Kept on Wikipedia:Parallel (operator)

Graphical interpretation of the parallel operator with $a\Vert b=c$

The parallel operator $\Vert$ (pronounced "parallel", following the parallel lines notation from geometry) is a mathematical function which is used especially as shorthand in electrical engineering, but also used in kinetics and fluid mechanics.

It represents the reciprocal value of a sum of reciprocal values and is defined by:

$\begin{array}{rlcl}\Vert :& \overline{\mathbb{ℂ}}\times \overline{\mathbb{ℂ}}& \to & \overline{\mathbb{ℂ}}\\ & (a,b)& \mapsto & a\Vert b=\frac{1}{\frac{1}{a}+\frac{1}{b}}\end{array}$ with $\overline{\mathbb{ℂ}}=\mathbb{ℂ}\cup \{\mathrm{\infty }\}$ being the extended complex numbers (with corresponding rules).

That is, the operator gives half of the harmonic mean of two numbers a and b.

As a special case, for $a\in \overline{\mathbb{ℂ}}$:

$a\Vert a=\frac{a}{2}$.

Further, for all $a,b\in \overline{\mathbb{ℂ}}$:

$a\ne b⟺|a\Vert b|>\frac{1}{2}\min (|a|,|b|)$

with $|a\Vert b|$ representing the absolute value of $a\Vert b$.

With $a$ and $b$ being positive real numbers follows $|a\Vert b|<\min (a,b)$.

Rules

For addition the parallel operator follows the associative law:

$(a\Vert b)\Vert c=a\Vert (b\Vert c)=a\Vert b\Vert c$

and the commutative law:

$a\Vert b=b\Vert a$.

Further, the parallel operator has $\mathrm{\infty }$ as neutral element and for $a\in \overline{\mathbb{ℂ}}$ the number $-a$ as inverse element. Hence $(\overline{\mathbb{ℂ}},\Vert )$ is an Abelian group.

In the absense of parentheses, the parallel operator is defined as taking precedence over addition or subtraction.

Applications

In electrical engineering, the parallel operator can be used to calculate the total impedance of various serial and parallel electrical circuits.

For instance, the total resistance of resistors connected in parallel is the reciprocal of the sum of the reciprocals of the individual resistors.

$\frac{1}{R_{\mathrm{e}\mathrm{q}}}=\frac{1}{R_1}+\frac{1}{R_2}+\cdots +\frac{1}{R_n}.$

Likewise for the total capacitance of serial capacitors.

The same principle can be applied to various problems in other disciplines.

Examples

Question:

Three resistors ${}_{R_1=270\mathrm{k}\mathrm{\Omega }}$, ${}_{R_2=180\mathrm{k}\mathrm{\Omega }}$ and ${}_{R_3=120\mathrm{k}\mathrm{\Omega }}$ are connected in parallel. What is their resulting resistance?

Answer:

$R_1\Vert R_2\Vert R_3=270\mathrm{k}\mathrm{\Omega }\Vert 180\mathrm{k}\mathrm{\Omega }\Vert 120\mathrm{k}\mathrm{\Omega }=\frac{1}{\frac{1}{270\mathrm{k}\mathrm{\Omega }}+\frac{1}{180\mathrm{k}\mathrm{\Omega }}+\frac{1}{120\mathrm{k}\mathrm{\Omega }}}\approx 56,84\mathrm{k}\mathrm{\Omega }$

The effectively resulting resistance is ca. 57 kΩ.

Question:

A construction worker raises a wall in 5 hours. Another worker would need 7 hours for the same work. How long does it take to build the wall if both worker work in parallel?

Answer:

$t_1\Vert t_2=5\mathrm{h}\Vert 7\mathrm{h}=\frac{1}{\frac{1}{5\mathrm{h}}+\frac{1}{7\mathrm{h}}}\approx 2,92\mathrm{h}$

They will finish in close to 3 hours.

Implementation

File:Hp30bwp34s.jpg

WP 34S with parallel operator (∥) on the g+÷ key.

The parallel operator is implemented as a keyboard operator on the Reverse Polish Notation (RPN) scientific calculator WP 34S (and a number of derivatives).

See also

• Process algebra ("merge" operator in computer sciences)

References

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