Uncertainty when multiplying two measured values

Question

Until now I have been using this rule:

"The result of the multiplication or division must not have more significant figures than the operands."

when dealing with uncertainty in multiplification. Now let's say that I have:

\begin{align} a &= 20 \pm 0.2=20\,(1\pm 0.2/20)=20\,(1\pm\color{blue}{0.01}) \longleftarrow \text{the rule holds}\\ b &= 10 \pm 0.1=10\,(1\pm 0.1/10)=10\,(1\pm\color{blue}{0.01}) \longleftarrow \text{the rule holds} \end{align}

In this case I sucessfuly aplied the rule when dividing. I calculated a relative uncertainties which have the same number of significant figures (1) than the operands do (1). Now I want to show that if we multiply the two measured values $a$ and $b$, the relative uncertainties must sum up like the rule says it:

"If we multiply two measured values, their relative uncertainty must sum up."

So I calculate the scalar product of the average values, maximum and minimum values:

\begin{align} \text{average:}~&\overline{a}\cdot\overline{b} &&= 20 \cdot10=200 &&{\longleftarrow \text{the rule holds}} \\ \text{maximum:}~&a_{max}\cdot b_{max} &&= 20.2 \cdot 10.1 = 204.02 = 204 &&{\longleftarrow \text{the rule holds}} \\ \text{minimum:}~&a_{min}\cdot b_{min} &&= 19.8 \cdot \underbrace{9.9}_{2~s.f.} = 196.02 = \color{red}{20\cdot 10^1} &&{\longleftarrow \text{?}} \end{align}

In the first line the rule holds out of the box, because result 200 has 1 significant figure just like the operands 10 and 20. In the second line I had to round up the calculated value 204 which has 3 significant figures just like operands 20.2 and 10.1.

At this point the second quoted rule starts to show as the maximum is 204 and that's 4 higher than average and we could almost say that $a \cdot b = 200 \pm 4 = 200\,(1\pm 4/200) = 200\,(1\pm \color{blue}{0.02})$. And it would imediately became obvious that relative uncertainties sum up like $\color{blue}{0.01} + \color{blue}{0.01} = \color{blue}{0.02}$.

But when I go and calculate the minimum I just can't get the result 196 which would be 4 lower than 200. That's because first rule is forcing me to have 2 significant figures in the result so 196.02 is round up to $\color{red}{20\cdot 10^1}$ and not 196 which I need. This is because operand 9.9 has only 2 significant figures!

Can anyone help me out here? Where did I mess up?

Answer 1

This is a problem with subtraction where you loose some significant figures as a result of subtraction. Treat the problem differently:

\begin{align} \text{min:}~~a_{\text{min}} \cdot b_{\text{min}} &= 19.8 (10.0 - 0.1) \\ &= 19.8 \cdot 10.0 - 19.8 \cdot 0.1 \\ &= 198 - 1.98 \\ &= 196.02 \end{align}

now apply significant figures to the results by multiplication rule:

\begin{align} 198 \cdot 10.0 &\longrightarrow \text{result is 3 s.f.}\\ 198 \cdot 0.1 &\longrightarrow \text{result is 1 s.f. because of the last value} \end{align}

by subtraction rule follow number with least decimal point, which is 198, so the final result is 196 which is still 3 significant figures.

Answer 2

It's a good rule but you have to use judgement as well. Your analysis is pretty good. But you can go deeper and realize that your + or - 0.2 or 0.1 is actually a statistic. Typically it represents about 1 std deviation or a 67% probability that most results are within this bound. Every measurement is has an average and a std deviation. The more measurements the better.

Answer 3

All the maths involved are just simple maths operations like addition, subtraction , multiplication or division. All these are, of course, scalar.

Another point to note is to use a bit more digits to calculate and truncating only at the final steps. This is to avoid cummulative truncation errors.

The rules are universal. But we have to be careful of lost of significant digits during a subtraction process. If it is just the end result of a subtraction, there is no problem. But your case involves a multiplication of a large number with a result with a small difference. The alternative approach still follows the rules and maintains the sf of the larger number before subtracting the error quantity with lesser sf.

You may wish to look up destructive cancellation of two numbers differing by a very tiny amount.

QH