Mathematics behind parallel resistors

Question

Could someone explain the mathematical logic behind the summation of total resistance for parallel resistance? $$\frac{1}{R_{tot}} = \frac{1}{R_1}+\frac{1}{R_2}+\dots+\frac{1}{R_n}$$ I saw the similar way of computing many times with other physics phenomena, for example when calculating spring coeficient of more springs connected in series, but I don't really understand it.

Answer 1

The electrical conductance is given by $\frac{1}{R}$, meaning the inverse value of resistance. The bigger conductance is, the better it conducts electricity. If you plug two resistors in parallel, their conductance is summed together, which is quite intuitive.

Answer 2

When resistances are connected in parallel, their two electrodes are perforce respectively at the same potential. The current that flows in them is (by definition of the resistance) inversely proportional to their resistance. Hence the total current, which is inversely proportional to the equivalent resistance, is proportional to the sum of the inverses of the individual resistances.

$$I=\sum_k I_k,\\\frac UR=\sum_k\frac{U}{R_k}.$$

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With series resistances,

$$U=\sum_k U_k,\\RI=\sum_kR_kI.$$