Variables: Connecting Letters and Numbers

Date: 12/06/2001 at 12:48:12
From: Elizabeth
Subject: Understanding the connection between letters and numbers in 
algebra

No matter how many books I read, I can't understand how x = whatever. 
I'm still trying to know how you make a connection. Is there some 
sort of table you can use to show what letter is equal to what number?

Date: 12/06/2001 at 14:10:56
From: Doctor Ian
Subject: Re: Understanding the connection between letters and numbers 
in algebra

Hi Elizabeth,

A variable is just a name you use to talk about a number whose value 
you don't know yet. 

Suppose I tell you that if you add John's age (in years) to Joan's age 
(in years), you get 27, and if you multiply their ages, you get 140. 

Now, you know that John _has_ an age. You just don't know what it is.  
And you know that Joan has an age, but again, you don't know what it 
is. 

However, you know some facts about their ages, and in order to express 
those facts, you need some way to refer to their ages. You could write 
something like

  John's age + Joan's age = 27

  John's age * Joan's age = 140

Now, this is more writing than you probably want to do, so you might 
choose shorter names.  For example, you might use 'J' to stand for 
John's age, since the J would remind you of 'John'. But this means you 
can't use 'J' for Joan's age, so you need to think up another name.  A 
somewhat natural choice would be the next letter of the alphabet, 'K'.  
Now we can write

  J + K = 27

  J * K = 140

At this point, it's important to realize that these are just arbitrary 
names. And a couple of important points follow from that:

1) We could use any other names without changing the meaning of the 
equations. For example, we could use 'X' instead of 'J':

   X + K = 27

   X + K = 140

And we could use 'Y' instead of 'K':

   X + Y = 27

   X * Y = 140

The particular variable names that we use don't make any difference, 
so long as within a given problem, the same variable name always has 
the same meaning. (It's sort of like in a novel. It doesn't matter 
what name is given to any character, so long as each name stays with 
the same character throughout the book.)

For that matter, you don't even have to use letters for names. You 
could use pictures if you wanted to. But letters have a few nice 
things going for them. For one, most people can recognize them, and 
can agree on how they should be pronounced. Also, most people already 
know how to write them, or how to type them with a keyboard. Also, 
they aren't used for operations (like addition or multiplication), so 
if you use a letter in an equation, people reading the equation can 
have a lot of confidence that it is a variable name.  

2) The value that turns out to be associated with a particular 
variable name can be different from problem to problem. For example, 
in this problem, it turns out that J and K have the values 20 and 7:

  20 + 7 = 27

  20 * 7 = 140

But here is another problem that uses the same variable names:

  John has twice as many cookies as Joan. If he gives her two 
  cookies, they'll have the same number of cookies.  

  Let J stand for the number of cookies that John has, and let
  K stand for the number of cookies that Joan has. 

          J = 2K          John has twice as many cookies

      J - 2 = K + 2       A transfer of two cookies makes them equal.

In this case, it turns out that J has the value 8, and K has the 
value 4:
          8 = 2 * 4

      8 - 2 = 4 + 2

So there is no table, or other device, that you can use to simply 
'look up' the value of a variable based on the name that has been 
given to it. Is J equal to 29, or 7, or 8?  It depends on the 
particular problem being solved. 

(Again, it's sort of like with books. If you see the name 'Chris' 
being used as a character in one book, and you see the same name being 
used in another book, you don't assume that they refer to the same 
character. And you don't know anything about the character until 
you've started reading the book.)

Because there is no connection between the name of a variable and the 
value that it takes, it's common to just use the same names over and 
over - x, y, and z, for example; or n, or k; or a, b, and c.  

As a way of getting used to the idea of variables, you might consider 
using entire names instead of letters. For example:

   A garden is 6 feet longer than it is wide. The perimeter of the
   garden is 30 feet. What are the dimensions of the garden?

     1. Write an expression for the perimeter:

        30 = perimeter

           = length + width + length + width
   
           = 2*length + 2*width

           = 2(length + width)

      2. Write an expression for length in terms of width:

         length - width = 6

                 length = 6 + width

      3. Substitute and solve

          30 = 2((6 + width) + width)

             = 2(6 + 2*width)

             = 12 + 4*width

          18 = 4*width

         18/4 = width

Now, when you see something like this, you don't get the feeling that 
you can just 'look up' the value of 'width', do you?  It's pretty 
clear that the width could be any value, and your job is to find out 
what value makes the equations true. 

Once you get used to working with words as variable names, you'll get 
more comfortable with the idea of variables in general; and eventually 
you'll decide, in your own time, to start using letters instead of 
whole words.  It's a little like learning to stand up before learning 
to walk. 

You might also find it useful to read these answers from the Dr. Math 
archives:

   What is Algebra?
   http://mathforum.org/dr.math/problems/jason.07.20.01.html   

   Letters for Variables
   http://mathforum.org/dr.math/problems/spencer.11.19.01.html   

I hope this helps.  Write back if you'd like to talk more
about this, or anything else.

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