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Why significant digits are counted the way they are in multiplication and division?

Anonymous Sat 26 Dec 23:54:24 2020 No.12511332 View Reply Original Report

Quoted By: >>12511355 >>12511734

Significant Digits: Why in multiplication and division, is the smaller, more accurate measurement, still considered the 'weakest link'?
Example: 33.58*1.007/0.00705 = 3 SD = 4.80x10^3

Please explain in practical everyday language, not just formulas.

And no, this isn't homework so no moderating control-freak bullshit please. If I had a real teacher I wouldn't have to ask this online, and in real homework situations they don't give a shit if you know why, they just care that you know how to do the calculation, which I do.

I just have no chance in hell of getting this question answered in WSR, and I tried asking in in Quora where there is less bullshit but it needs you to word the question as a single sentence which, I just can't, or else I won't get an answer I understand.

Anonymous

Anonymous Sun 27 Dec 2020 00:02:36 No.12511355 Report

Quoted By: >>12511360

>>12511332
Put all of those in reference to a power of ten. Then you choose the one with the fewest number of digits, because the error is going to be the square root of the sum of squares, therefore it will be greater than the error on any that have more SF than the smaller one.

(2+/-1)*(2+/-0.15)
Is some value between 1*1.85 and 3*2.15
The difference between those two bounds, means that the error is only reasonable to express wrt to the fewer sf as our error is over an order of magnitude (a sf) larger than 0.15

Anonymous

Anonymous Sun 27 Dec 2020 00:05:16 No.12511360 Report

Quoted By: >>12511480 >>12511512

>>12511355
Sorry I should have just said 0.1 and it would be
2+/-1 and 2.0+/-0.1

We drop the .0 as the multiplicative error essentially subsumes the larger number of sigfigs.

Anonymous

Anonymous Sun 27 Dec 2020 00:45:09 No.12511480 Report

Quoted By:

>>12511360
So it's because we can't be sure of anything beyond the smaller digits, despite the leading zeroes indicating that whatever I used to measure that second number, is more accurate than the first, kinda deal?

Sorry for being a smooth-brain, this has just been bugging me. I'm trying to figure it out practically.

Putting them all in reference to a power of 10 made a ton of sense and made it really clear mathematically, thanks for your answer.

I'm still trying to wrap my head around why the least significant numbers (the 0.00705) makes my answer 4800, as the more practical answer, and not 4,796.46

The error is too insignificant beyond the whole digits (the 4800) but I'm having trouble understanding why, when my most accurate measurement is the 0.00705.

When you say it's subsumed, I'm thinking that means anything generated beyond the 0.00705 is sucked into the answer kinda thing, like it can't/won't go past the 0.00000 mark, or if it does it's not reliable for some reason, or.. I don't know how to ask this properly.

I'm really sorry for being slow. And I understand if you don't want to answer further.

At least for practical purposes you helped me with what you've already answered, thank you Anon.

Anonymous

Anonymous Sun 27 Dec 2020 00:58:07 No.12511512 Report

Quoted By:

>>12511360

Putting them in reference to power 10 makes it super clear mathematically, but I still don't truly understand it.

I'm really sorry for being a smooth brain. I just watched an intro to chemistry video and its this last example that really made me confused, because I can't understand it in reality.

The example was: 33.58*1.007/0.00705
= 3 Significant digits
= 4.80x10^3 (i.e.4800)

I have no background in math, or chemistry, so I want to understand this for real.

Say I have 3 scales, my least accurate scale is the 33.58, so why does the 0.00705 (the least amount of significants, 3) determine the answer as 4800? Why isn’t the answer 4,796.46, since the least accurate measurement is to two decimal places?

I know you already answered me, but, I'm very sorry I don't understand the answer in real world terms.

I understand fully if you or someone else doesn't wish to answer further, and I thank you for your answer and trying.

Anonymous

Anonymous Sun 27 Dec 2020 01:47:20 No.12511622 Report

Quoted By:

Update: I think I semi-found my answer by searching online. Someone said this:

[...] ultimately, the details of the sigfig rules are arbitrary, and are only a rough approximation of proper error propagation considerations.

So that kind of answers what I'm asking, but I still don't know why it was chosen, I assume it's because it's more accurate, but I don't know why it's more accurate. I guess that's what I'm asking.

I suppose I would have to understand something about error math to truly understand what I'm asking. Anyway, sorry if I bothered anyone.

Anonymous

Anonymous Sun 27 Dec 2020 02:16:53 No.12511697 Report

Quoted By:

Honestly the 'significant digits' to include when reporting any computation is largely a matter of context. In your example you give a number 3*10 + 3 + 5:10 + 8:100 and wish to multiply this number by another number, 1 + 7:1000. At this point you ought to ask the question, why is one of these numbers reported to 3d while the other is reported to 2d. How could this arise if both were the result of a similar physical measurement? Its like measuring 33 meter, 58 cm and then saying, now I want to multiply this number by 1 meter and 7 mm to compute the resulting area. Resulting area? What are your supposed units of the resulting area m^2, cm^2 mm^2? Are square mm significant in an area on the order of square meters? These are questions that ought to be asked when computing anything to some precision (significant digits). Figure out the answer to the first operation (multiplication) before asking anything about the second.

Anonymous

Anonymous Sun 27 Dec 2020 02:42:58 No.12511734 Report

Quoted By:

>>12511332
Let f(x,y)=x*y
Then the significant figures of the result is equal to significant figures of (x,y) - $\log_{10} \left( \frac{||J(x,y)||}{||f(x,y)||/||(x,y)||} \right)$ where J(x,y) is of course the Jacobian matrix of f(x,y). The derivation is elementary.

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