Positives and Negatives with Infinity and Zero

Date: 10/05/1999 at 18:20:41
From: Chris King
Subject: Negative zero theory

I have a theory - a negative zero theory - but I must first build up a 
story to get to it...

First I will tell you my 1/0 theory.

We all know that if the numerator of a given fraction stays the same, 
the answer will get larger when the denominator of that fraction gets 
smaller: 1/5 = .2, 1/4 = .25, 1/2 = .5, 1/1 = 1, 1/.5 = 2; 1/.25 = 4, 
1/.2 = 5.

So, what happens when the denominator reaches zero?

The calculator says error. My math teacher says that it's "undefined." 
But isn't "undefined" often used interchangeably with infinity?

Also, if you graph out a whole bunch of 1/x's, the graph will move 
upward toward infinity as you get closer to x = 0.

Note also that with the tangent function, the graph will also move 
towards infinity in the direction of 90 degrees (and away from 
negative infinity when you move past 90 degrees. Perhaps they are the 
same? I'll get back to this later.)

So, if you believe that 1/0 = infinity, then we can move on to my main 
theory - Negative Zero.

If 1/0 = infinity, then 1/infinity must equal 0, by all mathematical 
laws.

Since it seems to be accepted that there is a negative infinity (no 
one has challenged that yet), then 1/-infinity must equal -0. Why? For 
the sole purpose of doing the reverse: 1/-0 = -infinity. Otherwise, 
the important mathematical law of dividing by the reciprocal (I'm not 
sure of the exact name of it) would be broken.

Now, my abstract theory, for which I have not much proof:

Not only is there negative zero, there is also positive zero and 
unsigned zero, which is what we're all used to using right now.

And considering that zero and infinity seem to be cousins, infinity 
also comes in those same three forms.

Each three forms of zero all have the same value, as is with the three 
forms of infinity. Remember the tangent graph that I was talking about 
before? The tangent of 90 degrees is actually unsigned infinity, not 
the positive infinity that it appears to be.

So, is negative zero (and all my other theories) correct? I've found 
that my peers are more accepting of these (and other) theories of mine 
than most adults are, perhaps because they are not bound so tightly to 
current mathematical "laws" as adults are. I ask you think more 
abstractly than you normally would; forget the "law" that you cannot 
divide by zero, and from that perspective, judge my theories! For 
this, I would thank you very much.

Date: 10/06/1999 at 11:54:56
From: Doctor Peterson
Subject: Re: Negative zero theory

Hi, Chris. You have some good ideas that just need refinement.

>First I will tell you my 1/0 theory.

If you haven't read our FAQ on dividing by zero and on infinity, they 
will be worth reading, because they deal with many of the issues you 
talk about.

   Dividing by 0
   http://mathforum.org/dr.math/faq/faq.divideby0.html   

   Large Numbers and Infinity
   http://mathforum.org/dr.math/faq/faq.large.numbers.html   

>isn't "undefined" often used interchangeably with infinity?

There is certainly a sense in which we can say that 1/0 is "infinite"; 
but we prefer to call it "undefined" (that is, it's not any real 
number) because of some problems that arise when you try to treat 
"infinity" as a number. Some of this will show up as we continue with 
your thinking, and other problems are discussed in our FAQ.

>Since it seems to be accepted that there is a negative infinity (no 
>one has challenged that yet), then 1/-infinity must equal -0. Why? 

One of the problems with talking about "infinity" is precisely this, 
that there is an ambiguity as to whether there is only one infinity or 
both + and -infinity. Think back to 1/0. You let the denominator 
approach 0 from above (positive), and saw the quotient "increase 
without bounds" as we say in math. But why not also let the 
denominator approach 0 from below? Then you'll get larger and larger 
NEGATIVE numbers, so that 1/0 would seem also to be -infinity. That 
means that + and -infinity must be the same thing. In fact, the same 
thing is happening here as in the tangent; both graphs look very 
similar near their points of discontinuity:

                          |* toward +infinity
                          |*
                          |*
                          | *
                          |  *
                          |   *
                          |     *
                          |        **** toward +0
    ----------------------+----------------------
    toward -0 ****        |
                    *     |
                      *   |
                       *  |
                        * |
                         *|
                         *|
        toward -infinity *|

This shows 1/x approaching "+infinity" as x approaches 0 from above, 
and "-infinity" as x approaches 0 from below; it also shows 1/x 
approaching "+0" or "-0" as x approaches "+ or -infinity."

Now, you're assuming that + and -infinity are different, and finding 
that +0 and -0 must be different. The question is, how would +0 and -0 
be different? If they're different numbers, then what is +0 - -0? If 
you can't do anything with these numbers - if all they are is 
meaningless names that have nothing to do with the rest of math - 
then you really haven't said anything. The fact is, we already know 
what -0 is: it's the same as 0. (That's certainly a law we don't want 
to break.) Therefore, by reversing your reasoning, + and -infinity 
must be the same in this case. (And this, of course, is the "unsigned 
infinity" you suggest.)

If we think of a single infinity rather than positive and negative 
infinities, then we're essentially thinking of the number line as a 
circle, with both "ends" meeting at a single "point at infinity." For 
some purposes this idea works; for other purposes it seems better to 
think in terms of + and -infinity. Since neither idea always works, we 
have to be extremely careful how we handle them. That's why we just 
call it undefined, and then use other techniques to deal with it more 
safely.

>I've found that my peers are more accepting of these (and other) 
>theories of mine than most adults are

Your peers accept this because they don't have the experience to see 
the difficulties raised by your ideas. But in fact, mathematicians 
have dealt with the issues you are raising, and have found that 
although treating infinity as a number results in contradictions such 
as +0 and -0 being different, they can still work with these concepts 
in other ways. For example, in calculus you will learn that we can 
work with "limits," which deal with how a function (such as tan x or 
1/x) behaves as x approaches some value. In cases like these, we 
actually use terminology like

            1                        1
      lim  --- = -inf   or    lim   --- = 0
     x->0-  x               x->-inf  x

Here "inf" would be written as the infinity symbol, and the + or - is 
written after the 0 to show that it doesn't mean a "negative zero," 
but rather the direction of approach, "from above" or "from below." 
The term -infinity is sometimes used (as a shorthand way to say "the 
function decreases without bound," without treating it as a number), 
but we never use -0 as a value, because it's the same as +0.

You may find this short article from Eric Weisstein's World of 
Mathematics interesting:

   Infinity
   http://mathworld.wolfram.com/Infinity.html   

You'll notice this says that "informally," 1/inf = 0, but that a 
"rigorous" (careful) treatment of this concept requires limits. In 
other words, your concepts are largely valid, but simply need to be 
treated very carefully in order to be useful and to avoid confusion.

Keep thinking! Many good ideas have come from ignoring what everyone 
"knows" and seeing what happens. Some parts of math are pretty hard to 
believe. Just remember to come back to reality from time to time and 
see what parts of your ideas work and which don't.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/