Gifted Issues Discussion homepage Forums General Discussion Semi-wonkish question about significant figures

There's only one problem; when the problem is considered as mathematics, the answer you want them to give is wrong.

Interesting. I think I must have used a calculator then rounded to the lower number of decimal places. Could they just do vertical addition, line it up carefully and leave that space blank? Or add horizontally from the left?

Not seeing why this isn't mathematically correct-- Colinsmum, can you enlighten me?

(this might be something totally dorky that I'm just failing to take into account, but in terms of significant figures-- that is, WHEN significant figures are supposed to be dealt with via rounding, this seems the correct answer any way that I try looking at this-- using before-or-after rounding, I mean)

I do know that sometimes the "pre-rounding" answer isn't identical to the "after-rounding" one, and this is why scientists tend to use the carrying of additional significant figures as a means of not introducing additional error (caused by "clipping" the value more than intended).

Anyway.

Curious about this.

Let's have a multiple choice test.

2+2=
[A] 3
[B] 4
[C] 5
[D] all of the above

for large-- or small-- values of 2?

This is absolutely right, says DH scientist - astrophysicist extraordinaire. For scientific research anyway, you use the number with the lesser amount of sig figs and so either rounding it first, or rounding after (I'm pretty sure they will always end up the same so it shouldn't matter the order), will give you the correct answer.

Precision matters in math and science. My 2.0 cents

Well, clearly we have good consensus among scientists... but I'm curious about the mathematician's perspective.

Scientists are trained that 3.2 is 3.2??? not 3.2000 or anything else. Depending on how much you know about the next digit (via statistical analysis) you might know that 3.2 is actually 3.18 to 3.22 or something. Personally I prefer +/- notation for that, but it does get you back to why significant figures are important.


My experience is also that one doesn't always wind up the with precisely the same result, depending upon WHEN you round, which is why one does carry an additional (and sometimes two additional) significant figures through calculations, rounding at the end so that you don't accidentally introduce rounding errors as you go. So that is WHY you add that zero, basically. Kind of. It's certainly why I taught college students in STEM to use that subscript designation. In my classes students would have been using the following notation:

3.14160
+2.71828
5.85988

because neither the zero (artificially used as a placeholder) nor the 8 in the resultant sum is significant.





This method does mean that you have to understand how to manage significant digits through all kinds of calculations and transforms, so that you don't wind up confused about how many are legit in a final result. Addition is the easiest case, clearly.

It just seems better to introduce some ideas early on-- like the notion that unknown values aren't necessarily zero if they haven't been determined.

I think the mathematical idea comes looking at proofs and truth. If you set out to prove two things equal and in the end one number is 0.00000002 less, then they are not equal and the proof fails. PI does not equal 22/7.

This.

This is pure mathematics, and these are just numbers. 3.1416 and 3.14160 are exactly the same number.

Looking at the sum 3.1416+2.71828 and seeing some issue with measurement/precision/significant digits, is a bit like looking at a slice of burnt toast and seeing an image of the Virgin Mary. Its pedagogically unsound.

3.1416 and 3.14160 are exactly the same number.


Well, perhaps they are-- and perhaps they are not. It's ambiguous without additional information.

As a scientist, though, I have to say that I think that the practice of teaching children to ASSUME infinite trailing zeros is probably unsound, at least when looking at the larger picture, but thanks for the burnt toast aside. The non-theoretical universe seems to be primarily composed of burnt toast, ethanol, and mateless socks, by the way.

The vast majority of students (future psychiatrists, attorneys, deputies, teachers, and yes, engineers, scientists and physicians) might be better served to have a clear and intuitive understanding that measured values are not infinitely precise. That's my opinion, but I guess we're each sharing our biases freely at this point. It's easier to explain "the value is exact-- assume an infinite number of significant figures" than to try to explain things the other way round-- for students, I mean.

Clearly math pedagogy concerns itself with such things. Why else would one bother teaching students about significant figures to begin with? The assumption that 3.67 is numerically identical to 3.67000000 means that the idea of significant figures is somewhat extraneous as a math concept to begin with. It is an assumption. Why assume that particular thing rather than something else?

Why teach the topic of significant figures at all if such a thing should be unilaterally assumed in math?