# Meaningful Numbers and “Significant Figures”

. Isn’t the idea of owning a hammer whose weight you know to five significant figures thrilling? Too bad only Spanish and French speakers get that much information, while we English speakers have to get by with only one figure! Ahem. Of course, the additional “information” in the Spanish and French versions is almost certainly meaningless.

Not long ago, I taught Brief Survey of Calculus, a one-semester 100-level course, as a part-time instructor at a university. The only major requiring the course was business, and naturally, I was supposed to teach a very “applied” slant on calculus, heavily emphasizing problems with only approximate solutions. Okay, but I soon discovered that most of my students had no idea of what a reasonable approximation was! I finally wrote something up on significant figures—see the attached document—and I added the following text to the Course Policy. (I realize that changing the Course Policy during the course isn’t a great policy, but I felt this was too important to let it go.)

In this course, please use at least 4 significant digits in all calculations, and give answers to at least 3 but no more than 5 figures. On one problem in Exam #1, looking for a function to fit the data given, one student wrote “2 / 1.5 = 1.3”, which is lower than the correct value of 4/3 by about .0333. That may not sound like much, but it’s a large relative error: .0333 is over 2% of 1.3. That’s enough to lead to the wrong answer in many situations, including that problem!

But why not give answers to more than 5 significant figures? Because in situations involving mathematical modeling, it can be very misleading. A question on Exam #1 said a company’s sales were “\$257 million” in a certain year, stated they’d gone up by “at least 5%” every year since, and asked for the minimum sales in a later year to the nearest million dollars. Several people answered “\$361,624,809”, or even “\$361,624,808.6”. These look like very accurate figures, but how could the last five or six digits mean anything? “\$362 million” is a much better answer.

Naturally, when a problem specifically calls for more accuracy or for an exact answer, these rules don’t apply.

I’ll bet I’m not the only teacher who’s run into this problem! Feel free to use some or all of my verbiage, and by all means the picture of the hammer label. (If you’d like a 226.72 gram hammer of your own, I’ve seen this model for sale in a number of places, including CVS drugstores. You can write yourself a reminder to get one on a 5.08 cm. square PostIt note.)