Can Even and Odd Numbers Be Negative?

Date: 04/11/2006 at 07:12:07
From: Sarah
Subject: definition of even and odd number

Can even or odd numbers be non-positive?  For example, is 0 even?  Is
-4 even?  Is -5 odd?

Some teachers say that even and odd numbers do not include negative
integers, but some teachers say they are included.

Date: 04/11/2006 at 08:04:09
From: Doctor Rick
Subject: Re: definition of even and odd number

Hi, Sarah.

No problem arises when the definition of even and odd is extended from 
the positive integers to all integers, so I see no reason not to do 
so.  An even number is an integer that is divisible by 2 (that is, it 
can be written as 2 times some integer); an odd number is an integer 
that is not divisible by 2.

In contrast, if we try to extend the concept to non-integer rational 
numbers, we encounter difficulties; any definition that is self-
consistent is not particularly useful.  Therefore we restrict evenness 
and oddness to integers; the number 1/2 is neither even nor odd.

I'd like to hear the reasons that those teachers give for not 
including negative integers.  This question comes up a lot (usually in 
the form "is zero even, odd, or neutral?"), and I have yet to see a 
good reason.

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

Date: 04/12/2006 at 07:06:55
From: Sarah
Subject: definition of even and odd number

Some teachers say that we only need the concept of even and odd 
numbers when we count.  It is meaningless to extend our discussion of
even and odd to non-positive integers.  Are these teachers right?
What is the use of extending the discussion of evenness and oddness 
to non-positive integers?

Date: 04/12/2006 at 08:38:27
From: Doctor Rick
Subject: Re: definition of even and odd number

Hi, Sarah.

I think we must distinguish between "meaningless" and "useless". 
Extending the definition of evenness and oddness to non-positive 
integers is not meaningless, because it is self-consistent; as long as 
a definition doesn't break anything, it imparts meaning "by 
definition".  "Useless," on the other hand, is a word that pure 
mathematicians don't care for; many mathematical concepts were 
developed without thought of whether they had any use, but often 
applications of the concepts have appeared long afterward.

The terms "even" and "odd" do have their origin in counting: if you 
count 22 objects into two piles, the piles are the same size (even), 
whereas if you count 23 objects into two piles, you get one left 
over (odd).  The most frequent usage of the words "even" and "odd" is 
certainly in the domain of non-negative integers.  (I want to include 
0 as an even number, because if you have no apples and you divide 
them between two people, you don't have any apples left over.)

I could point out other ways in which mathematicians use the words 
"even" and "odd", without reference to numbers at all: there are even 
and odd functions and even and odd permutations.  Both grow out of the 
non-negative-integer sense of even and odd: a polynomial consisting of 
only even powers of the variable (including 0) is even; a permutation 
that can be expressed as a product of an even number of transpositions 
(including no transpositions) is even.

My principal difficulty with those who ask, "Is -2 an even number?" 
and answer "no", is that once a negative integer is brought into the 
discussion, there is no reason to say it isn't even or odd.  The only 
reason to restrict the even and odd numbers to positive integers is 
when no other numbers are in view; for instance, when counting or when 
factoring numbers.  Then, even numbers mean *positive* even numbers, 
simply because positive numbers are the only numbers that exist, for 
the present purpose.  But when you ask about a negative number, it is 
natural to apply the definition to that domain: an even integer is one 
that can be expressed as the product of 2 with some integer (not only 
a positive integer).  What would be gained by saying that there are 
three classes of integers: even integers, odd integers, and negative 
integers?

One interesting thing about the even integers is that this set is 
closed under addition and multiplication: if you add two even 
integers, you get an even integer, and if you multiply two even 
integers, you get an even integer.  The set of even integers is thus 
a useful example in abstract algebra.  Looking through my abstract 
algebra textbook, I find this: "Let E = {even integers} = {2n | n is 
in Z}.  Then (E, +, *} is an infinite commutative ring without 
identity where + and * are ordinary addition and multiplication." 
There is one example of mathematicians applying the term "even" to 
all integers, and also one demonstration of the usefulness of the 
concept to mathematicians.  The set of even integers is interesting. 
To me, that is sufficient justification.

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

Date: 04/13/2006 at 07:54:21
From: Sarah
Subject: Thank you (definition of even and odd number)

Thank you very much for answering my questions in detail.  I learned a
lot!