In a thread I posted recently about significant figures, someone said that they are defined as follows. I will use an example to save time trying to generalize the definition.
The person said, for example, that 2.34 has 3 significant figures not because we can count three digits in the usual way, but because significant figures are defined by the ratio of the uncertainty to the number. Here, we have $2.34\pm 0.005$, and the required ratio is $\frac{0.005}{2.34}$ which is roughly $\frac{10^{-3}}{10^0}=10^{-3}$. Thus, the number of significant figures is $3$ (as per the exponent of the result, which is -3).
Another example is 0.00002. Here the ratio is $\frac{0.000005}{0.00002}$ which is roughly $\frac{10^{-6}}{10^{-5}}=10^{-1}$, hence there is 1 significant figure in 0.00002.
My problem is that this seems to be a much more complicated definition than the usual one which is defined as "the number of digits ignoring all leading zeros". Since definitions should be simple, it seems to make more sense to define significant figures in terms of the simpler definition. Of course, both would suffice as a definition, since they are equivalent: one can show that each definition implies the other.
So what is the actual definition of significant figures? Is it the simple one related to simply counting the digits, or is it the complicated one involving ratios?
Also, another question that is still bugging me is regarding the motivation behind the definition of significant figures. How was the definition discovered? Ie what was the motivation for defining significant figures in this way?
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