# Significant figures

Math teachers and science teachers never agree about significant figures — neither about their importance nor about how to use them. Math books and competitions tend to ignore the issue, or else they promulgate arbitrary rules, such as “answers must be rounded correctly to at least four significant digits,” regardless of the inputs. In physics, however, there are rules that are supposed to make sense. But do they? Here is what one physics teacher has to say:

I’ve long disliked the physics practice of rounding numbers and relying on reporting of “significant figures” rather than explicitly stating the precision.  The “significant figure” convention, which only allows expressing a precision of ±1/2 the least-significant digit, is a rather awkward and (I believe) stupid convention.

Here’s the difficulty, in plain English. If you report an answer as 6.4×103, you are saying that the actual answer is somewhere between 6350 and 6450. If you report it as 6.42×103, you are saying that the actual answer is somewhere between 6415 and 6425. So your choice is between an uncertainty of ±50 or ±5. But what if your actual uncertainty is ±20? There’s no way to express that if you rely on significant figures rather than reporting uncertainty. So why not explicitly state your uncertainty rather than use the implicit conventions of sig figs? It would be clearer and more informative to say what you mean.

Math teachers have some more issues with significant figures. First, aside from the fact that many of our math problem numbers are theoretically exact — often a sore point with those who are of an applied bent, since measurements cannot be exact — we have the question of where one should round. If you use 9.8 (or worse, 10) as the acceleration due to gravity on earth, there is no point in having three sig figs in your answer. (Note the lack of units, another complaint that is often levied against math teachers and math textbooks. But more on that in a different post.) Also, there are rules in science books about how to figure out the number of significant figures to leave in one’s answer, depending on the precision of the original data. But it’s fairly easy to show from concrete examples that these rules, though handy, don’t always work. It’s all a can of worms. Even though it’s necessary in chemistry, math teachers would prefer to avoid the issue.

Finally, another good point from the original :

Rounding in mid-calculation, as is done in examples in so many books, is both unnecessary and dangerous, increasing the error in the final result for no good reason.  (Exception: mental calculations and quick back-of-the-envelope order-of-magnitude estimates often benefit from keeping only 1 or 2 significant figures to reduce the load on working memory and mental calculation time.)

This is why I keep telling students not to round until reporting the answer.

Categories: Math, Teaching & Learning