In my article Variables, I gave a basic explanation of what a variable is, which is that it is a name for a value, where we haven't yet decided what the value is.

When you start studying algebra, this idea of unknown values called \(x\) or \(y\) might seem like something strange and new.

But, actually, we do talk about unknown values of things even when we are not talking about mathematics. The only difference is that we don't use the idea of variables with one letter names, like \(x\) and \(y\).

Instead the English language (like many other languages) has its own conventions for the discussion of unknown values, and the best way to show this is to give some examples, and show how the examples could be translated into Algebra-speak.

When a Man Loves a Woman ...

First convention is the use of the word "a". "a" has the technical name of indefinite article, and a phrase like "a man" means something like "some man, and we haven't yet specified which man".

The Algebra-speak version of "When a man loves a woman ..." is as follows:

• \(x\) is a man
• \(y\) is a woman
• When \(x\) loves \(y\) ...

Introducing "the"

The indefinite article "a" is often used to introduce a previously unknown something into a discussion, but "the" is then used to refer to the same something a second time.

For example:

In this case we know that "the man" is the same man as the preceding "a man". Also note that the first "a man" might not be the same man as the second "a man" (although usually we would think that it is).

So the Algebra-speak translation of this proverb is as follows:

• \(x\) is a man
• \(y\) is a man
• If you give \(x\) a fish, you feed \(x\) for a day. But if you teach \(y\) how to fish, you feed \(y\) for a lifetime.

Pronouns

Actually, the proverb would normally be stated without repeating the word "man" so much. We would say something like:

In other words the word "him" is standing in for the second reference to "a man".

A Detective Story

In the Variables article, I pointed out that different variables don't necessarily have different values. But in normal conversation, if we refer to different unknown people, we don't usually leave our speaker uncertain as to whether or not two different references are to the same person.

To demonstrate a situation where we can be uncertain about the equality of two different unknown people, consider a murder story.

The details are:

• Judy Smith was murdered at 3.50am on Tuesday in her apartment.
• Jenny Robertson was murdered at 5.00am on the same Tuesday in her apartment, just two blocks away.

When the police detectives discuss these cases, they assume that the same person might have murdered these two victims, given the closeness in time and space, but they can't really be sure. So a detective might report something like:

• Judy's murderer left footprints from muddy size 9 hiking boots.
• Jenny's murderer was seen by a witness running away at 5.15am, wearing a black leather jacket.

In Algebra-speak this would be:

• \(x\) murdered Judy Smith at 3.50am on Tuesday
• \(y\) murdered Jenny Robertson at 5.00am on Tuesday
• \(x\) was wearing muddy size 9 hiking boots.
• \(y\) was wearing a black leather jacket at 5.15am.

The point is that \(x\) and \(y\) are different unknowns (i.e. variables), but they don't necessarily have different values.

At some point in the investigation, the police might discover that the two victims really were murdered by the same person, and the Algebra-speak for this would be:

• \(x=y\)

This equation tells us that Judy Smith's murderer was the same person as Jenny Robertson's murderer, even though we don't yet know who that person was.