Is Zero Really a Number or Just a Concept?

Date: 09/15/2005 at 20:24:42
From: Swapnil
Subject: Zero: Just a concept like infinity?

Hi, for a long time my teachers have hammered into my brain that 
infinity is not a number, it's just a concept.  Now that I understand 
that, I wonder if zero is just a concept too?  I guess this is both a 
philosophical and a mathematical question. 

Philosophically, you can't really have a zero of things, because the 
thing which is zero doesn't really exist.  In fact, if you do count 
things that don't really exist, you will have an infinite number of 
those things since there is no limit to things that you can imagine 
existing (which is rather an interesting fact).

In mathematics, many definitions and rules go haywire when they try 
to deal with zero (such as 0^0, 1/0, 0/0).  The two most used bounds 
for evaluating the limit of a function are in fact 0 and infinity.

So it seems obvious that there is a deep intricate relation between
zero and infinity.  And like infinity, it seems that zero should just
be considered a concept.  Furthermore, by making zero a concept we
would be able to avoid writing those exceptions to rules where we have
to specify that the equation is true as long as something is not zero. 

I just thought of few more examples:

1) If in a game, your opponent has 1 point and you have two points 
then you can say that you have twice as many points as him.  Whereas 
if your opponent has zero points and you have two points, you would 
technically have to say that you have infinite times as many points 
as him.
2) You can't really have a circle with radius zero.
3) Zero is neither positive nor negative and neither it is even or 
odd.

Date: 09/15/2005 at 20:56:31
From: Doctor Peterson
Subject: Re: Zero: Just a concept like infinity?

Hi, Swapnil.

It's true that infinity and zero have many interesting relationships.
But zero is far less troublesome than infinity, and far more 
important.  We couldn't do much algebra without zero, because we
couldn't define the additive inverse (negative) without the additive
identity.  For example, in solving x+5=1, we add -5 to each side,
getting x+5-5 = 1-5.  The left side becomes x because 5-5=0, and 
x+0=x.  We couldn't do that without zero! There would be ways to get 
around it, I suppose, but they would be very awkward.  You'd lose a 
lot if you refused to do any arithmetic with zero, which is what we 
mean when we say that infinity is not a number.

Now, it's true that zero is "just a concept"--but the same can be said 
of all numbers.  You've never seen a three, have you?  You see three 
boys, three sticks, and so on, and you generalize from that to the 
concept of "three".

You can't have zero apples--but you also can't have -3 apples. 
(Actually, you can have both, if you think of it the right way.)  You
can't have a circle with radius 0, or with radius -3, either--so do
negative numbers not exist?  Zero and negative numbers are just
extensions of the concept of counting, taking the idea of "number"
beyond where it began.

The reason zero is neither positive nor negative is that positive
MEANS greater than zero, and negative means less than zero.  Zero is
not an exception there--it's what the whole idea of positive and 
negative revolves around (almost literally!).

Zero IS even--I hope no one taught you that it isn't!  And the fact
that it DOES fit neatly into the integers in ways like that is one
reason it is obviously one of them.

In general, zero works in MOST settings, while infinity works in
almost none of them.  You can't divide by zero, but you can add it; 
you can't even add infinity without getting in trouble!  So it makes 
good sense to work with the few peculiarities of zero, but to treat
infinity as something else, just a direction to go when you take a
limit, rather than an actual destination.

If you have any further questions, feel free to write back.


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

Date: 09/15/2005 at 21:20:14
From: Swapnil
Subject: Thank you (Zero: Just a concept like infinity?)

I would like to thank you, Dr. Peterson, for responding to my question
in such a short period of time and with such a good answer.  I really
appreciate your help.  Keep up the good work! :)  

-Swapnil