数学英语 36 How to Convert Units

（单词翻译）

by Jason Marshall

In the last article, we began discussing how to convert¹ from one system of units to another—for example from miles to kilometers—and how all of this relates to the multiplicative identity. But we didn’t quite have enough time to get to the punch line, so without further ado let’s finish up and get to the bottom of exactly how to convert between units.
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Review of Measurements and Units
We started the last article off by talking a bit about the power of measurement—in particular how quantitative² measurements (that is, measurements with numbers) give us information about the world. We also talked about the units we use when making measurements. For example, when you measure the length of something, perhaps the diagonal size of your precious new television (just to make sure you got what you paid for), you can’t just say the length is 50. That number 50 alone doesn’t tell us anything about the size of your TV since we don’t know 50 of what. Are we talking miles? Meters? Thumb lengths? You have to give the units you used to make the measurement for it to make any sense. In this case, you measured 50 inches—which now makes perfect sense!
Review of the Multiplicative Identity
We also spent some time in the last article talking about the multiplicative identity. We found out that the multiplicative identity is really just a fancy word for the number 1. The multiplicative identity property says that you can multiply any number by the multiplicative identity, 1, and the answer will be the same number. No big surprise there, I know—so how is this all related to converting³ units? Well, let’s find out.
How to Use the Multiplicative Identity to Convert Units
Let’s imagine that after measuring the size of your new TV screen, 50 inches, you can’t help but wonder if that’s bigger than your little brother. So you ask your brother how tall he is, and he tells you that he’s 4 feet tall. That doesn’t immediately give you the answer though—you still need to figure out how to convert 4 feet into inches so that it can be directly compared to the 50 inch size of your TV. (I know that many—or most—of you will already know how to do it; that’s fine—it’s the method that’s important here, not the specific numbers or choice of units we’re using.)
So here’s what we’re going to do: Let’s write a fraction that’s equal to the number 1. What does that look like? Well, any fraction where the numerator and denominator are the same number must equal 1. For example, 2/2 and 4/4 are both equal to 1. And that makes perfect sense since putting together 2 halves or 4 quarters of something must give a whole. Okay, how else can we make a fraction that’s equal to 1? How about something like the fraction: 1-foot / 1-foot? The numerator and denominator are the same, so even though there are units in it this fraction must equal 1 too! Okay, here comes the important part. How many inches are in 1 foot? 12 inches-per-foot, right? Indeed, which means that the fraction 12-inches / 1-foot is equal to 1!

Now this is getting interesting. Since 1 is equal to the fraction 12-inches / 1-foot, and the multiplicative identity property says that we can multiply any number by 1 without changing its size, that means we can multiply the height of your little brother, 4 feet, by the fraction 12-inches / 1-foot to get his height in inches. In other words, let’s write the problem as 4-feet times the fraction 12-inches / 1-foot.