What is Algebra?

Date: 07/20/2001 at 13:43:47
From: Jason
Subject: Algebra

What is algebra?

Date: 07/20/2001 at 15:45:08
From: Doctor Ian
Subject: Re: Algebra

Hi Jason,

Algebra is like arithmetic, but some of the numbers have names instead 
of values.  

For example, if I ask you something like

  3 + 4 * 5 - 6 / 3 = ?

you can just apply the operations in the correct order to get

  3 + 4 * 5 - 6 / 3 = ?

     3 + 20 - 6 / 3 = ?

         3 + 20 - 2 = ?

             23 - 2 = ?

                 21 = ?

Now, suppose that instead, I ask you something like

  (x + 3) * (x + 4) = 42

Now you can't really apply the operations, because you don't know the 
values of the numbers. What's x + 3? It depends on what the value of 
x is, doesn't it? 

In this case, you might start guessing possible values for x that 
would make the equation true:

  x = 1?     (1 + 3) * (1 + 4) = 4 * 5 = 20    (No.)

  x = 2?     (2 + 3) * (2 + 4) = 5 * 6 = 30    (No.)

  x = 3?     (3 + 3) * (3 + 4) = 6 * 7 = 42    (Yes!)

However, suppose the problem changes to 

  (x + 3) * (x + 4) = 35.75

Now it becomes a lot harder to guess an answer. What 'algebra' gives 
you is a set of tools for figuring out the answers to problems like 
this, without having to guess. 

This becomes more and more important as you start using more 
complicated equations, involving more than one variable.  

So why should anyone care about this stuff?  Take a look at the Dr. 
Math archives:

  What Are Variables For?
  http://mathforum.org/dr.math/problems/cindy.02.09.01.html   

to see what makes variables so useful. 

Does this help?  Write back if you'd like to talk about this some 
more, or if you have any other questions. 

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