Resistors in Parallel: A Rational Expression

Connect two resistors side by side and current has two paths, so the total resistance drops below either one. The rule adds reciprocals, a complex fraction that simplifies to a product over a sum.

How it works

Complex fractions and rational expressions

In one sentence

A rational expression is a quotient of polynomials. A complex fraction such as $\frac{1}{1/R_1 + 1/R_2}$ simplifies by multiplying numerator and denominator by the least common denominator $R_1R_2$ , giving $\frac{R_1R_2}{R_1 + R_2}$ for $R_1, R_2 > 0$ .

General formula

\frac{1}{\htmlData{sym=0}{{R}}} = \frac{1}{{\htmlData{sym=1}{R_1}}} + \frac{1}{{\htmlData{sym=2}{R_2}}} \quad\Longrightarrow\quad {\htmlData{sym=0}{R}} = \frac{1}{\frac{1}{{\htmlData{sym=1}{R_1}}} + \frac{1}{{\htmlData{sym=2}{R_2}}}} = \frac{{\htmlData{sym=1}{R_1}}{\htmlData{sym=2}{R_2}}}{{\htmlData{sym=1}{R_1}} + {\htmlData{sym=2}{R_2}}}

Reciprocals measure how easily current flows (conductance), and those simply add. Flip back to get the resistance.

What the symbols mean

$R$
Total resistance
The resistance of the pair as seen by the rest of the circuit.
$R_1$
First resistor
Resistance of one path, in ohms.
$R_2$
Second resistor
Resistance of the other path, in ohms.

In parallel, the resistors carry currents $V/R_1$ and $V/R_2$ , and the total current is their sum, so $\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}$ .
Solving for $R$ gives a complex fraction. Multiply its top and bottom by $R_1R_2$ to clear the small fractions: $R = \frac{R_1R_2}{R_1 + R_2}$ .
As a function of $R_2$ this is rational: it starts at 0, rises and levels off toward $R_1$ , a horizontal asymptote.
The total is always smaller than both resistors: adding a path can only make it easier for current to flow.

Learn the math

Rational ExpressionsAlgebra · Lesson 6, Preliminaries Rational FunctionsAlgebra · Lesson 33, Common Graphs Kirchhoff's Laws and Circuit AnalysisE&M · Lesson 14, Current and DC Circuits