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# How to Convert Units Using the Multiplicative Identity

Learn what the multiplicative identity is and how it’s used to convert between different units of measurement.

By
Jason Marshall, PhD
4-minute read
Episode #35

In the last two articles we’ve talked about the meaning of multiplication and how to use this meaning to make multiplying fractions easier. Today, we’re going to put everything we’ve learned about multiplication to good use by learning how it can help us convert between different units of measurement. For example, how to convert from miles to meters, meters to kilometers, kilometers to light-years, and even more.

## Why Do We Make Measurements?

Before we start talking about how to convert between various units of measurement, we should probably talk a bit about why we make measurements in the first place. The answer to “Why?” is pretty simple: We make measurements to quantify the world. It’s one thing to stand on the beach and note that the water occasionally crashes ashore as waves, but it’s an entirely different thing to note the height of the waves, how frequently they occur, how fast they are moving, and so on.

Making quantitative measurements (that is, measurements with numbers) allows us to understand why things happen—not just that they happen. And they even help us to predict what will happen next. Measurements are powerful! Of course, we make measurements for far simpler things too, like figuring out what size shoes or pants we wear, or whether or not the new sofa we just bought will fit through the front door (probably should’ve checked on that first). Quantitative measurements are a fact of life. So, how exactly do we make them?

## What are Units of Measurement?

It doesn’t really matter what units you use to make a measurement because you can always convert from one system of units to another.

Well, that depends on what we’re measuring. If it’s length, we use something like a measuring tape. If it’s time, we use a clock. Or, if it’s weight, we use a scale. All of these measurement devices are calibrated to give quantitative measurements in some generally accepted “unit.” For example, a measuring tape could be calibrated to give lengths in inches, feet, meters, miles, or even light-years (though that would be a very long tape).

So, which of these distance units is the right one to use? Again, that depends on what we’re measuring. If it’s something relatively small like the size of a sheet of paper, then inches is probably the most appropriate since the answer will be a reasonable number of inches—perhaps 8.5” by 11”. If we used miles instead, the size of the same sheet of paper would be 0.00013 miles by 0.00017 miles—pretty awkward to work with! But if we’re measuring something much larger, like the distance from New York to Los Angeles, then miles or kilometers would be much more appropriate. The good news, however, is that besides issues of convenience, it doesn’t really matter what units you use to make a measurement because you can always convert from one system of units to another. How does that work?

## 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.