User:FireflySixtySeven/Parallelism operator

Graphical interpretation of the parallelism operator with ${a\|b=c}$

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 kΩ.