Resistors

Every electrical appliance (such as a light bulb) has a certain resistance. If the appliance is connected to a given voltage, the higher its resistance, the lower the current flowing through the appliance. The unit of measurement for resistance is the ohm. To avoid round-off errors that can affect floating-point numbers, we will use rational numbers (fractions of positive integers) to represent the resistance of an appliance numerically. There are two basic ways to connect two or more appliances into a configuration: in series (Figure 1) or in parallel (Figure 2). Two or more configurations can be further connected in series or in parallel to yield yet another (more complex) configuration, and this process of constructing more complex configurations from existing ones can be repeated any number of times (Figure 3). In general, a configuration is either a single appliance, a series connection of two or more configurations, or a parallel connection of two or more configurations. The resistance of a configuration of resistors can be computed according to the following two rules:

1. The resistance of a series configuration is the sum of the resistances of its component configurations.

2. The resistance of a parallel configuration is the reciprocal of the sum of the reciprocals of its component configurations.

In Figure 1, the resistance of the configuration is 3/2 + 3/4 + 1/4 = 5/2 ohms.

In Figure 2, the resistance of the configuration is 1/(1/(3/2) + 1/(1/2) + 1/(1/4)) = 3/20 ohms.

In Figure 3, first calculate 1/(1/(1/2) + 1/(2/3)) + 2/5 = 24/35 and 1/2 + 1/(1/(2/3) + 1/(2/5)) + 3/2 = 9/4. Adding the reciprocals of these two values and taking the reciprocal of the sum, we get $72/137$ ohms.

A configuration can be represented in text format.

• A single appliance is represented by the numerical value of its resistance (without parentheses).

• A configuration that is a series connection of several configurations is represented as a list of the representations of its components, separated by the ampersand character ("&") and enclosed in a pair of parentheses.

• A configuration that is a parallel connection of several configurations is represented as a list of the representations of its components, separated by the vertical bar character ("|") and enclosed in a pair of parentheses.

Thus, Figures 1, 2, and 3 are represented in text format by the following expressions:

(3/2 & 3/4 & 1/4)(3/2 | 1/2 | 1/4)(((1/2 | 2/3) & 2/5) | (1/2 & (2/3 | 2/5) & 3/2))