**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.

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 \(x\):

\[x+3\]

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.

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.

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:

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

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 \(y\):

\[x+y\]

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 \(y\).

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.