Why Do We Use Variables in Algebra?

Date: 01/31/2004 at 19:38:16
From: Chrystyne
Subject: Use of letters for numbers

Why do we use letters to represent numbers in algebra?  I think that
the letter represents a number that we do not have the answer to.

Date: 01/31/2004 at 23:36:55
From: Doctor Peterson
Subject: Re: Use of letters for numbers

Hi, Chrystyne.

There are several slightly different reasons for using variables,
which are all variations on the idea you suggested:

  1. We don't know the value, and we want to find what it is.

     For example, we might want to solve the equation

       3x = 6

     to find the height of a rectangle whose width is 3 units and
     whose area is 6 square units.  We are finding what value makes
     that equation true.

  2. We don't YET know the value, and want to be able to make a
     formula so we can use the value when we get it.

     For example, we might be preparing to find the areas of a set
     of different rectangles whose width is 3 units but whose
     heights will vary.  We can write the expression

       A = 3x

     to remember how to find the area of each rectangle when we get
     its height.  This is how computers usually use variables; when
     you enter a value for a variable, a program can calculate a
     result using an expression like this.

  3. We know the value, but want to do all the algebraic
     manipulation (such as simplifying an expression) without using
     actual numbers, to save work.

     For example, rather than calculate (sqrt(1.234))^2 -- that is,
     the square of the square root of 1.234 -- I might first think
     of it as (sqrt(x))^2, replacing the known value with a variable
     temporarily so I can simplify the expression to "x", and not
     have to take a square root at all.  That's too simple an
     example, but this sort of thinking is useful in many big
     problems, where solving the problem in terms of variables
     rather than numbers can save a huge amount of work and errors.

  4. We want to prove that something is true regardless of the value
     of the numbers involved.

     For example, we can show that

       ab = ba

     for all numbers a and b; we don't need to ever know what those
     values are in order to talk about this fact.

In summary, any time we want to talk about a number without having to 
know its value at the moment, we can give it a name and talk about it 
using that name.  It's not much different from using your name rather 
than a picture of you when you write about yourself; or using the 
pronoun "she" or the noun "occupant" or "user" instead of your name 
in a document that might have been written with reference to anybody. 
It allows us to talk about things that are very hard to talk about in 
any other way.

Here are some places where we have explained the usefulness of variables:

  Understanding Variables
    http://mathforum.org/library/drmath/view/57311.html 

  Variables: Connecting Letters and Numbers
    http://mathforum.org/library/drmath/view/57312.html 

  Why Use Letters in Math?
    http://mathforum.org/library/drmath/view/57020.html 

  The Why of Algebra
    http://mathforum.org/library/drmath/view/57015.html 

If you have any further questions, feel free to write back.

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