What Are Variables? Part 2
How do variables work? How do you work with variables? And what do they have to do with algebraic expressions? Math Dude has the scoop!
Jason Marshall, PhD
Listen
What Are Variables? Part 2
Over the past few weeks we’ve learned about the nature of algebra and the nature of the variables that we use in algebra. Now that we have a firm grasp on these key ideas, it’s time for us to spend some time figuring out why all of this stuff matters in the first place. With that in mind, let’s start today by talking about how variables work and how you can work with variables..
How Do Variables Work?
In our first episode on variables, we decided to use a symbol that looks like a little box as our variable…at least for a while. As we talked about, this decision is completely arbitrary—it really doesn’t matter what the symbol looks like. What matters is what we can do with it. So, what can we do with our little box? The short answer is: anything that we can do with numbers. For example, we can add our variable to the number 3:
3 + ?
Afterwards, we can subtract our variable from this sum
3 + ? – ?
Of course, when we first add and then subtract our variable from the number 3, the net result is doing nothing…just as it would be if we instead added and then subtracted a number like 2. In other words, 3 + 2 – 2 = 3, and similarly
3 + ? – ? = 3
The important thing to realize is that the answer is always 3…no matter what value the variable represents.
What else can we do? Among other things, we can add our variable to itself:
? + ?
We can multiply it by a number:
9 • ?
We can even multiply our variable by itself:
? • ?
How to Work With Variables
Some of these things may seem kind of strange, but they’re all legitimate things to do with a variable. And I can prove that to you. How? By using the fact that we don’t have to leave all of these problems written in terms of our variable. We’re free to assign an actual number to our variable whenever we please, so let’s do it—let’s set the value of our variable to the number 2.
To make this as visual as possible—so that you can really picture what’s happening—let’s imagine that our variable is no longer an empty box, but is instead a box that contains the number 2. Now, when we add a number to our variable (which actually isn’t so “variable” anymore), we get
This now looks exactly like ordinary everyday arithmetic. The only difference is the outline of a box that reminds us that the number 2 is actually the value of a variable. What happens when we add our variable to itself? In that case we get
Again, that’s exactly what we’d expect from ordinary arithmetic. How about multiplying our variable by a number?
Or multiplying it by itself?
Yes, those give perfectly reasonable answers, too. I know that this might all seem kind of obvious to you, but as we’ll soon see it’s actually a key ingredient when it comes to understanding the really big ideas behind solving algebra problems.
To sum things up, the quick and dirty tip is that variables are just like ordinary numbers: You can add them, subtract them, multiply them, divide them, and do anything else with them that you can do with numbers in arithmetic. The only difference is that variables don’t have numerical values assigned to them. Admittedly, this little difference makes them seem kind of weird, but it also makes them extremely powerful.
What Are Algebraic Expressions?
Why are variables so powerful? It’s all about what we can make with them. In particular I’m talking about the fact that we can use them to make something called algebraic expressions. What’s that? An expression in algebra is like a phrase in English. It’s a statement that’s made of numbers, variables, parentheses, and ordinary arithmetic operators like “+” and “–”. Notice that I did not say that an expression in algebra is like a sentence in English—it’s like a phrase. As we’ll see in the coming weeks, the honor of being an algebraic sentence is reserved for something else: the all-important equation.
Wrap Up
Okay, that’s all the math we have time for today. If you want to learn more about algebra, please check out my book The Math Dude’s Quick and Dirty Guide to Algebra.
Remember to become a fan of the Math Dude on Facebook where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too. Finally, please send your math questions my way via Facebook, Twitter, or email at mathdude@quickanddirtytips.comcreate new email.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
About the Author
Jason Marshall is the author of The Math Dude’s Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.