In a previous video, we learned about the most common units for measuring distances.

In this video, we're going to talk about how you actually use some of those units when making a measurement in real life.

Making measurements is something you would typically learn how to do in science class, but since measurement uses math to get the job done, it's often taught in math class too.

So here we go…

Suppose we're given an object like this pencil, and we're asked to measure its length in centimeters.

To do that, we'd need some method or device that will tell us how many centimeters long the pencil is.

Remembering that a centimeter is roughly the width of a pinky finger,

I could just use my finger as that device and see how many finger widths it takes to get from one end of the pencil to the other, but something a little more accurate would be nice.

And that's where a ruler comes in handy.

I, the great King Rob, ruler of all the land, declare that the span of my royal hand shall henceforth be the measure of all things in my kingdom, great or small.

Uh… not that kind of a ruler.

In math and science, a ruler is a flat piece of material that has markings on it that correspond to standard units of distance.

For example, this ruler has a series of markings that correspond to inches on one side and centimeters on the other.

That's cool, we'll just ignore the inches side and use the centimeters side for this measurement.

We start by moving the ruler so that the zero centimeter mark is aligned as closely as we can with one end of our pencil.

Then we'll see where the other end of the pencil lies on the scale of centimeters.

Looking at the numbers, you can see that the other end lines up nicely with the 19th centimeter mark.

So this pencil is 19 centimeters long.

But what if the pencil gets sharpened and then used and sharpened and used again so that it gets shorter?

Now if we re-measure it with our ruler, you'll see that we have a small problem.

The tip of the pencil doesn't line up with any of the centimeter marks anymore.

It lies somewhere between the 17 and 18 centimeter marks.

We could just say that it's between 17 and 18 centimeters long, but it would be nice if we could be a little more accurate than that.

Being more accurate means making a measurement that is closer to the true value.

Fortunately, most rulers divide the space between each centimeter mark into ten equal parts that represent millimeters which are exactly one-tenth of a centimeter.

Because the millimeter marks are so much smaller, they don't have numbers on them.

But if you look closely, you'll be able to count that there are nine smaller lines that divide the centimeter into ten equal parts.

The middle of these nine lines is usually a little longer than the rest so that it's easier to tell where the halfway point is.

Using these subdivision marks, we can get a more accurate measurement of the length of our sharpened pencil.

Do you see how the pencil's tip almost lines up with the third subdivision line that comes right after the 17th centimeter mark?

That means that the length of the pencil is 17 centimeters plus 3 millimeters, or 17.3 centimeters.

Well, that's a pretty close measurement, but remember the tip of the pencil didn't line up exactly with the third subdivision line.

It went just a little bit past it.

Our ruler doesn't have markings smaller than a millimeter, so it will be hard for us to make a measurement more accurate than that.

But we could make an estimate.

For example, it looks like the pencil tip goes past the 3 millimeter mark by a very small amount… maybe just a tenth of a millimeter.

So we could estimate that its length is closer to 17.31 centimeters.

If you really did need a more accurate measurement, you'd need to use a better measurement device that could provide that level of accuracy.

For example, calipers and micrometers are devices that can measure distances as accurate as a tenth or even a hundredth of a millimeter.

And certain measurement techniques using lasers can achieve even higher accuracies, down to extremely small units like nanometers.

Those more advanced types of measurements are beyond the scope of this video, but hopefully they'll help you realize something fundamental about the nature of measurement.

You can't measure the EXACT value of something.

There's always a limit to the accuracy you can achieve based on your measurement device.

Accuracy is basically how close a measured value is to the true value.

And in measurement, the idea is to get as close as possible, or at least as close as you need for your purposes.

Luckily, I didn't really need to know the length of this pencil down to the nearest tenth of a millimeter.

In fact, I didn't really need to know its length at all, since I'm just gonna write with it.

Okay, now that you understand what accuracy is, and we've made a measurement that was accurate to the nearest centimeter as well as one that was accurate to the nearest millimeter, let's try making a measurement with the non-metric side of our ruler, which measures inches.

Suppose we want to know the length of this toothbrush in inches.

To measure that, we first line up one end of the toothbrush with the start of the inches scale on the ruler, which represents zero inches.

Then we see where the other end of the toothbrush lies on that scale.

Notice it's somewhere between 7 and 8 inches.

Can we get a more accurate measurement than that?

Yup!

Fortunately, as was the case with centimeters on the other side of the ruler, inches are usually subdivided into fractions of an inch also.

On this particular ruler, each inch is subdivided into 8 equal parts.

That means our toothbrush is not quite as long as 7 3⁄8 of an inch, but it's a little longer than 7 2⁄8 of an inch, which would be equivalent to 7 ¼ inches.

Whoa, whoa, whoa! What are you talking about with all these fractions?

I thought this video was about measurement, not fractions!

Well yes, but when things don't line up exactly with a particular unit, to get more accuracy, you need to use fractions of that unit.

The metric system makes that look easy because things are always divided by 10, so the fractions match up really nicely with our base 10 decimal system.

But English or American units have traditionally been divided up differently, so the fractions you use for them are a little bit trickier.

So… you're saying this is all America's fault?

Well, America didn't invent the system.

It's just a traditional system of units that goes way back in history to the days of monarchies, so I guess it's probably some king's fault.

How dare you!

Anyway, it's true that the traditional way of subdividing inches is kinda messy when compared to the metric system, so it will help if we take a closer look.

Basically, there's two ways that inches are commonly subdivided.

One is based on dividing by 10, and the other is based on dividing by 2.

We'll start with the system that's based on dividing by 10 because that sounds a lot like the metric system, doesn't it?

It turns out that an inch can be divided up in a metric-like way even though an inch is NOT a metric unit. Here's how that works.

You start with an inch and then divide it into 10 equal parts.

Each mark represents a tenth of an inch, so you can express the fractional parts easily with decimal digits, just like you do with the metric system.

For example, if a measurement came out to be 1 and 2 tenths inches, you'd just say it's 1.2 inches.

Or if a measurement came out to be 5 and 8 tenths inches, you'd just say it's 5.8 inches.

And when you need more accuracy, you can keep subdividing by 10 so that you'd get hundredths of an inch, thousandths of an inch, ten thousandths of an inch, and so on.

This way of dividing up inches is commonly used in American engineering since it has many of the benefits of the metric system even though it's based on inches.

The other way of dividing up inches, which is still commonly used in American construction, is to divide them up by two. Here's how that system works.

You start with an inch and then divide it into two equal parts.

That means you can now measure to an accuracy of half an inch.

Then you divide those half inches by two so you can measure to an accuracy of a quarter of an inch.

Then you divide those quarter inches by two so you can measure to an accuracy of an eighth of an inch.

And you keep going like that.

Dividing by two again lets you measure to an accuracy of a sixteenth of an inch.

And dividing by two again lets you measure to an accuracy of a thirty-second of an inch, and so on.

The bases of these fractional parts of an inch are all different, but they're all powers of two.

It's actually a really logical system if you think about it.

But it has the disadvantage that it's harder to convert to decimal values when you need them.

It's also harder to add and subtract measurements if the fractions don't have the same base.

When things are divided by ten, it's easy because we use decimal number places that are specifically designed for counting fractions like tenths, hundredths and thousandths.

But we don't have number places for halves, quarters and eighths.

Instead, it's very common for people who use a lot of these traditional fractions of an inch to just memorize some of the most common equivalent decimal values and use a calculator to convert the rest.

For example, they might memorize things like one-half equals 0.5, one-quarter equals 0.25, and one-eighth equals 0.125.

Now that you know the two main ways of dividing up inches, let's go back to our toothbrush example.

If we use a ruler that has divisions of a sixteenth of an inch, you can see that its length is about 7 and 5 sixteenths of an inch.

That should be plenty accurate for brushing my teeth!

So now you know the basics of how you use a ruler to make measurements, but you may need to brush up on your fractions to be successful at it.

See what I did there?

Last of all, I want to briefly introduce you to some low-tech devices that are commonly used to measure longer distances.

For example, a tape measure is sort of a long, flexible ruler that can be wound up on a spool to make it more compact.

It's a really handy device and carpenters use them all the time.

When you unwind it, you can measure much longer distances in the same way you would with a ruler.

And it's a lot more convenient than carrying around a 10 meter stick.

There's another cool device, often called a measuring wheel, that can be used to measure even longer distances by rolling a wheel along the ground, or any surface.

It has a counter on it that tallies up each foot or meter that you roll it past so you can measure the total distance traveled.

Of course, there are lots of high-tech methods for measuring distances nowadays too, and your phone can probably keep track of how many miles you've traveled in a day, and where you went, and who you talk to, and what you might like and might want to buy, and who you have a crush on, and…

Alright, that's the basics of measuring distance.

Remember, the way to get good at math is to actually practice it.

So get on out there and start measuring stuff!

As always, thanks for watching Math Antics and I'll see ya next time.

You heard the man… Practice!

Learn more at www.mathantics.com