I've found a rule that says, "When two quantities are multiplied, the error in the result is the sum of the relative error in the multipliers."
Here, why can't we use absolute error? And why do we've to add the relative errors? Why not multiply them?
Please give me an intuition to understand the multiplication of two uncertain quantities.
It basically comes from calculus (or more generally just the mathematics of change).
If you have a quantity that is a product $z=x\cdot y$, then the change in this value based on the change of $x$ and $y$ is$^*$ $\Delta z=x\Delta y+y\Delta x$. So then it is straightforward that
$$\frac{\Delta z}{z}=\frac{x\Delta y+y\Delta x}{xy}=\frac{\Delta x}{x}+\frac{\Delta y}{y}$$
The reason you don't use absolute uncertainty or multiply the relative uncertainties is the same reason why $(a+b)^2\neq a^2+b^2$. It's just not the result you get when you do the math.
$^*$We are neglecting the term $\Delta x\cdot\Delta y$ in $\Delta z$, since ideally $\Delta x<x$ and $\Delta y<y$ to the extent that $\Delta x\cdot\Delta y\ll xy$ such that $\Delta x\Delta y/xy$ is much less than both $\Delta x/x$ and $\Delta y/y$.
You get the best intuition if you just take two easy numbers with a possible error and multiply it. Choose 100±1 and 200±4 the relative errors are 1/100 or 1% and 4/200=2%.
Now multiply and you get for the positive error 101*204=20604=20000+604 or an error of about 3%. Multiplying the absolute error would give you 1*4 instead of 604, multiplying the relative errors would give you 2/10000 or 0.02%.
Try it with any two other numbers.
