Natural Numbers, Positive Integers

Date: 04/07/97 at 10:25:22
From: Eric Manalastas
Subject: Natural Numbers, Positive Integers

My Algebra book says that there is a difference between the set of 
natural (counting) numbers, i.e. {1,2,3,...} and the set of positive 
integers, which should be {1,2,3,...}.  The book doesn't elucidate, 
saying that that discussion would be too difficult for the reader.  I 
can detect no obvious difference.  I know zero is neither positive nor 
negative so it's not an element of the latter.  What gives?

Date: 04/07/97 at 15:36:06
From: Doctor Wilkinson
Subject: Re: Natural Numbers, Positive Integers

The distinction is one that isn't going to matter even to 
mathematicians most of the time.  It matters only when we're concerned 
with what's called "the foundations of mathematics."  A number of 
mathematicians in the late nineteenth century started to wonder about 
how mathematics could be built up from the simplest possible 
foundations.  The starting point was what we call the "natural 
numbers."  They're called "natural" because they're so universal and 
apparently so obvious that they seem to be given to us by nature.  
Everybody in every culture has always known how to count.  But not 
every culture has always had negative numbers, fractions, etc.  

Now if you start with just the natural numbers, you don't have any 
zero or any negative numbers. Zero and negative numbers were really 
invented by mathematicians so they could do subtraction without having 
to think about whether or not the number they were subtracting was 
smaller than the number they were subtracting from. All this happened 
a very long time ago, of course. But these nineteenth-century 
mathematicians started worrying about where these new numbers came 
from and they figured out a way to manufacture the new numbers out 
of the natural numbers. The new set of numbers they constructed were 
the integers (positive, negative, and zero). And lo and behold, the 
positive integers, that is those new numbers which happened to be 
greater than zero, turned out to form a system that looked exactly 
like the old system of natural numbers. It had exactly the same 
properties, and its smallest member behaved exactly like the natural 
number 1, even though it was built by a somewhat complicated 
construction and was by no means identical to the natural number 1.

Similar methods were used to build rational numbers out of integers 
and real numbers out of rational numbers.  At each stage of the 
process the numbers of the previous stage acquire images in the new 
set of numbers which behave exactly like their predecessors but are 
technically not identical to them.

Once this whole structure has been built up, we can pretty much forget 
about it and go on with our lives, treating the natural numbers, the 
integers, and the rational numbers as all just parts of the bigger 
structure of real numbers.

-Doctor Wilkinson,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/