Science teachers — and science textbooks — generally insist on careful attention to significant figures. Math teachers — and math textbooks — generally pay no attention to them. Here are two representative examples:
- Our Algebra II textbook contains a word problem in which a speed is given as 40 mph. Never mind the the details of the problem; the answer in the Teachers’ Edition is given as 199.5 miles. Why not 200, I hear you ask.
- Many math contests tell students to round answers off to four digits after the decimal point, regardless of the given information. Why four? Well, why not four?
Science teachers will correctly point out that there is no excuse for the 199.5 or the four-digit requirement. The former implies much more precision that the data justify; the latter might imply either more or less, depending on the data.
So why don’t we math teachers follow the rules? There are several good answers. One is simply that the information given in a problem rarely comes from actual measurements. No one clocked the speed of the car and recorded it as 40 mph — not to mention our ignorance about the precision of that figure anyway, which might represent 4.0×102 or perhaps just 4×102, or even possibly 4.00×102. From a mathematician’s point of view, 40 is pretty much just a pure number and is therefore not susceptible to the sig figs rule. I still don’t think that that justifies an answer of 199.5 rather than 200, but at least there’s some rationale there.
Oh — look what I’ve just done. I wrote 199.5 and 200 without units! Shame on me.
Actually there’s a good reason why math teachers only pay lip service to units and rarely pay attention to them. But we’ll save that issue for a different post. Let’s wrap up sig figs in this one.
Anyhow, the second reason is that the rules simply don’t work very well. Here’s are excerpts from a typical statement of “the rules”:
When measurements are added or subtracted, the answer can contain no more decimal places than the least accurate measurement.
When measurements are multiplied or divided, the answer can contain no more significant figures than the least accurate measurement.
Note the subtle difference between the two rules, which we miss if we read too quickly.
Let’s try to apply these rules to a concrete example given by John S. Denker: “4.4 × 2.617 – 9.064”. According to the rules, we should express the answer of 2.451 as 2.5. But each of the three numbers is taken to be a measurement where the least significant digit is uncertain. So the first one, for example, might represent anything from 4.35 to 4.45 [we’ll ignore issues of when a digit of 5 rounds up and when it rounds down]. Taking one extreme, we might really have 4.35 × 2.6165 – 9.0645. At the opposite extreme, we might really have 4.45 × 2.6175 – 9.0635. We’ll do all three calculations:
Our approved answer of 2.5 suggests that the “real” answer lies between 2.45 and 2.55, and yet both the second and third possibilities lie outside that range.
Finally, things really go haywire when we encounter sensitive dependence on initial conditions. For instance, Nagai Tosiya examines the function f(x) = 4x(1 – x). In generation 2, as a result of prior calculations we might be rounding to four sig figs, so 0.15360 and 0.15357 would both be expressed as 0.1536. But in the 15th generation the former value results in an answer of 0.13561, the latter in an answer of 0.00180. Sig figs can’t save us! We need all the precision we can get. In fact, we need more than we can get. The sig figs rule provides the comfort of all clearly expressed rules — but it’s cold comfort when it can’t give correct answers. That’s the most important reason why we math teachers pay little attention to significant figures.