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I understand the sig fig rules that pertain to addition and multiplication problems, however, in multi-step problems, I have not been able to find a systematic approach to getting the right amount. Everybody who I have asked just said "it's tricky so be careful".

If the problem is: (5.01 + 4.1) / (1.00 x 1.00) Is the answer 9.11 or 9.1? The addition part indicates 2 sig figs because the tenths place is the last accurate decimal, while the multiplication part indicates 3 sig figs. Does it depend on whether the result inside the parentheses is added/subtracted or multiplied/divided from the other result?

What about a problem like: (2.00 x 30) + 5.01 + 2.0 Is the answer 60. or 67.0? The multiplication indicated the former, the addition indicates the latter.

Problems where the numerator has numbers being added and the denominator has numbers being multiplied like: (5.01 + 2.0 - 3) / (60.01 x 3.0) are even more confusing, especially when scientific notion is mixed in. I want to know an approach that goes step-by-step and works 100% of the time.

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  • $\begingroup$ Related: Significant figures addition vs multiplication $\endgroup$ – user7951 Sep 6 '19 at 14:57
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    $\begingroup$ First, note that there is no statistically correct way a single number can represent both a best estimate plus its error, so the rules for significant figures are very useful in practice, but they are neither perfect nor without some exceptions. For your first problem, the answer is 9.1. For your second question, is 30 an integer or is it 30. (2 sig. figs.) or is it 3. times 10 to the first power (1 sig. fig.). These kinds of ambiguities are yet another good reason to use scientific notation. Finally, forget working 100% of the time: not happening! $\endgroup$ – Ed V Sep 6 '19 at 17:26
  • $\begingroup$ For physicist John Denker's 100 page discussion of "Uncertainty as Applied to Measurements and Calculations", see this: av8n.com/physics/uncertainty.htm . Personally, I think he goes way too far in his criticism of significant figures, but I post this to show that not everyone agrees about significant figures. Good for starting arguments, though. ;-) $\endgroup$ – Ed V Sep 6 '19 at 18:41
  • $\begingroup$ In the second question, 30 is an integer. If I wanted to represent it as having 2 significant figures, I would write 30. At least, that's what i'm assuming you're "supposed to do" to represent 2 significant figures and hope my exam does. Should scientific notation be used in every single possibility where they can be used? Should I never write 30, but instead write 3 x 10^1? $\endgroup$ – Albert Goldman Sep 6 '19 at 22:49
  • $\begingroup$ Unfortunately, people usually write 30 (which looks like the integer) when they really mean 2 sig. figs. and hope that context will resolve ambiguity. But it is a problem. Scientific notation is best and 30. is fine if (and it is a big if) the decimal point is not mistaken for a "full stop". And notice that 30. at the end of a sentence causes the issue to pop up again. All in all, very annoying, especially if points are lost on an exam or homework. $\endgroup$ – Ed V Sep 6 '19 at 23:33
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For the following, I take 3 to have one significant figure, 67 to have 2 significant figures, and 100 to have one significant figure. If you want to avoid confusion with exact integers (such as a stoichiometric coefficient), use scientific notation for numbers lacking a decimal point, as in $\pu{0.3e1}, \pu{6.7e1} \text{ and }\pu{0.1e3}$.

Pencil and paper

The step-by-step approach is to consider each step separately (observing the correct order of priority), keeping track of significant figures of the intermediate results. Then, calculate the number without any rounding of the intermediate results, and record it with the appropriate number

(5.01 + 4.1) / (1.00 x 1.00)

  1. The sum will be significant to a tenth, so 2 significant figures.
  2. The product will have 3 significant figures because that is the minimum of the factors
  3. For the division, the numerator has 2 significant figures and the denominator has 3, so the result has 2 significant figures.
  4. The arithmetic result without any rounding is 9.11, so you write down 9.1.

(2.00 x 30) + 5.01 + 2.0

  1. The product has one significant figures, least significant figure is the tens.
  2. Adding 5.01 will have least significant figure as the tens.
  3. Adding 2.0 will have least significant figure as the tens.
  4. The arithmetic result (unrounded) is 67.01, so you write down 70 or $\pu{0.7e2}$.

(5.01 + 2.0 - 3) / (60.01 x 3.0)

  1. The sum of three numbers will be significant to the ones, giving one significant figure.
  2. The product will have 2 significant figures because 3.0 does too.
  3. The overall answer will have 1 significant figure because the numerator does too.
  4. The arithmetic result (unrounded) is 0.02227406543..., so you write down 0.02.

Using a tool

I wrote a calculator in a browser that knows about significant figures (and units), described in J. Chem. Educ.2015, 9211, 1953-1955. For the first example, it gives the following output (input is:"(5.01 + 4.1) / (1.00 * 1.00)"):

$$\mathrm{result} = \dfrac{5.01 + 4.1}{1.00 \cdot 1.00}$$

$$\ \ \ =\dfrac{9.11}{1.000}$$

$$=9.1\ \ $$

The intermediate results are written with one extra digit ("guard digit") so that no information is lost (in the pencil and paper section above, I kept all digits until rounding at the very end). If you write down and then reuse intermediate results, you should follow that practice (unless your teacher tells you otherwise). The calculator is available at http://pqcalc.pythonanywhere.com/. Behind the scenes, it uses a slightly more sophisticated error propagation model, so there are times when it shows a digit more than you would expect. Also, different from most textbooks, the calculator interprets numbers without decimal points as exact integers, and numbers with a trailing decimal point (67. or 100.) as significant to the ones position.

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    $\begingroup$ For the second question you wrote "the product has two significant figures, least significant figure is the ones", but because 30 only has one significant figure, shouldn't the intermediate result have 1 significant figure as it is the minimum of factors? How does the number of significant figures go from 1 to 2? $\endgroup$ – Albert Goldman Sep 6 '19 at 23:09
  • $\begingroup$ @AlbertGoldman I'll fix it. $\endgroup$ – Karsten Theis Sep 6 '19 at 23:35
  • $\begingroup$ Oh that makes a lot more sense. My rounding was off. Thank you. $\endgroup$ – Albert Goldman Sep 7 '19 at 0:03
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The valid digit number approach is simplification of evaluation of absolute and relative errors of values.

Digits

For addition/subtraction, the order (position) of last valid digit of the result keeps the higher of the orders of the last valid digits:

$$12.3 + 4.56 = 16.9$$ $$12.34 + 4.56 = 16.90$$

For multiplication/divisions, the number of valid digits is minimum of the number of valid digits

$$2.0 \times 2.3 = 4.6$$ $$2.00 \times 2.3 = 4.6$$ $$2.00 \times 2.30 = 4.60$$

Errors

The last valid digit of the result should match the order of the first valid digit of estimation of standard deviation or confidence interval.

E.g. $10.2 \pm 0.1$

Only for big number of measurements can be SD known with 2 valid digits,

like $10.23 \pm 0.45$

Errors of values follow additiveness of squares of standard deviations.

Absolute ones for additions/subtractions,

$$s^2_{A+B}=s^2_{A}+s^2_{B}$$

relative ones for multiplications/divisions.

$$s^2_{r,A+B}=s^2_{r,A}+s^2_{r,B}$$

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  • $\begingroup$ I am not given standard deviation in pure math problems. If there is some way of determining standard deviation or "confidence level" with just math, I'd like to know it. It would be helpful if you looked at my example problems and tell me how to think about them. $\endgroup$ – Albert Goldman Sep 6 '19 at 15:53
  • $\begingroup$ You can always take 5.01 as 5.01 +- 0.01. It would have more valid digits with better precision. $\endgroup$ – Poutnik Sep 6 '19 at 15:53

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