Negative Ratios and Dividing by Zero

Date: 8 Jun 1995 12:53:31 -0400
From: Andrea Berni
Subject: Question

Question which has been hovering in my mind for a while...

1 : -1 = -1 : 1

Then how can a bigger number be to a smaller number as a smaller
number is to a bigger one ?

Another little question to which I never had a satisfying answer...

X divided by 0 = infinity

Why not ?

Thanks for wasting your time with me.

Ciao !    -    -   -   -  -  - - ---> Andrea !

Date: 8 Jun 1995 13:27:08 -0400
From: Dr. Ken
Subject: Re: question

Hello there!

Well, hmm.  The concept of a ratio can of course be dealt with by dividing
the two numbers, i.e. 1:-1 = 1/-1 = -1, and -1:1 = -1/1 = -1, so they're the
same thing.  But what does it mean to have a ratio of a positive number to a
negative number?  I'm not sure I have a good answer, but I'll try.  

Look at this: Let's say Bob is in debt $5 and Yolanda is in debt $10.
What's the ratio of Yolanda's money to Bob's money?  Well, -10/-5 = 2.  So
does Yolanda have 2 times as much money as Bob?  Not really, she's in more
debt than he is.  So the ratio of two numbers only gives you a good idea of
how big they are (in the positive direction) when you can be sure both of
the numbers are positive.  I think that's the best I can do.

>X divided by 0 = infinite
>Why not ?

I've got a better answer to this one. You see, it's kind of a gray area.
The way most people think about it, x/0 is kind of infinite, as long as x
isn't also 0.  But nobody can really say that, because there are a few
problems with it that you'd better be prepared to deal with.

1) When you divide one number by another, you expect to get yet another
number, not some abstract concept like infinity.  Infinity isn't a real
number because it doesn't act like one.  For instance, you can have
infinity + infinity = infinity, not 2(infinity) like you'd expect. 

2) If you treated infinity as a number, and you said that 3/0 was infinity,
then what would 6/0 be?  Would it be twice as big?  The same size?

3) Would 3/0 be positive infinity or negative infinity?  The sequence 
1/(1/2), 1/(1/3), 1/(1/4), ... goes to 1/0, and you can simplify the
sequence to be 2,3,4,..., which goes to positive infinity.  But the sequence
1/(-1/2), 1/(-1/3), 1/(-1/4), ... goes to 1/0 too, and it's better known as
the sequence -2,-3,-4,..., which goes to negative infinity.

But yeah, go on thinking that in an abstract sense, x/0 really does go
infinite.

-K

Date: 9 Jun 1995 09:02:19 -0400
From: Andrea Berni
Subject: Re: question

 Ke> 1) When you divide one number by another, you expect to get yet
 Ke> another number, not some abstract concept like infinity.  Infinity
 Ke> isn't a real number because it doesn't act like one.  For instance, you
 Ke> can have infinity + infinity = infinity, not 2(infinity) like you'd
 Ke> expect.  
 Ke> 2) If you treated infinity as a number, and you said that 3/0 was
 Ke> infinity, then what would 6/0 be?  Would it be twice as big?  The same
 Ke> size? 

     Well, Cantor suggested there are different infinites (i.e.
     infinites of different sizes), and according to what I
     understood from reading his explanation, 6/0 and 3/0 would give
     the same sized infinite, however, in this case maths is mainly
     an opinion...

 Ke> 3) Would 3/0 be positive infinity or negative infinity?
     Well, the best I can think of is that it behaves as a square
     root: 3/0= +- infinity
 Ke> But yeah, go on thinking that in an abstract sense, x/0 really does go
 Ke> infinite.
 
     I guess this is the only way out.
     However the positive or negative infinity you mentioned is
     really a good argument against this.

Thanks again !

Ciao

Date: 9 Jun 1995 09:34:48 -0400
From: Dr. Ken
Subject: Re: question

Hello there!

Yes, you're right, Cantor suggested different orders of infinity, but the
sense he was talking about was in the sense of _counting_ elements of a set,
i.e. Cardinal numbers.  For instance, the size of the set {1,2,3,...} is the
same as the size of the set {2,4,6,...} and they're both the same size as the
set of all rational numbers, but none of them is as big as the set of all
real numbers.  It's important to make the distinction between numbers that
are used to count elements of a set and numbers that are used in equations.
In fact, infinity is the only context in which this really makes a
difference.  The infinity that you get when you talk about 3/0 is just kind
of a vague, general infinity, whereas the infinity you get when you talk
about the number of elements of the set {1,2,3,...} is a very specific kind
of infinity, called "countable" infinity or "Aleph 0."  For what that's
worth.

-K

Date: Wed, 25 Feb 1998 16:02:58 -0500
From: Ken Shoemake
Subject: Ask Dr. Math: Negative Ratios ...

Howdy. I was browsing through the old "Ask Dr. Math" stuff to find
something interesting for my young niece, and came across the "Negative
Ratios and Dividing by Zero" item.

May I suggest another kind of answer?

Would you agree that if both numbers in a ratio, like 2:3, are
multiplied by the same number then the meaning of the ratio should be
the same? We will avoid zero for now. Then we can proceed.

We can multiply both parts of 1:-1 by -1, which gives -1:1. So that's an
easy answer. But do you know why -1 times -1 gives 1? Think about (2-1)
times (2-1), and look at each of the four pieces it makes:
1 = (2-1)(2-1) 
  = 4 - 2 - 2 + (-1 * -1)
  = -1 * -1               

So -1 times -1 must equal 1.


As for X/0 and infinity, we can use a similar approach. Let us say that
two fractions, a/b and p/q, are equal whenever a*q equals b*p. We get
this from multiplying both fractions by b and by q.

Now suppose X/0 is some definite fraction, let's say a/b. If X/0
equals a/b, then we expect X*b equals 0*a, which is 0. It doesn't
matter what a is, and b must be 0 since X isn't. Is this really what
we want? If we try to make X/0 be some kind of number then anything
over zero equals anything else over zero. So we usually don't do it.
We say that X/0 is undefined; it is not any number at all. Otherwise
it has to be some special new kind of number, with its own rules.

But what about dividing by 1/2, 1/3, ..., approaching zero? We can get
as close as we like to zero without a problem. One clever trick is to
say "I have a number that acts like an integer, except that it is bigger
than any integer." We should not call this number "infinity", expecting
that it is the only "number" that will be that big. But we can now say
that one over that big "number" is "infinitesimal"--closer to zero than
any ordinary fraction, but still not zero. This is the idea used in
calculus, as described by A. Robinson.

There are many different ideas about what the word infinity can mean.
Don't let that bother you. Have you heard the joke that "Noses run and
feet smell"? We use all kinds of words in different ways!

 -- Ken