Why Use Letters in Math?

Date: 12/01/1999 at 12:52:34
From: Jamie Funk
Subject: Algebra 1 question

I've been in Algebra for lots of years and I haven't quite grasped the 
concept of why we need to use letters in math.

Date: 12/01/1999 at 16:45:18
From: Doctor Rick
Subject: Re: Algebra 1 question

Hi, Jamie.

We don't really need to use letters, but we need to use some sort of 
symbols in order to write down what we're thinking. Numerals are 
symbols for numbers that we know. We usually use letters as symbols 
for numbers that we don't know - which we call variables. (Sometimes 
we use a letter as a name for a number that we do know, for instance 
the Greek letter pi for 3.14159...)

Letters in other alphabets (Greek, Hebrew, or old English) are also 
sometimes used for variables in math. In computer programming, we 
often use whole words as names for variables. You could make up your 
own symbols, but then we wouldn't know how to say them.

Are you wondering why we need variables in the first place? There are 
two big reasons. First, they let us write formulas that we can use 
over and over. For example, we have this formula for the area of a 
rectangle:

     A = lw

where A is the area of a rectangle, and l and w are its length and 
height. One formula tells us how to find the area of any rectangle. 
What would we do if we could only write numbers?

Second, variables harness the power of the properties of numbers. This 
is the power: since every number has the same properties 
(commutativity, associativity, etc.), you don't need to know what a 
number is in order to know how it will behave. So, for instance, if we 
know that the area of a rectangle is 120 square inches and its length 
is 10 inches, we can plug these numbers into the formula:

     120 = 10 * w

Now we can use properties of numbers to solve for w:

     120/10 = 10 * w / 10
     12 = w * 10/10
     12 = w * 1
     12 = w

I used the property of equality that equals divided by equals are 
equal. Then I used the commutative property that w * 10 = 10 * w, 
which is true no matter what number w stands for. Finally, I used the 
identity property that w * 1 = w, no matter what number w stands for. 
It's because of these properties that I can use variables to solve 
equations.

You asked a broad question, so I don't know if I have hit what you are 
really wondering about. If not, please write back.

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