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What are Absolute Values?

Learn what absolute values are and how to find them.

By
Jason Marshall, PhD
5-minute read
Episode #76

There aren’t many terms in the world of math that sound more serious and menacing than “absolute value.” And sometimes serious-sounding things are difficult to understand. So is that true for absolute values? Thankfully, no. First of all, they aren’t nearly as serious as they sound (and they’re certainly not menacing). And second, as you’ll soon see, understanding absolute values is easy…and it turns out to be very important too.

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What are “Small” Numbers?

We’ll talk about exactly what absolute values are in a minute, but to get an idea of why they’re important, let’s take a moment to talk about really teeny-tiny small numbers. Have you ever noticed that it’s easy to get tripped up when using the word “small” to describe numbers? While it’s true that a really tiny number like 0.001 is “small,” it’s still a lot bigger than a negative number like –1,000,000…just think about where those numbers are located on the number line if you need convincing!

But there’s another meaning of the word “small” in which a number like 0.001 is indeed actually much smaller than the number –1,000,000. So what is it? Well, it has something to do with what’s called the “magnitude” of the numbers. And, as you might have guessed, that has something to do with today’s main topic: absolute values. So what are they?

What are Absolute Values?

The Quick and Dirty way to think about absolute values is that the absolute value of a number simply tells you how far away it is from zero. Start by picturing the number line in your head—zero in the middle, negative numbers to your left, and positive numbers to your right…you know the drill by now. The absolute value of a positive number like 2 is just equal to 2 since that’s how far away it is from zero on the number line. The absolute value of the number 1,000,000 way out there to your distant right on the number line is just 1,000,000 since, once again, that’s just how far away it is from zero. And the absolute value of zero itself? Well, that’s just zero, right? It had better be since zero is exactly zero steps away from itself on the number line!

What’s the Absolute Value of a Negative Number?

But things get slightly more complicated when we talk about the absolute values of negative numbers. For example, what’s the absolute value of –3? Well, how far away is the number –3 from 0? If you think about the number line, you’ll see that –3 is 3 steps away from 0. Which means that the absolute value of –3 is equal to 3. It doesn’t matter if the steps were in the positive or negative direction, all that matters is the total number of steps away from zero.

In other words, the absolute value of a number only tells us about its size—also known as its magnitude. It doesn’t tell us anything about its direction away from zero…which means it doesn’t tell us anything about the sign of the number. And speaking of magnitude, we can now see that while the number 0.001 is certainly greater than –1,000,000, its magnitude is much smaller. So, at least in this one sense, 0.001 is indeed a “small” number.

How to Write Absolute Values

A quick word on notation: The absolute value of a number is indicated in writing by putting the number between a pair of vertical bars. For example, the absolute value of the number –2 is written |–2| and the absolute value of the number 1,000 is written |1,000|. So whenever you see something that looks like that, you now know that we’re talking about an absolute value. In other words, we’re only interested in the magnitude of the number, not its sign.

How to Quickly Find Absolute Values of Numbers

[[AdMiddle]In practice, the easiest way to find the absolute value of a single number is to ignore any negative sign in front of it. So, |5| = 5  (there’s no negative sign to ignore here) and |–1| = 1 (this time we did ignore the negative sign). If you’re instead looking to find the absolute value of an expression containing multiple numbers and operators—something like |3+2–7|—all you have to do is simplify the expression and then ignore any negative signs in front of the result. For example, the expression |3+2–7| simplifies to |–2|, which is just equal to 2.

Why does that work? Because ignoring any negative signs in front of the number at the end is the exact same thing as figuring out how far away that number is from zero. That’s all there is to absolute values! As you can see, despite their serious sounding name, absolute values really are easy to understand and work with. And, as we’ll see in the coming weeks, they’re very important for solving lots of different types of problems too.

Number of the Week

Before we finish up, it’s time for this week’s featured number selected from the various numbers of the day posted to the Math Dude’s Facebook page and to QDT’s new blog, The Quick and Dirty. This week’s number is 3 billion. Why? Well, if I’m lucky, I might live to be 90 years old. If I’m really lucky, I might even make it to 100. And if I do, a little unit conversion math will show you that my lifetime will have contained around 3 billion heartbeats.

Second after second, while we work, walk, run, sleep, and do or don’t do everything else in our lives, our trusty tickers keep on ticking away a lifetime of 3 billion heartbeats. Which makes me wonder: How does that number of heartbeats in a human lifetime compare to the number of heartbeats in the lifetime of something small like a mouse? Or huge like a whale? Check out next week’s article to find out! Or, if you just can’t wait, head on over to The Quick and Dirty blog and find out now!

Wrap Up

Okay, that’s all for today. Remember to become a fan of the Math Dude on Facebook where you’ll find a new number of the day and math puzzle posted each and every weekday. And if you’re on Twitter, please follow me there too. Finally, if you have math questions, feel free to send them my way via Facebook, Twitter, or by email at mathdude@quickanddirtytips.com.

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, PhD

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.