Here is the 'simplified' derivation of S.F.. I'm not sure about your background, but I assume you have relatively strong mathematics which is why you are asking about this. There isn't any point for me to keep discussing this with you, S.F. in my opinion is quite useless once you are outside of high-school (some undergraduate work still use it, but many use better methods).
1. Why S.F. is not a good method.Suppose we want to deliver a volume of 90 mL with a 10.00 mL pipette. The implicit error of the 10.00mL pipette is
mL. Since we need to do this 9 times, the final result should be
_{exact}\times (10.00\pm 0.005\, mL)_{4\, sf} = 90.00 \pm 0.045 \approx 90.00 \pm 0.05 \equiv (90.0 mL)_{3\, sf})
. The final answer should only be accurate to 3 S.F. given the relative uncertainty of the initial measurement.
However, by the rules of S.F., we have
_{exact} \times (10.00 mL)_{4\, sf} = (90.00\, mL)_{4\, sf})
This is one of the reasons why we don't use S.F. for proper treatment of uncertainties.
2. Derivation of S.F. lawsSuppose we have a quantity
, then its significance can be approximated by
\approx -\log_{10} \frac{\Delta x}{x})
(this is a rough approximate of what the sf actually does. Doing the following calculations become a lot more messy if we use the proper definition, as the proper definition involves a discontinuous function)
i. Multiplication and division by an exact quantity = cx \pm c\Delta x)
 \approx -\log_{10} \frac{c \Delta x}{cx} = -\log_{10} \frac{\Delta x}{x} = sf(x))
ii. Multiplication of two quantities with uncertainties(y\pm \Delta y) = xy \pm (x\Delta y + y \Delta x + \Delta x \Delta y))
Ignoring the second order uncertainties, we get
 \approx -\log_{10} \frac{x\Delta y + y \Delta x}{xy} = -\log_{10} \left( \frac{\Delta y}{y} + \frac{\Delta x}{x} \right))
Here, two things can happen. Firstly,
=sf(y)\implies \frac{\Delta y}{y}\sim \frac{\Delta x}{x})
, therefore
\approx -\log_{10}(\gamma) +sf(x))
where
and
 \approx -\log_{10} 2 \approx -0.3)
, so
\sim sf(x)\sim sf(y))
Alternatively,
>sf(y))
, implying
} \ll 10^{-sf(y)})
, thus
, thus
\sim sf(y))
, in other words, we take the sf of the quantity with fewer sf.
Thus, we arrive at the final rule
 = \min (sf(x),sf(y)))
iii. Taking the reciprocal_{s=x} = \frac{1}{x} \pm \frac{\Delta x}{x^2})
The significance is
 \approx -\log_{10} \frac{\Delta x/x^2}{1/x} = -\log_{10} \frac{\Delta x}{x} \equiv sf(x))
iv. Division of two quantities with uncertaintiesCombining ii. and iii., we arrive at
 = \min (sf(x),sf(y)))
v. Addition and Subtraction + (y \pm \Delta y) \approx (x+y) + (\Delta x + \Delta y))
Applying the scale argument, we see that absolute errors are important here, not relative errors. S.F. is not used here.
vi. Logarithms \approx \log_{b}(x) \pm \frac{\Delta x}{x\log_e{b}})
It can be shown for reasonably sized values of
, the digit-significance (i.e. the decimal place) of the absolute error is
) \approx \log_{10} \log_{e}(b)-\log_{10} \frac{\Delta x}{x})
, since
)
is small for small
, we have thus shown that
)\sim sf(x))
, that is, the significance of the error (i.e. the decimal place) is equal to the number of sf in x.
vii. ExponentialsA similar argument to vi. can be constructed to show that
 \sim -\log_{10}(\Delta x))
, that is, the decimal place of the initial error is equal to the number of sf in the exponential.
viii. Other transcedentals) \approx -\log_{10} \left( \frac{f'(x)}{f(x)} \Delta x \right) = -\log_{10} \left( \frac{x f'(x)}{f(x)}\right) + sf(x))
For range of values of
that are of interest, if
}{f(x)} < 10)
, then
) \sim sf(x))
.
This really depends on what kind of function you have. If
 = \sin(x))
, then
}{f(x)} = \frac{x}{\tan(x)})
, which goes to
and
at certain x values.
On the other hand, the power functions
=x^n)
will always behave nicely for reasonable values of n. (Which makes sense in terms of rules ii, iii and iv)
The following are methods I use.
3. Method of relative errorsThis is more popular at lower level undergraduate studies.
Suppose we have a bunch of quantities
and so on. Define the relative error to be
=\frac{\Delta x}{x})
Rule of addition: Use absolute uncertainties
Rule of multiplication:
 \approx \epsilon(x)+\epsilon(y))
Rule of transcendentals:
) \approx \frac{f'(x)}{f(x)}\Delta x)
4. Method of standard deviationsThis is the most popular method for treating uncertainties of physical quantities, as everything is almost always normally distributed. Since the normal distribution is a nice function, the rules for propagation is exact. For some non-linear functions, the errors need to be relatively small.
Suppose we have a bunch of quantities
with standard deviations
)
and so on.
Rule of addition: Use absolute uncertainties
Rule of multiplication:
}{xy}\right)^2 \approx \left(\frac{\sigma(x)}{x}\right)^2 + \left(\frac{\sigma(y)}{y}\right)^2)
Rule of transcendentals:
)^2 \approx \left(\frac{df}{dx}\right)^2\sigma(x)^2)
, can be generalised to multivariable functions using partial derivatives and the covariance matrix.
I'm out.
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