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### 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

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/
```
Associated Topics:
High School Basic Algebra
Middle School Algebra

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