Is Zero a Number?

Date: 07/05/2003 at 15:37:20
From: Joe
Subject: Zero and infinity

If infinity is not a number, then is zero really a number?  I see that 
it is not recognized as a real number but as a whole number, integer, 
etc. It seems as though zero has been accepted as a number and 
infinity has been accepted as a concept. This question stems from an 
argument about 1/0 = infinity.

Date: 07/05/2003 at 20:27:30
From: Doctor Jaffee
Subject: Re: Zero and infinity

Hi Joe,

Zero is a number; in fact, it is a real number.  It is on the number 
line right between 1 and -1. You can add, subtract, and multiply with 
0 and get real answers. You can divide numbers into zero and get a 
real answer, zero.

You can't say anything like that about infinity.  It is not on the 
number line and you can't do computations with it.

Now, consider 1/0. You know that 1/1 =1, 1/0.1 = 10, 1/0.01 = 100,
1/0.001 = 1000, etc... Pick a power of 10 as large as you want and I 
can find a number larger than 0 that I can divide into 1 and get your 
number as a result.

In other words, as we divide numbers into 1 and those numbers get 
closer and closer to 0, the quotient gets larger and larger with no 
boundary. We conclude then, that 1/0 = infinity.

However, that is just a shorthand notation.  Actually, division by zero 
is undefined.  It is more precise to say that

  Limit 1/x = oo            As x gets closer to zero, the value of 1/x
   x->0                     grows without bound (i.e., approaches infinity)

Unfortunately, often people will use the shorthand, without making it clear
that this is what's going on.  So other people see what they've written, and
think that '1/0 = infinity' is an actual statement of fact, when it's not. 

In the same way, people will often write '1/infinity = 0', instead of 
the more precise

  Limit 1/x = 0             As x grows without bound (i.e., approaches
   x->oo                    infinity), the value of 1/x gets closer to 0. 

But '1/infinity = 0' is also untrue.  

For more about dividing by zero, see the Dr. Math FAQ:

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

I hope this explanation helps.  Write back if you want to discuss the 
problem any more or if you have other questions.

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