User:FireflySixtySeven/Parallelism operator



The parallelism operator $${\|}$$ (Read "parallel") is a mathematical function that is used primarily as a shorthand notation in electrical engineering. It computes the reciprocal of a sum of reciprocal values, ​​and is defined as


 * $$\begin{matrix}\|:\ & \overline{\mathbb{C}} \times \overline{\mathbb{C}} &\to& \overline{\mathbb{C}}\\

& (a,b) &\mapsto&a \| b = \frac{1}{\frac{1}{a} + \frac{1}{b}} \end{matrix}$$ where $$\overline{\mathbb{C}} = \mathbb{C}\cup\{ \infty\}$$ is the set of extended complex numbers (with its usual rules of operation).

Properties

 * 1) $$a\|a = \frac{a}{2}$$.
 * 2) $$a \neq b \iff \left| a\| b \right| > \tfrac{1}{2} \min(|a|,|b|) $$, where $$ \left| a\| b \right|$$ denotes the absolute value of $$a\|b$$.
 * 3) When $$a$$ and $$b$$ are positive real numbers, $$\left| a\| b \right| < \min(a,b)$$.
 * 4) The parallelism operator is commutative: $$a \| b = b\| a$$.
 * 5) The parallelism operator is associative: $$ \left( a \| b \right) \| c = a \| \left( b \| c \right) = a \| b \| c$$.
 * 6) The parallelism operator has $$\infty$$ as its identity operator, and, for $$a \in \overline{\mathbb{C}}$$, $$-a$$ is the inverse element. Thus, $$(\overline{\mathbb{C}},\|)$$ is an Abelian group.

Examples
Example 1
 * Problem:
 * A bricklayer can build a brick wall in 5 hours. A second bricklayer can build the same wall in 7 hours. How long does it take if both bricklayers work on the wall simultaneously?
 * Solution:
 * $$t_1 \| t_2 = 5\,\mathrm h \| 7\,\mathrm h = \frac{1}{\frac{1}{5\,\mathrm h} + \frac{1}{7\,\mathrm h}} \approx 2.917\,\mathrm h$$
 * Thus, it takes just under 3 hours.

Example 2
 * Problem:
 * Three resistors of resistances $${R_1 = 270\,\mathrm{k\Omega}}$$, $${R_2 = 180\,\mathrm{k\Omega}}$$, and $${R_3 = 120\,\mathrm{k\Omega}}$$ are connected in parallel. What is the total resistance of the circuit?
 * Solution:
 * $$R_1 \| R_2 \| R_3 = 270\,\mathrm{k\Omega} \| 180\,\mathrm{k\Omega} \| 120\,\mathrm{k\Omega} = \frac{1}{\frac{1}{270\,\mathrm{k\Omega}} + \frac{1}{180\,\mathrm{k\Omega}}+ \frac{1}{120\,\mathrm{k\Omega}}} \approx 56.842 \,\mathrm{k\Omega}$$
 * Thus, the circuit has a total resistance of about 57 k&Omega;.