -1 inside the limit, we get 0/0.  That doesn't really tell us
anything. To figure out what 0/0 is in this problem, you have to use
something called L'hopital's rule which you will learn about in calculus.
As it turns out, in this problem, 0/0 = 4.  Pretty neat, huh?

Anyway, the point of all of this is that 0^0 is just another indeterminite
form. Why is this, you might ask?  Let the limit as x goes to 0 be 0 and 
the limit as y goes to 0 be 0.  Then the limit as x and y go to 0 of x^y is
0^0, if we plug in 0 for x and for y, right?  But this is indeterminate.
Why?  Well, x^y = e^ln(x^y), right?  And, e^ln(x^y) = e^(ylnx).
As x goes to 0, lnx goes to negative infinity, right?  So the limit as x and
y go to zero of ylnx is going to be 0 times negative infinity, which is
indeterminite.  e raised to an indeterminate form is indeterminate, so, 
x^y, itself, is indeterminate.  So, we have 0^0 is indeterminate.  

Does that make sense to you?  I'm not sure how much you have worked 
with  natural logs, so if you have any questions about what I did, feel 
free to write back.  

On to your last question about infinity.  What an excellent question!  I
love thinking about concepts dealing with infinity as, it also seems, you
do.  Mathematicians usually make a distinction between 2 different 
"kinds" of infinity.  The first kind of infinity is called COUNTABLY 
infinite.  The counting numbers, for instance, are countably infinite.  
This means there exists a 1 to 1 correspondance between the counting 
numbers and the natural numbers.  The set of all rational numbers is 
also countably infinite.  Can you figure out why?  

This is kind of tricky, so I'll give you a hint:  you are looking for some 
way to write down the rational numbers so that there is a specific order 
in your arrangement...you are looking for an order such that if I were 
to ask you what the 328th rational number in the order you selected was, 
you could tell me (though it might be kind of painful!).  

The second kind of infinity is called UNCOUNTABLY infinite.  This 
infinity is, in a way, bigger than the other infinity because it is impossible 
to find a one to one correspondence with the natural numbers.  In these 
sets, there are simply not enough natural numbers to cover all of the 
elements of the set.  This is kind of a strange concept to get used to, I 
think.  The real numbers are an example of an uncountably infinite set 
of numbers.  

But, on to what you were asking about...there are an uncountably 
infinite number of numbers between 1 and 2, and there are also an 
uncountably infinite number of numbers between 1 and 50, but are there 
more numbers between 1 and 50 than between 1 and 2.  Thinking about 
it in one way, I would agree with you, that there are more numbers 
between 1 and 50 than there are numbers between 1 and 2.  After all, 
all the numbers between 1 and 2 are also between 1 and 50, but there 
are lots more numbers betwen 1 and 50 that aren't between 1 and 2.  
However, if you think about it in terms of the fact that there are lots and 
lots of numbers between 1 and 2 -- an uncountably infinite amount, and 
there is an uncountably infinite amount of numbers between 2 and 50, 
and you add the two infinities, you are just going to get another 
uncountably infinite amount.  Is this infinity bigger than the original 
infinities?  What do you think?  Remember that infinity is NOT a 
number!!

I hope this helps you understand some of these concepts better, and I 
hope also you keep thinking about these ideas and write us with any 
other questions.  I'm impressed you are thinking about these things in 
high school!  Keep it up.!!

--Sydney, Dr. "math rocks" Foster


Date: 5 Jan 1995 14:10:50 -0500
From: Dr. Ken
Subject: Re: Infinity

Hello there!

It just so happens that parts of your question have come up before, so 
I'm going to paste some pertinent stuff in here.  First, about your 
dividing by zero question:

Date: Wed, 2 Nov 94 14:20:49 EST
From: "Terry Strohecker"
Subject: Question!

I am in need of a detailed answer to the following question:
Why can't you divide a number by 0?
Thanks for your time.

Terry
----------------------

From: Dr. Ken
Subject: Re: Question!
Date: Wed, 2 Nov 1994 16:53:27 -0500 (EST)

Hello Terry!

There are sort of two reasons.  For one thing, when you divide one 
number by another, you expect the result to be another number.  So in 
particular, look at the sequence of numbers 1/(1/2), 1/(1/3), 1/(1/4), ... .  
Notice that the bottoms of the fractions are 1/2, 1/3, 1/4, ..., and that 
they're going to zero.  So if there's a limit to this sequence, we would 
take that number and call it 1/0.  So let's see if there is.

Well, the first sequence i've written down turns out to be 2, 3, 4, ..., and
that goes to infinity.  Since infinity isn't a real number, we don't assign
any value to 1/0.  We just say it's indeterminate.

But let's say we did.  Let's say that infinity is a real number, and 1/0 is
infinity.  Then look at the sequence 1/(-1/2), 1/(-1/3), 1/(-1/4), ..., and
notice again that the denominators -1/2, -1/3, -1/4, ..., are going to zero.
So again, we would want the limit of this sequence to be 1/0.  But looking 
at the sequence, it simplifies to -2, -3, -4, ..., and it goes to negative
infinity.  So which would we assign to 1/0?  Negative infinity or positive
infinity?  Instead of just assigning one willy nilly, we say that infinity
isn't a number, and that 1/0 is indeterminate.

I hope this helps.

-Ken "Dr." Math

About your second question, Sydney gave a good description of why there 
are so many problems about raising zero to the zero power.  What she 
forgot to tell you, though, is that you actually _can_ raise zero to the zero 
power. You get 1.  Pretty rad, huh?  You probably see the conflict:  
anything to the zero power is 1, and zero to any power is zero.  So what 
do we call zero to the zero power?  In this case, the exponent wins.  If 
you want to know more about why this is the case, please write us back.  
The reasons for defining 0^0 this way are kind of subtle, and some would 
say arbitrary.  We think it's great that you're interested in this stuff.


Your third question is the deepest of these three deep questions.  As it
turns out, there is kind of a wierd answer to it.  See, there are exactly
the same number of points between 1 & 2 as there are between 1 & 50, 
but the set of points between 1 & 50 is still a bigger set.  Here's how it 
works.

The reason we can say there are exactly the same number of points in these
two sets (I'll call these sets A & B from now on) is that we can set up a
one-to-one correspondence between the points in the two sets.  Here's such a
correspondence: take the value of any point in A, multiply it by 49, and
subtract 48.  Then you'll get a point in B (alternatively, we could have
said take the value of any point in A, square it, multiply it by 49/3, and
subtract 46/3; there are lots of functions that work).  This is the kind of
one-to-one correspondence that Sydney talked about in her message, and the
one that's usually used to compare the size of two infinite sets (or, come
to think of it, finite ones, too).

But in another sense, the set B is a bigger set.  This is because it has a
greater _measure_.  See, if we picked a number between 1 and 100, we'd 
be way more likely to pick one from set B than from set A, right?  That's 
the sense in which B is bigger.  It's really a funny notion.  I mean, there 
are the same number of points in each set, and B is just A blown up some.  
But it's not like the points in B are packed together more loosely than those 
in A, because if you look at the layout of points anywhere inside B, it'll 
look exactly like the layout of points anywhere inside A.  But B is bigger.  
That means that there really _must_ be something fundamental about the 
way the points are arranged that makes them spread out more.

Measure is a concept that's pretty recent in mathematics, and it's pretty
subtle stuff.

Anyway, here's another reply I gave to someone else with a related
question.  I didn't delve into measure theory, but I did more with that
correspondence stuff (which is in the part of math called Set Theory).  
I hope you enjoy it!  Wow, this is a long message!
_______________

Dear Dr. Math,

Can one infinity be larger than another? When you reply
please use my name in the subject line.

Sincerely,
Gina Miller

Date: 21 Nov 1994 19:27:52 GMT
From: Ken "Dr." Math
Organization: The Math Doctors
Subject: Re: Gina's problem

Hello there, Gina!

Great question.  Yes, one infinity can be larger than another, a wierd
kind of concept.  One standard way of dealing with the different sizes of
infinities is to take one kind of infinity as a "normal" infinity or a
"base" infinity, and then compare its size with other infinities.  What we
do in practice is take the number of counting numbers 1,2,3,4,5,... as our
normal infinity, and we call it "countably infinite."

The way we compare the sizes of two infinite sets is to see whether we can
pair the elements of the two sets up in a one-to-one correspondence.  For
instance, if we match up the even positive integers and the odd positive 
integers like this:

2,4,6,8,10,12,14,16,18,...
1,3,5,7, 9,11,13,15,17,...

We can see that there are exactly the same number of even positive integers 
as odd positive integers.  What if we try to match up the even positive 
integers and the counting numbers?

2,4,6,8,10,12,14,16,18,...
1,2,3,4, 5, 6, 7, 8, 9,...

Again, we can do it, so there are exactly the same number of even positive 
integers as counting numbers, so we say there are a "countably infinite" 
number of even positive integers.

Notice that all I've done above is write the even numbers in a list,
making sure that I list them all.  This is the usual way to show that
there are countably many (a synonym for countably infinite) numbers in
some set: try to find a way to list them all.

What about the rational numbers?  Remember that the rational numbers are
numbers that can be written as the quotient of two integers.  I'll write a
list of all the positive rational numbers; can you find the pattern?  If
you're really ambitious, you could try to find a formula for the n'th
number in the sequence!

1/1, 2/1, 1/2, 3/1, 2/2, 1/3, 4/1, 3/2, 2/3, 1/4, 5/1, 4/2, 3/3, 2/4, 1/5, ...

Here's a hint about how to look at the numbers in that sequence: look at
the numerators seperately, then the denominators.  Do you believe that
I'll get all the positive rational numbers this way?  If so, I've just shown 
that there are countably many positive rational numbers.  How could
you use a similar sequence to show that there are countably many 
rational numbers (both positive and negative)?

It is an interesting fact that you cannot write such a listing of all the
real numbers (which includes both the rationals and the irrationals).  If
you're interested, write back and I'll help you show that.

If that's true, what does that say about how many real numbers there are?
There's a lot of them, more than the number of counting numbers.  So 
there are two infinite sets that really do have two different sizes.

I hope this helps you.

Ken "Dr." Math
    
Associated Topics:
High School Analysis
High School Sets

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