General approach to finding number of significant figures in mixed operations

Question

There are ways to doing operations with significant figures, but not really for mixed operations. I have looked all over the Internet, Stack Exchange, and the textbook I use. I'm surprised, but in any case, I would like a general approach for dealing with the precision, uncertainty, and accuracy of a calculation.

Each operation has its own set of rules. See below:

1. Addition/subtraction - Use the significant figures of the number with the fewest decimal places (digits) for the calculation. $$1.145 + 1.5 = 2.6$$
2. Multiplication/division - Use the significant figures of the number with the fewest significant figures for the calculation. $$1.145 × 1.50 = 1.72$$

However, problems arise in these cases:

1. Trigonometry $$sin(\fracπ{3.1}) = 0.018$$ Here, I used the significant figures of the number with the fewest significant figures for the calculation. In other words, I took 3.1 (2 sig figs) to calculate 0.018.
2. Logarithms $$\log_{10}(0.00002734) = \log_{10}(2.734) + \log_{10}(10^{-5}) = 0.46379 + 0.00001 = 0.4630$$ Here, the significant figures of addition turned out to be how you would do significant figures of multiplication/division, but in general, you calculate by calculating the operation as if it were multiplication/division.

But I’m just generally confused. What if you have a situation as I saw above with the logarithms? Or something like this:

$$\frac{[(0.23 × 5) + (8.521)^5]}3$$

There is not much consensus.

Any clarification would be golden.

Answer 1

Good question. Don't mix significant figures with the error propagation concept and the output of transcendental functions like $\log x, \sin x, \cos x$ etc. These are three different things.

For example $\sin \left(\pi/4\right) = 0.70710678118...$(never ending). There is no experimental or measurement "error" here. The concept of significant figures is meaningless here. Similarly, the output of a logarithm for pure numbers is also exact (or rather it never ends). There is no error. No need to apply significant figure concept there. The easy trick is that you use the maximum numbers in all steps which your calculator is showing and round off only at the end.

Significant figures is just a common sense approach of retaining answers to a realistic experimentally possible value. For example, if we weigh a few crystals of salt on an analytical balance, which can measure up to 4 decimal places of grams, we should record the weight as 0.1074 g. It would be a bad practice to record it as 0.11 g or 0.10740 gram. Both are wrong because we did not retain the four significant figures.

So the Rules 1 and 2 are just qualitative common sense type statements.

In real cases, we must estimate the errors as standard deviations of our measurements by repeatedly making the same measurements.

Scenario I: In reality, if you wish prepare a solution of 1 M NaCl in one liter. You would know the standard deviation of the mass you weighed, and you would also know the standard deviation of your 1 L volumetric flask, then the uncertainty in the molarity of 1 M NaCl would be

$$ (1 M \pm ?)=\frac{58.4427\pm0.0034}{(1.000\pm0.050)(58.4427)}$$

Scenario II: Suppose you measure the hydrogen ion concentration of a solution by titration 10 times. The hydrogen ion concentration is $0.058 \pm0.002$ M. Now I wish to calculate the pH of this solution, which is given by

$$-\log [H^+]=-\log[0.058 \pm0.002]$$.

What is the pH with its experimental error?

How do we determine the error in both scenarios?

As you stated problems start when experimental measurements become more complex involving more complex functions. You need to know partial derivative formula for error propagation.

Suppose your function is $$y = f(X, Z, \ldots \, )$$, where $X$, $Z$ are measured numbers with their associated standard deviations $s_x$, $s_z$ and so on. The the standard deviation of the final calculation should be

$$s_y = \sqrt{ \left( \frac{\partial y}{\partial X} \right)^2 s_x^2 + \left( \frac{\partial y}{\partial Z} \right)^2 s_z^2 + \cdots +}$$

This formula becomes even more involved when the measurement errors depend on each other. See the NIST page
https://www.itl.nist.gov/div898/handbook/mpc/section5/mpc55.htm

Answer 2

In such a case, I would try the methods of extreme incertitudes. I would compare the result of the calculations made first with the exact calculation, and also of the calculation made with highest possible or maximum value. Here, it would be by replacing $0.23$ by $0.24$ and $8.521$ by $8.522$. As the first (exact) result is $14974$ and the second is $14982$, I would give the result $(1.497±0.001) 10^4$