Is Zero a Signed Number?

Date: 01/09/2002 at 10:27:50
From: Intisar hussain
Subject: Zero

Please tell me, Dr. Math,

Is zero a directed number? What is the direction of zero? If zero is 
not a directed number, why do we use it in the set of directed 
numbers?

Example: When we place directed numbers in line:

   -4, -3, -2, -1, 0, +1, +2, +3, +4

Date: 01/09/2002 at 13:07:05
From: Doctor Peterson
Subject: Re: Zero

Hi, Intisar.

Probably you are using "directed numbers" to mean something like 
"signed numbers"; it appears that this is a British school term for 
what we would properly call "integers" or "real numbers," with the 
emphasis on the presence of a sign. It is meant to be descriptive, not 
a precise definition, and I think you may be taking the term too 
seriously.

I would say that "the set of directed numbers" refers to numbers in 
which we are allowing both a length and a direction (sign). In the 
case of zero, the direction is meaningless, since +0 and -0 are the 
same; but that does not mean that it has no sign, only that the sign 
makes no difference. 

The important point about the set is not that each member of the set 
_must_ have a significant "direction," but that directions are 
_allowed_, so that the set does not consist only of positive numbers. 
That is, no claim is made that you can separate out the size and 
direction for every such number, so that each number (including zero) 
should have a specific direction; rather, numbers in the set are built 
by combining a "size" and a "direction" (sign), and there is nothing 
wrong with both +0 and -0 turning out to be the same number.

Moreover, I would not even say that any particular number is "a 
directed number"; rather, a number like 1 (or 0) may be treated either 
as a mere number (by children who have not yet learned about negative 
numbers, or when only positive numbers make sense), or as a "directed 
number" in contexts where signs are meaningful. It is really only "the 
set of directed numbers," or "operations on directed numbers," that 
are significant, not the individual numbers.

If we tried to formally define the "integers" (directed whole numbers) 
or the "real numbers" as numbers that combine size and direction, we 
would have difficulty in stating clearly what we mean. But if you are 
using "directed numbers" just to indicate that you are working with 
numbers with (optional) signs, and not as a formal definition of a 
set, then there should be nothing wrong with accepting that zero 
belongs in this set.

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