Properties of Algebra

Date: 5/8/96 at 19:2:31
From: Justin March
Subject: Properties of Algebra

What are the properties used for algebra? 
What about the "distributive property"?

Date: 10/16/96 at 19:16:47
From: Doctor Lani
Subject: Re: Properties of Algebra

Dear Justin,

When we're working with real numbers (integers, fractions, 
decimals, roots, and numbers like Pi), we like to say that 
certain things will always be true.

When we find these things, we give them the name "property."  The 
example you give, the distributive property, is something that 
works for *any* set of real numbers.  Because of this, properties 
are often stated with variables, which say that you can put any 
number in and that  I'll use a, b, and c in this answer.

Now, for some examples:

Some properties true for multiplication and addition:
  
 THE COMMUTATIVE PROPERTY: a + b = b + a
                           a * b = b * a
 Basically, this says that you can switch the numbers that you 
are adding or multiplying and your answer stays the same: 
2 + 3 = 3 + 2; 57 * 22.3 = 22.3 *57


 THE ASSOCIATIVE PROPERTY: (a + b) + c = a + (b + c)
                           (a * b) * c = a * (b * c)
 Since parentheses mean "do this first," this property just tells 
you that you can add or multiply in any order without changing 
your answer: (2 + 3) + 6          2 + (3 + 6)
              \  /                     \  /
              = 5    + 6        = 2 +   9    
              = 11              = 11

Sometimes students mix up these names.  I just think of the 
words: "commute" means going from one place to another and 
"associate" means hanging out with. Numbers commute when they 
switch places and they associate in parentheses.  

Also, students sometimes think that properties are silly or 
obvious.  Well, test these properties with subtraction and 
division!

Now, finally, I'll explain distribution.  The official name is:

DISTRIBUTION OF MULTIPLICATION OVER ADDITION:

a * (b + c) = a*b + a*c

This one is best explained by a couple of examples.

(1) Have you ever had to multiply something like 4 * 52 in your 
head? If so, you might have used the distributive property!  
Like this:

4 * 52 = 4 * (50 + 2) = 4 * 50  +  4 * 2 = 200 + 8 = 208

It's easier to multiply 4 by 50 and 4 by 2 and then just add the 
answers.  If you know that trick, you know the distributive 
property!

(2) Why does it work?  Well, let's look at another example:

     (a)  3 *(2 + 5) = (2 + 5) + (2 + 5) + (2 + 5)
This is true because multiplying by 3 means adding the number three 
times.
   
     (b)  (2 + 5) + (2 + 5) + (2 + 5) = 2 + 5 + 2 + 5 + 2 + 5
This works because the associative property tells me it doesn't 
matter which order I add in, so I don't need the parentheses.

     (c)  2 + 5 + 2 + 5 + 2 + 5 = 2 + 2 + 2 + 5 + 5 + 5
This is true because the commutative property tells me I can move 
stuff around without changing the answer.

     (d)  2 + 2 + 2 + 5 + 5 + 5 = 3*2 + 3*5
This is true because adding a number three times is the same as 
multiplying by 3.

If you look at the first line of (a) and the last line of (d) 
you'll see that 3*(2 + 5) = 3*2 + 3*5.  This is called a proof.  
It should convince you that the distributive property works!

There are a lot more properties (identity, reflexive, 
distribution of multiplication over subtraction...).  You can 
find a good list in any algebra book.

Good luck!

-Dr. Lani  The Math Forum
 Check out our web site  http://mathforum.org/dr.math/