Adding an Even and an Odd Number

Date: 03/26/2003 at 19:21:39
From: Debbie
Subject: Addition

Can you think of any two numbers, one even and the other odd, whose 
sum is even?

My son is in the third grade, and this was one of his homework 
questions. We have tried different combinations of even and odd 
numbers and can't come up with an even number answer.

Date: 03/27/2003 at 00:58:37
From: Doctor Ian
Subject: Re: Addition

Hi Debbie, 

It depends on how tricky you want to get. If you're just using 
standard arithmetic, you're right. No such pair of numbers exists:

   19 Odd Number Puzzle
   http://mathforum.org/library/drmath/view/57207.html 

However, see:

   Can the Sum of 5 Odd Numbers Equal an Even Number?
   http://mathforum.org/library/drmath/view/61945.html 

Probably, though, the point of the exercise is to help your son get
the idea that it's not enough to just try a bunch of examples. If you
want to make a claim about _all_ numbers, then you have to construct
more general explanations. 
 
For example, suppose I ask you this:

   I'm thinking of a number, say 3.  I'm going to multiply
   it by itself: 3 * 3 = 9.  And then I'll multiply it by
   itself twice: 3 * 3 * 3 = 27.  

   Can you think of any number for which the second product
   will be less than the first?

Now, you might try a bunch of examples:

  3 * 3 < 3 * 3 * 3

  5 * 5 < 5 * 5 * 5

  21 * 21 < 21 * 21 * 21

and you might conclude, from these examples, that there is no number
for which the second product is less than the first.  But you'd be 
wrong:

  -1 * -1 > -1 * -1 * -1

Also, 

  1/2 * 1/2 > 1/2 * 1/2 * 1/2

So you can _never_ base a general statement about all numbers on a
finite number of examples using particular numbers, because there is
always a chance, no matter how small, that you have forgotten to look 
at the numbers that would prove you wrong.  And understanding that is 
the first step towards understanding what proofs are, and how they 
work. 

Does that make sense? 

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

Date: 04/02/2003 at 21:05:36
From: Debbie
Subject: Thank you (Addition)

Thank you.  I love your site.  It is very informative, and I use it 
very often when helping my son with his homework.  Again, THANKS!