Why is Zero Not a Natural Number?

Date: 03/14/2001 at 00:28:33
From: Laura
Subject: Why is Zero not considered a natural number?

Hello!

I don't understand why zero is not considered to be a natural number.  
Natural numbers are our typical counting numbers and I have learned 
that in mathematics we count beginning at zero. Correct? Can you 
explain why the natural numbers are {1, 2, 3.......}?

Also, is there a way to prove that zero is an integer?

Thanks!
Laura

Date: 03/14/2001 at 13:14:09
From: Doctor Ian
Subject: Re: Why is Zero not considered a natural number?

Hi Laura,

Here are a couple of Dr. Math references for you to check out before I 
go on:

  Natural Numbers, Positive Integers - Dr. Math archives
  http://mathforum.org/library/drmath/view/55753.html   

  Integers, Rational and Irrational Numbers - Dr. Math FAQ
  http://mathforum.org/dr.math/faq/faq.integers.html   

Actually, counting from zero is _not_ a very natural thing to do!  It 
takes a certain amount of sophistication to realize that you can have 
a set containing zero elements, or that you need a number to describe 
such a set in the first place. Consequently, most numeral systems from 
history (e.g., Roman numerals) don't contain a symbol for zero, or any 
way to represent the concept.  

The most 'natural' way to count is by making a mark for every item 
counted. If you have seven sheep, you write

  |||||||

If you have two sheep, you write

  ||

If you have no sheep at all, what do you write? Nothing at all! If 
there's nothing to keep track of, why would you need a number to keep 
track of it? 

The concept of 'zero' may seem natural to you, but that's only because
you're growing up surrounded by people who already know about it. 
Similarly, it probably seems natural to you to think of the earth as 
being round, but it took a long time for people to figure that out, 
too.

As for 'proving' that zero is an integer, it's not really necessary, 
since the set of integers is whatever we say it is. If we say that the 
set includes zero, then it includes zero. (Proving that zero is an 
integer would be like proving that people are members of a club. Are 
they on the membership list?  Okay, then they're members.)

And anyway, all the other integers are _defined_ in terms of zero and 
one, so zero had better be an integer, or we're in big trouble!

Having said all that, I should point out that agreement on this is
not universal.  One of our other math doctors, Doctor Floor, points
out that in the Netherlands, it is generally taught that the natural
numbers are {0, 1, 2, ...}.  Probably the best thing to do is to 
avoid the term 'natural' altogether, and use the term 'positive' 
or 'non-negative' to specify unambiguously which set you're referring
to in any given context. 

I hope this helps.  Write back if you'd like to talk about this some 
more, or if you have any other questions. 

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