Absolute Value

Imagine we are playing a competitive game of gravity racer golf, where the objective is to get your racer to stop as close as possible to a target.

In this example, the target is set at ninety three centimeters. One racer stops at seventy eight centimeters, and another stops at one hundred seven centimeters. Which racer is closer to the target, and by how much?

To figure this out, we need to see how far away each racer is from ninety three. Let's subtract the racer's distance from the target distance of ninety three.

Let's start with the racer at seventy eight centimeters. If I subtract ninety three minus seventy eight, that equals fifteen. So this racer is fifteen centimeters away from the target. That's pretty straightforward.

Now, let's try the racer at one hundred seven centimeters. If I subtract ninety three minus one hundred seven, that looks pretty strange. How do we do this using pencil and paper?

Well, first, we know the answer will come out negative because we're taking away a bigger number from a smaller number. And here's the thing. Subtraction is really about finding the difference between two numbers. How far apart they are.

But the traditional pencil and paper setup doesn't work if you put the smaller number on top. So to use this method, we need to switch it around and write it as one hundred seven minus ninety three. That gives us fourteen.

But remember, since the original problem was ninety three minus one zero seven, the answer needs to be negative. That's how we subtract a bigger number from a smaller one. But here's the problem. Can a distance really be negative? No. Distance is always positive.

Think about it on a number line. Seventy eight is fifteen away from ninety three. One hundred seven is fourteen away from ninety three.

In math, there's a term that tells us how far apart two numbers are on a number line, no matter the direction. It's called absolute value, and it's written using this symbol. These symbols mean we are finding the distance between these two numbers.

You can solve it just like before. The absolute value will make the end result positive.

So to answer our initial question, the racer that traveled one hundred seven centimeters was actually one centimeter closer to the target than the racer that traveled seventy eight centimeters.

Absolute value is really about distance. It turns negatives into positives because distance is always positive.