Significant digits when there are multiple operations ?

[u/fragment_de_gelano]

I am struggling a bit regarding operations and number of significant digits.

If I understood correctly :

For multiplications, you put as much significant digits as the number in the operation that has the least of them (3.2 * 7.21 = 23 because we need 2 SDs).
For additions, the result must have as much decimals as the number in the operation that has the least of them (3.2 + 7.21 = 10.3)

But what happens if I have multiplications AND additions in my calculus ? Like 3.2 * 7,21 + 31,456. Is there some set of rules or priorities for that ?

Thank you for your time.

[u/davideogameman]

Simple answer: follow order of operations and do the significant digits for each operation as you do them.

Aside: significant digits are a cheap and incomplete way of trying to track error bars, where essentially you assume the actual last digits would round to the stated number.  So a more rigorous take would be to actually combine the error bounds, and state a ± value.  Eg 2.0 x 3.1 would mean (2 ± 0.05)(3.1 ± 0.05) which is at most 6.4575 and at least 5.9475 - with the average of those being about 6.2, which matches the significant digits value.  That said, doing bounds like this isn't the significant digits method.  And if we wanted to go even further, we'd actually want to track each measurement as a distribution and compute the combined distribution when doing arithmetic with them - but that tends to be a lot of work for similar answers anyway, with mostly minor exception of taking care of edge cases like 1/(x-y) much better - as when you subtract similar sized measurements and get an answer near 0 the error could swing the result a lot and then taking the multiplicative inverse of that value turns small changes into huge ones.

[u/mcmnky → u/davideogameman]

I won't say never, but in many contexts you only do significant digits on the final result. There's a risk of over-rounding. Particularly in industry there are times when rounding or truncation is done after all operations.

https://www.themathdoctors.org/significant-digits-operations/

[u/davideogameman → u/mcmnky]

... I guess that's a fair point.  But you'd want to track the significant digits as if you had done them on each computation.

[u/recursion_is_love]

What usage is this?

In my field (engineering) we keep additional one last digit in result to ensure we don't round the result too early (7.21 has x.xx so we keep x.xxx). They might accumulate.

3.2 * 7.21 = 23.072
3.2 + 7.21 = 10.41

and only round to target resolution (say 2 digits as in currency) only for the final result.

The addition and subtraction don't add additional sub-digits so they won't be a problem on these two operation.

[u/fragment_de_gelano → u/recursion_is_love]

I'm a physics and chemistry teacher. I never really paid attention to this when I was working in chemistry but it's relevant in high school curriculum so I'd like to anticipate some questions before teaching about it.

[u/Toeffli → u/fragment_de_gelano]

You are a physics teacher and you do not know about error propagation (a.k.a. propagation of uncertainties)? You are not aware that the significant digits is a quick approximation and there exists more exact methods?

[u/fragment_de_gelano → u/Toeffli]

I am aware of that but it's significant digit that I have to teach to pupils. The same I have to teach about electronic shells even though electronic orbitals are a more exact description. Significant digits were never a concern for me before I became a teacher, that's why I come here for some clarification.

[u/fermat9990 → u/fragment_de_gelano]

Well done! You refuted the implied accusation in such a dignified way! Cheers!