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### Additive Identity and Other Properties

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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?
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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;
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/
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Associated Topics:
Elementary Definitions
Elementary Multiplication