Whole Numbers and Multiples

Date: 09/11/2001 at 23:29:00
From: Julie Wimberly
Subject: Whole Numbers and Multiples

Dr. Math,

I am a 6th grade math teacher working on common factors and multiples. 
In my advanced class my students and I have been trying to figure out 
what numbers are in the set of "Whole Numbers."  I thought I knew the 
answer {1, 2, 3, ..} but then I questioned myself (and the kids 
questioned me) about the number 0. Is 0 a whole number?

Why do we need to know?

Well, the definition of multiples in our book says that multiples are 
products of any given whole number and another whole number. Then it
gives an example with the first multiple of 3 being 3. BUT if 0 is a 
whole number, wouldn't the first multiple of all numbers be 0 and not 
the number itself? Then what about the LCM's?  They would all be 0. I 
have found different sources that say 0 is the first multiple of all 
numbers... so which is right??      

I looked in several math books and found no definition of whole 
number.  Then I went to a "very mathematical" math dictionary - it 
said that whole numbers were  1) the numbers in the set {1, 2, 3, ...}   
2) the numbers in the set {0, 1, 2, 3, ...}  or  3) the numbers in the 
set {... -3, -2, -1, 0, 1, 2, 3, ...}.  So... which is it? Are there 
many different interpretations of whole numbers? If so how do you 
decide which one to adopt?

Please help my class and me figure out this problem!

Thank you,
Julie Wimberly

Date: 09/12/2001 at 09:24:08
From: Doctor Peterson
Subject: Re: Whole Numbers and Multiples

Hi, Julie.

It's not uncommon in math to have slightly different definitions 
depending on context. The definition of "whole number" is notoriously 
variable; that's why mathematicians don't use it formally. (Educators 
use some terms that mathematicians don't even know, much less defend.) 
The most popular definition is probably that the natural numbers are 
1, 2, and so on, while the whole numbers also include zero. The 
integers (your third case) are not formally called whole numbers in my 
experience, but the root meaning of the word "integer" is "whole," and 
I sometimes call them "signed whole numbers."

Your problem is not really whether zero should be considered a whole 
number, but whether your definition of LCM should use that word, or 
the word "first." Mathematicians are careful in their definitions, and 
don't resort to such sloppy terms. I'll give my definitions:

First, a multiple is in fact any product of an INTEGER and the given 
number; there is no "first" or "smallest" multiple.

Next, the Least Common Multiple is defined in the Harcourt Academic 
Press Dictionary (listed in our FAQ as

   http://www.harcourt.com/dictionary/browse/19/   

   "the smallest positive integer that is divisible by each integer
    in the given set."

Note how much more careful this is than "the first multiple." Also, 
note that we don't use "whole number," but "positive integer," which 
is defined unambiguously. If your text defines either "whole number" 
or "natural number" clearly with this meaning, then it can use it in 
place of the phrase; but we tend to avoid it in general. (The trouble, 
of course, is that younger students haven't yet been introduced to the 
term "integer," so they have to use terminology at their level. I 
would much prefer to go ahead and teach about integers from the start, 
rather than have to learn better terms later.)

You may note that this dictionary (which is far from complete, 
especially at an elementary level) does not even define "whole"!

This sort of problem occurs surprisingly often here at Dr. Math. 
Although some of us are teachers and know the current educationese 
jargon, many are mathematicians or others who use math, and we know 
mostly the math terminology actually used by adults. Kids are taught 
words, or meanings for words, that are deficient, invented only for 
temporary use in teaching, by people who may not have a good grasp of 
the real mathematical significance of what they are doing. 
Inconsistencies such as you have found are the natural result. We need 
to find ways to be clear and accurate, yet without overwhelming kids 
with overly technical language, and that's not easy.

Back to whole numbers: Looking through our site, I find both 
definitions in use. In the Dr. Math FAQ, we seem implicitly to take it 
as excluding zero:

   http://mathforum.org/dr.math/faq/faq.integers.html   

   Integers are the whole numbers, negative whole numbers, and zero. 

Here, we include zero:

   Tree Diagram for Math Numbers
   http://mathforum.org/dr.math/problems/hallock10.5.97.html   

   Numbers
   http://mathforum.org/dr.math/problems/greeno2.12.10.97.html   

   Is Zero a Number?
   http://mathforum.org/dr.math/problems/tratclif10.23.html   

Here, we use my informal phrase "negative whole":

   The Real Number System in a Venn Diagram
   http://mathforum.org/dr.math/problems/amy9.8.98.html   

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