I don't understand. If you're going to specify two significant digits (0.10), then why would you round it at all!? 0.095 is also two significant digits and is more accurate!

Well, here's the thing. Significant digits are a fiction. They're a shorthand for something much more useful, but rather than deluging chemistry students (and let's face it, this is almost

*always* first introduced in chemistry) with yet another complex calculation, we simplify it.

The reality is that "one significant digit" has the same meaning as "+-~10%". Two significant digits is +-~1%, and so forth. These aren't exactly right, as 40 +- 5 is different from 90 +- 5, but they're close

So if you multiply 0.3 (1 sd) by 0.32 (2 sd), you get 0.096, but should only keep 1 sd. 0.1 and done, right? Well...

The reality is that these measurements really "mean" 0.3 +- 0.05, and 0.32 +- 0.005. Or, as bounds, 0.25 to 0.35, and 0.315 to 0.325. If we multiply the

*bounds*, we have a range in the final answer of:

0.25 x 0.315 = 0.07875 to 0.35 x 0.325 = .11375 = 0.09625 +- 0.0175

When chopped down to the maximum sd (2) from earlier, this is 0.096 +- 0.018

Now, here's where significant digits become a problem:

One sd is 0.1, which implies 0.1 +- 0.05

Two sd is 0.096, which implies 0.096 +- 0.0005

If we write 0.10, that implies 0.1 +- 0.005

The actual uncertainty is somewhere between the first and the third. What to use, then? Let's use the actual answer, 0.096 +- 0.018. That eliminates our uncertainty with respect to... uncertainty.

--------

So what's the real problem here, then? Well, there are two.

First, if you only have one significant digit in one of your measurements, you don't need to use significant digits anyway; your measurement is all but useless. In reality, "one significant digit" means "using the wrong measuring implement."

Second, significant digits are best used when all values being used have similar uncertainties. If we were multiplying 0.35 (0.345 to 0.355) and 0.47 (0.465 to 0.475), with typical sd usage the answer is 0.16 (0.155 to 0.165), and with properly described uncertainty, the range is 0.1604 to 0.1686, or 0.164 +- 0.004. Those ranges overlap a lot better than the ranges we had in the other problem.