# Hidden rules in uncertainties?

Lets start with the EMF equation, $$\epsilon = I(r+R)$$, where $$\epsilon$$ is EMF, $$R$$ is the resistance of the circuit, $$r$$, internal resistance and finally $$I$$, current. If we are given the values, $$I = 2 \pm 0.1$$ $$A$$, $$V = 1.5 \pm 0.2$$ $$Volts$$ where $$V$$ is the terminal potential difference, and $$\epsilon$$ $$= 4.5 \pm 0.2$$ $$Volts$$. From the EMF equation we can expand and get $$\epsilon = Ir + IR$$, and finally $$\epsilon = V + Ir$$, where $$V$$ is terminal pd. Rearranging for $$r$$ gives $$\frac{\epsilon - v}{I}$$. The rules we have are, if $$Z = A \pm B$$, the absolute uncertainty in $$Z$$ is just the sum of the absolute uncertainties in $$A$$ and $$B$$, and if $$Z = A *B$$, the percentage uncertainty in $$Z$$ is the sum of the percentage uncertainties in $$A$$ and $$B$$, the same rule goes for $$Z = \frac{A}{B}$$. Putting this all together we know the percentage uncertainty in $$r$$ will be the sum of the percentage uncertainties in $$\epsilon - V$$ and $$I$$, giving us $$\frac{100 *0.3}{1.5} + \frac{100*0.1}{2}$$ $$= 25$$. This is where the confusion starts. What if I had rearranged it to $$\epsilon - V = Ir$$, we can find the percentage uncertainty in $$\epsilon - V$$ to be $$20$$ just like before, but we know that the percentage uncertainty in $$\epsilon - V$$ will be equal to that of $$Ir$$, so $$20$$ equals the percentage uncertainty in $$r +$$ the percentage uncertainty in $$I$$, but this gives $$15$$ percent, not $$25$$. Where is my misunderstanding?

Let $$D=\epsilon-V$$, so you have $$D=Ir$$. The uncertainty propagation is NOT symmetric. i.e. uncertainty of D given $$I$$ and $$r$$ uncertainties is different from the uncertainty of $$r$$ given uncertainty of $$D$$ and $$I$$.
In more precise terms, $$\frac{\Delta D}D=\frac{\Delta I}I+\frac{\Delta r}r\qquad\text{does NOT imply}\qquad\frac{\Delta r}r=\frac{\Delta D}D-\frac{\Delta I}I$$but rather the one with the positive sign instead of negative. Which equation to use, depends upon which uncertainty is unknown and which are, so that the propagation defines a direction, and that direction will be the one the determines which equation to use.