Variable vs. Constant: Using Letters to Represent Numbers

If you are far along enough in your math education that you have encountered letters, then this post is for you.

There are cases in math where we want to work with a more abstract form than a number. See, to us, a number represents only one single quantity. We don't look at the number 4 and say "well that's the number of toes I have." (Assuming that we are human beings that have all of our appendages in the proper places…otherwise, we might say that.) A number to us is a concrete, undeniable amount.

But what if we want to talk about many different numbers? There are times in math when we want to think about a single number without knowing exactly what that number is. Or maybe we need to be able to think abstractly about a whole bunch of numbers at once. That's where letters come in.

We don't have to use letters. We could draw little pictures of elephants if we wanted to, and have the pink ones represent even numbers while the green ones represent odd numbers. The reason we use letters is simply convenience: they are right there, quickly accessible in our minds and we already know how to draw them.

There are two different uses for these letters, which I mentioned above. Now I will go into detail.

If we want to think about a single number without knowing exactly what that number is, we call that a constant.

It is fixed at a specific value, we can't just randomly choose a number for it. For example, I could say "a is the next number after 5." So then you could say to me, "a must be 6." It couldn't be 7 or 8 or 528,933. It could only be 6.

If we want to think about many different numbers all at once, we call that a variable.

It can vary and change because we don't always have to put the same number in for it. For example, I could say "let x be any positive number." Well, then I could put 8 in for x.

Or I could put 1,000 in for x.



But I can't put -2 in for x because I specifically said that x had to be a positive number and -2 is negative.

A few notes:

- If this is at all confusing, try this trick: Whenever you see a letter anywhere, put a box around it just as I have above. Forget about constants and variables and letters and everything else. You just have a box and a number, and you are putting the number into the box.

- In my experience x, y, and z are the most common variables while a, b, and c are the most common constants.

- While you can technically use any letter you want, some of them have special meanings just because that's the way the ancient mathematicians did things. For example, lowercase e is actually a predefined constant that stands for a number that is about 2.71. Lowercase i is actually imaginary, it's not a real number (I know that sounds fun, but it's extremely complicated, so you probably won't see it until 11th grade or later).

- In very high-level math, such as calculus and physics, we run out of letters because we make them all represent certain things. When this happens, we start using Greek letters, such as alpha, phi, and theta:

So, technically, you don't even have to use the Latin letter system, you could represent numbers with obscure Chinese characters if you want.