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:

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:

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 \(x\):

The Values of Expressions Containing Variables

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

The 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?

Why would we want to discuss the addition of the number 3 to an unknown number?

There are several situations where we might want to do this:

• 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.

For example, talking about one unknown value is hard. So let's talk about two unknown values, or three, or even more than that.

If we have more than one unknown, then we have to have more than one variable, and each variable has to have a name which is different from the names of all the other variables. Given that our first variable is called \(x\), we can follow a common mathematical rule of thumb, which is:

So following this rule, the second variable will generally be called "y", also written in italics, like so: \(y\).

And the third will be "z". And after that we have run out of letters, so we might start again at "x" and go backwards, choosing "w", "v", etc.

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

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

A subtle point here 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.