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Graphical interpretation of the parallelism operator with
a
‖
b
=
c
{\displaystyle {a\|b=c}}
The parallelism operator
‖
{\displaystyle {\|}}
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
‖
:
C
¯
×
C
¯
→
C
¯
(
a
,
b
)
↦
a
‖
b
=
1
1
a
+
1
b
{\displaystyle {\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
C
¯
=
C
∪
{
∞
}
{\displaystyle {\overline {\mathbb {C} }}=\mathbb {C} \cup \{\infty \}}
extended complex numbers (with its usual rules of operation).
Properties
a
‖
a
=
a
2
{\displaystyle a\|a={\frac {a}{2}}}
a
≠
b
⟺
|
a
‖
b
|
>
1
2
min
(
|
a
|
,
|
b
|
)
{\displaystyle a\neq b\iff \left|a\|b\right|>{\tfrac {1}{2}}\min(|a|,|b|)}
|
a
‖
b
|
{\displaystyle \left|a\|b\right|}
absolute value of
a
‖
b
{\displaystyle a\|b}
When
a
{\displaystyle a}
b
{\displaystyle b}
positive real numbers ,
|
a
‖
b
|
<
min
(
a
,
b
)
{\displaystyle \left|a\|b\right|<\min(a,b)}
The parallelism operator is commutative :
a
‖
b
=
b
‖
a
{\displaystyle a\|b=b\|a}
The parallelism operator is associative :
(
a
‖
b
)
‖
c
=
a
‖
(
b
‖
c
)
=
a
‖
b
‖
c
{\displaystyle \left(a\|b\right)\|c=a\|\left(b\|c\right)=a\|b\|c}
The parallelism operator has
∞
{\displaystyle \infty }
identity operator , and, for
a
∈
C
¯
{\displaystyle a\in {\overline {\mathbb {C} }}}
−
a
{\displaystyle -a}
inverse element . Thus,
(
C
¯
,
‖
)
{\displaystyle ({\overline {\mathbb {C} }},\|)}
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
h
‖
7
h
=
1
1
5
h
+
1
7
h
≈
2.917
h
{\displaystyle 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
k
Ω
{\displaystyle {R_{1}=270\,\mathrm {k\Omega } }}
R
2
=
180
k
Ω
{\displaystyle {R_{2}=180\,\mathrm {k\Omega } }}
R
3
=
120
k
Ω
{\displaystyle {R_{3}=120\,\mathrm {k\Omega } }}
parallel . What is the total resistance of the circuit ? Solution :
R
1
‖
R
2
‖
R
3
=
270
k
Ω
‖
180
k
Ω
‖
120
k
Ω
=
1
1
270
k
Ω
+
1
180
k
Ω
+
1
120
k
Ω
≈
56.842
k
Ω
{\displaystyle 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 . 
