Additive Identity and Other Properties

Date: 01/03/99 at 19:50:17
From: Elizabeth
Subject: Properties

I don't understand properties. Can you explain the identity of zero, 
the commutative property, and the multiplicative properties?

Date: 01/04/99 at 12:40:21
From: Doctor Peterson
Subject: Re: Properties

Hi, Elizabeth. I'm not sure which multiplicative properties you mean; 
we talk about the multiplicative property of equality, about the 
commutative and associative properties of multiplication, and about 
the distributive property of multiplication over addition, and you 
might mean any of those. I hope my explanation of the others will help 
you. Write back if you need more.

We call zero the additive identity because when you add it to anything, 
the result is identical to the original number - you haven't changed 
it. This means that

    n + 0 = 0 + n = n     for any number n

There's no other number for which you can say this, so zero is very 
special.

Similarly, there is a multiplicative identity. It is a number you can 
multiply anything by and not change it. What is that? It will be the 
number X in this statement (where "*" means multiply):

    n * X = X * n = n   for any number n.

The commutative property means that it doesn't matter which order you 
do things in. For addition, this means that

    a + b = b + a      for any numbers a and b.

You can do the same thing with multiplication.

These are things you've probably understood for years, but haven't 
thought of as "properties." You just knew that 5+6 and 6+5 are the 
same. They are important because not everything you can do has an 
identity, or is commutative. Suppose we picture operations like 
addition and multiplication as machines with two input pipes and one 
output pipe:

      a b
     in in
      | |
    +-+-+-+
    |     |
    |  +  |
    |     |
    +--+--+
       |
      out
      a+b

Commutativity says that if you switch the two input hoses, the same 
thing will come out. That's true for addition. It's also true if the 
machine is a paint mixer that takes blue and yellow paint, or yellow 
and blue paint, and makes green. But it's not true for every machine 
you can imagine. Maybe the left pipe has to take water and the right 
air. It probably wouldn't work right if you switched them! 

When we list properties like these, we're reminding ourselves of all 
the ways in which addition "works nicely," all the things we can depend 
on when we work with numbers. That's what a property is.

You might want to look through our archives. You can search for words 
like "commutative" and find lots of helpful explanations like this:

   Meanings of Properties
   http://mathforum.org/dr.math/problems/colleen7.23.97.html   

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