Properties of Real Numbers

Date: 05/20/97 at 19:46:14
From: Sarah Mahlstedt
Subject: Properties of real numbers

I have a problem that is all about properties of real numbers. I don't 
get it at all!  Help explain what they are!  Particularly, what are 
associative properties, commutative properties, and the zero product 
property?

Date: 05/22/97 at 19:59:18
From: Doctor Reno
Subject: Re: Properties of real numbers

Hi Sarah!

These properties are simply fancy names for mathematical ideas that 
you probably already know very well.  Let's look at them one at a time 
to learn more.

ASSOCIATIVE PROPERTY

There are two associative properties, actually, but don't let that 
upset you!  One is the associative property of addition, and the other 
is the associative property of multiplication.  These are sometimes 
also called "grouping properties," and you will see why soon.

Let's say that you want to add 3 numbers: 25, 75, and 30.  We can add 
these up two different ways.  We can add the first two: 25 + 75 = 100. 
Then we can add that sum (100) to 30 and have a new sum of 130 
(100 + 30 = 130).  But we could also add them in this way: 
75 + 30 = 105, and then 105 + 25 = 130.  The answer is the same. 

All that the associative property of addition says is that we will 
always get the same answer no matter how we "group" numbers when we 
add them!  That's it!  But we knew that, didn't we? It seems obvious. 
Sometimes mathematics is like that...it gives big, fancy names to 
things that seem easy and that we already know.  In the fancy language 
of mathematics, the associative property of addition says that if a, 
b, and c are any whole numbers, then (a + b) + c = a + (b + c).  We 
have shown that to be true in my example above.

The associative property of multiplication is written in the same 
fancy mathematical language: For any whole numbers a, b, and c, 
(a x b) x c = a x (b x c). 

Once again, we all know and understand this property. If we want to 
multiply three numbers together....let's say 3 x 5 x 8.....we can do 
this two ways:

 3 x 5 = 15
15 x 8 = 120...which is to say that 3 x 5 x 8 = 120.

OR...

 5 x 8 = 40
40 x 3 = 120...which is to say that 3 x 5 x 8 = 120!  The same answer!

And that's all there is to the associative properties!  We use these 
properties regularly to solve equations and arithmetic problems, but 
they are so familiar to us that we may not even realize that we use 
them!

COMMUTATIVE PROPERTY

The commutative properties are more fancy names for stuff you already 
know about math!  And once again, there are two of them..one for 
addition and one for multiplication. 

These are the easier properties to remember.  They simply say that 
if 2 + 5 = 7, then 5 + 2 = 7; and if 2 x 5 = 10, then 5 x 2 = 10. 
That's it! 

The fancy words go like this: If a and b are any whole numbers, 
then a + b = b + a.  For multiplication, it says: For any whole 
numbers a and b, a x b = b x a. That seems pretty obvious, doesn't it?

ZERO MULTIPLICATION PROPERTY OF WHOLE NUMBERS

This property is the easiest one of all! All it means is that whenever 
you multiply a number by zero, you get zero for an answer! That's all! 

Of course, we have to say it differently in math...For any whole 
number a, a x 0 = 0 = 0 x a.

That's all there is, Sarah! Relax and enjoy your math, and remember 
that we have to write these simple ideas in this fancy language in 
order to be exact and precise.  Thank you for writing to Dr. Math, and 
be sure to write again if you have any further problems!

-Doctor Reno,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/