statistics - Why is propagation of uncertainties quadratic rather than linear?

1) Up until now, during practical work sessions, I always used these formulas for uncertainty propagation:

if $C = A+B$ or $C = A-B$ $$\Delta C = \Delta A + \Delta B$$ if $C = AB$ or $C = \frac{A}{B}$ $$\frac{\Delta C}{C} = \frac{\Delta A}{A} + \frac{\Delta B}{B}$$ if $C = A^m$ $$\frac{\Delta C}{C} = |m|\frac{\Delta A}{A}$$

These formulas are derived from the expression for the differential of a function of multiple variables:

if $C = f(A,B)$ $$ dC = \frac{\partial C}{\partial A}dA + \frac{\partial C}{\partial B}dB \Rightarrow \Delta C = \frac{\partial C}{\partial A}\Delta A + \frac{\partial C}{\partial B}\Delta B$$

and I think that makes sense because the value C is just a function of two variables that happen to be measurements.

2) But this morning, I was told that this is wrong and I should actually use these instead:

if $C = A+B$ or $C = A-B$ $$(\Delta C)^2 = (\Delta A)^2 + (\Delta B)^2$$ if $C = AB$ or $C = \frac{A}{B}$ $$\left(\frac{\Delta C}{C}\right)^2 = \left(\frac{\Delta A}{A}\right)^2 + \left(\frac{\Delta B}{B}\right)^2$$

which are apparently derived from a general formula:

$$(\Delta C)^2 = \left(\frac{\partial C}{\partial x_1}\Delta x_1\right)^2 + \left(\frac{\partial C}{\partial x_2}\Delta x_2\right)^2 + \hspace{0.3cm}...$$

or

$$\Delta C = \sqrt{\left(\frac{\partial C}{\partial x_1}\Delta x_1\right)^2 + \left(\frac{\partial C}{\partial x_2}\Delta x_2\right)^2 + \hspace{0.3cm}...}$$

3) So:

• Why is this formula better?


• Where does it come from?


• What does it actually represent? (Do I recognize the shape of a norm in that last formula?)


• What is wrong with the formula of the differential?

I'm asking a lot, but the propagation of uncertainty always confused the ship out of me.