How Are Distances and Absolute Values Related?

 

In the last article, we learned exactly what absolute values are and how you can find the absolute value of a number. In today’s article, we’re going to put this knowledge to work and learn about the very practical skill of using absolute values to find distances between numbers and places.

Review: What are Absolute Values?

As we talked about last time, 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 on the number line. For example, since the numbers 5 and –5 are both 5 steps away from zero on the number line, they both have the same absolute value of 5.

What is Distance?

Does this idea that the absolute value of a number tells you how many steps away it is from zero on the number line remind you of anything in the real world…perhaps the idea of distance? The connection here is actually pretty straightforward, but let’s take a minute to look at an example that will drive home the relationship between absolute values in math and distances between objects in the real world.

As you know, the distance between two trees in your backyard is just a number that tells you how far apart the trees are. If you draw a coordinate system in your backyard (which is really just a number line) and set one of the trees at the origin of your coordinate system (the location marked zero), then the distance to the other tree is the absolute value of its location in your coordinate system. For example, if one tree is at the location marked 0 and the other tree is 7 steps away in whatever direction you choose to be the positive direction, then the distance to the tree is |7| = 7. If the second tree is instead at –7 in in your coordinate system (in the opposite direction), its distance to the first tree is |–7| = 7. In other words, independent of direction, the second tree is always 7 steps away.

How to Find the Distance Between Positive and Negative Numbers

So we know that the absolute value of a point on the number line (or the absolute value of the coordinate of a tree in your backyard) tells you the distance between that point (or tree) and the number zero at origin of your coordinate system. But how do we find the distance between any two numbers? In other words, what if the first tree in our example wasn’t located at the origin of the coordinate system? What if one tree is at 2 and the other is at –5? How do you find the distance between them in that case?

Let’s start by realizing that this problem with trees is the same as the problem of figuring out the distance between the numbers 2 and –5. We know that the distances from 2 to 0 and from –5 to 0 are each given by the absolute values |2| = 2 and |–5| = 5. And if we think about where 2 and –5 sit on the number line, we can see that the distance between these two numbers is equal to the distance from –5 to 0 plus the distance from 0 to 2. In other words, the distance between –5 and 2 is equal to |–5| + |2| = 5 + 2 = 7.

But is that always true? Is the distance between any pair of numbers always equal to the sum of their absolute values? For example, what if we want to find the distance between the numbers 2 and 5 instead of 2 and –5. Can we just add the distance from 0 to 2 to the distance from 0 to 5 to get |2| + |5| = 7. Is 7 the distance between 2 and 5? No! As you can easily see by looking at a number line, the distance from 2 to 5 isn’t 7…it’s 3!

How to Find the Distance Between Any Two Numbers

So what went wrong? Well, since the absolute value of a single number is its distance from zero, the sum of the absolute values of two numbers is not the distance between them, it’s the sum of each number’s distance from zero. When one number is positive and the other is negative, this is the same thing as the distance between the numbers. But it doesn’t work when both numbers are either positive or negative.

[[AdMiddle]Okay, but what’s the right way to calculate the distance between any pair of numbers? The quick and dirty tip is that the distance between any pair of numbers is given by the absolute value of their difference. To see what this means, let’s go back to finding the distance between the numbers 2 and 5. The absolute value of their difference is |5–2| = |–3| = 3…which, as you can check by looking at a number line, is exactly the distance between 2 and 5! And, as you can also check, this works for any pair of numbers. The absolute value of the difference of two numbers, two coordinates on a map, or two locations of trees in your backyard, always tells you the distance between them.

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 actually a conversion that you can use to figure out how fast a wide receiver in the football game you’re watching is running as he jets toward the end zone.

The trick is to know that a speed of 10 yards per second is about the same as 20 miles per hour. So if it takes a player one second to run 10 yards, he must be running at about 20 miles per hour. If it takes two seconds to cover those 10 yards, his speed must be only 10 miles per hour. That’s all there is to it! Now, the next time you’re watching a football game, you can impress your friends by telling them how fast everybody is running. It’s sure to give you a whole new perspective on what’s taking place on the field!

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!