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7th grader Meera from U.A.E. asks the following question: "why should you use significant digits?"
Good morning Meera. This is an excellent question. I remember way back when I was in school, learning about significant figures and thinking, "This is a waste of time to try to keep track of how many digits I should use. It's a hassle, and who really cares how many digits I put in my answer?"
Of course, even back then, I was much more interested in theoretical instead of practical math and science. If you're anything like me, you'd rather give an exact answer like 22/3 or 5√2 instead of converting that to 7.333.. or 7.071068... and then figuring out how many decimals you're allowed to use.
Really, though, significant figures are the intersection of the theoretical and the practical. If all you were ever going to do is theoretical science, you would probably leave your answers in exact fractional or radical form, and never worry about significant figures. But practical science (engineering) requires you to use significant figures. Choosing not to do sig figs can be costly. Let me give you an example.
Suppose I contract a machine shop to build for me a cylindrical container that is 850 cubic cm in volume, and has a base area of 120 square cm.
Can you figure out how tall that cylinder would have to be? As a fraction it's 850 / 120, or 85 / 12 cm. But that's not helpful; who has tape measures marked to the nearest 12th of a centimeter? So they'll have to convert that to a decimal: 7.08333...
Which is still impractical, because who has a tape measure with infinite precision? Hmmm...what to do? They're going to have to round that number. And who decides how they round it? They do! So maybe they say, "Ah, 7.08333... is pretty close to 7, so we'll call that 7 cm."
So they build my cylinder, and now, instead of having a container with 850 cubic cm of volume, it has 7(120) = 840 cubic cm of volume. Not a big deal? Maybe. But maybe I actually have a reason that I want my container to hold 850 cubic cm of something, and now it's going to overflow because he didn't make it tall enough. So I bring it home, start filling it up, and whatever I pour into it (hopefully not sulfuric acid!) starts going all over the floor, because the container isn't actually as big as I wanted it. So now I have to go back to the machine shop and tell them they built it wrong, and they have to start over. That's going to cost someone - either me, or or the machine shop, depending on whether one of us can convince the other that it's their fault.
Here's where significant figures are important. Significant figures take all the guesswork out of how the machine shop should round. They should have looked at the two measurements I gave them (120 and 850) and said, "Each of these has two significant figures, therefore, the result of the division will have two significant digits. Thus, instead of rounding to 7 (one sig fig) we should round to 7.1 cm."
Now, if my cylinder is not the right size, they can point at the measurements I gave them and say, "We performed this exactly to your specifications, using the rules of significant figures."
What it boils down to is this: using significant figures is a way of indicated how precisely our quantities are known, or have been measured. Significant figures keep us from claiming more accuracy than we really have, and they're a way of communicating more precisely what we want. If I said said to the machine shop, "I want my cylinder to have a volume of 850.0 cubic cm, and a base area of 121 square cm," they would know that they need to be much more precise, and the result needs to have three significant figures: 7.08 cm.
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