# Variables

Copyright © 2006 Philip Dorrell

Variables are the thing that make algebra different from other kinds of mathematics like, say, arithmetic.

In arithmetic, we deal with numbers, and each number has a value, which is itself. And if we want to be more complicated, we can make expressions, like:

\[2 + 3\]

What is the value of this expression? (This isn't a trick question.) Answer: 5.

## Definition of "Variable"

So what is a variable? One simple definition is:

A variable is something which could have a value, but we haven't decided yet what the value is going to be.

In elementary algegra, the as-yet undetermined values are almost always numbers.

Because we haven't yet decided what the value of a variable is, we can't write down the value, so we have to write the variable some other way, and the usual way to do this is to give the variable a name, and the most common name for a variable is the letter "x", which is usually written in italics, like so: \(x\)

Here is an example of an expression containing the variable x:

\[x+3\]

## The Values of Expressions Containing Variables

So what is the value of \(x+3\) ?

The basic answer is that it is 3 more than whatever \(x\) is. But since we haven't yet decided what the value of \(x\) is yet, we can't know what the value of \(x+3\) is.

## What's the Point of Talking About Unknown Values?

What is the point of taking some number that we haven't decided what it is, and then adding 3 to it, to get another number that we don't know what it is, but which must be 3 more than the number we started with?

There are actually a few different reasons for wanting to do this. The most common reasons are among the following:

- The number has a value, but we don't yet know what it is. We have to talk about the unknown value in order to find out what it is, and in order to talk about it, we have to give it a name, like, for example, "x".
- The number has a value, but we just don't care what it is.
- We want to say something about a number that is true for all numbers.

## Having More Than One Variable

Give a mathematician a hard problem, and sometimes they'll sit down and think up a harder problem, even before they have properly solved the first problem.

The way to do this with variables is to have more than one variable.

Each variable has to have a name, and different variables have to have different names. Following a common mathematical rule of thumb, which is:

If you have to use a different letter, then use the next available letter of the alphabet,

a second variable will generally be called "y", also written in italics, like so: \(y\).

Here's an example of an expression using two variables \(x\) and \(y\):

\[x+y\]

What this expression means, is take the unknown value of \(x\) and add it to the unknown value of \(y\), to get a third number, whose value is of course not known, but which is equal to the sum of the unknown values of \(x\) and \(y\).

## Different Variables Don't Necessarily Imply Different Values

I will highlight a subtle point here, which is that although \(x\) and \(y\) are different variables, they might actually have the same value. Or then again, they might not. I will give an example of this in my article on Variables in English.

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