The parallel operator ‖ {\displaystyle \|} (pronounced "parallel", following the parallel lines notation from geometry; also known as reduced sum, parallel sum or parallel addition) is a binary operation which is used as a shorthand in electrical engineering, but is also used in kinetics, fluid mechanics and financial mathematics. The name parallel comes from the use of the operator computing the combined resistance of resistors in parallel.
Overview
The parallel operator represents the reciprocal value of a sum of reciprocal values (sometimes also referred to as the "reciprocal formula" or "harmonic sum") and is defined by:
a ∥ b := 1 1 a + 1 b = a b a + b , {\displaystyle a\parallel b\mathrel {:=} {\frac {1}{{\dfrac {1}{a}}+{\dfrac {1}{b}}}}={\frac {ab}{a+b}},}
where a, b, and a ∥ b {\displaystyle a\parallel b} are elements of the extended complex numbers C ¯ = C ∪ { ∞ } . {\displaystyle {\overline {\mathbb {C} }}=\mathbb {C} \cup \{\infty \}.}
The operator gives half of the harmonic mean of two numbers a and b. As a special case, for any number a ∈ C ¯ {\displaystyle a\in {\overline {\mathbb {C} }}} :
a ∥ a = 1 2 / a = 1 2 a . {\displaystyle a\parallel a={\frac {1}{2/a}}={\tfrac {1}{2}}a.}
Further, for all distinct numbers a ≠ b {\displaystyle a\neq b} :
| a ∥ b | > 1 2 min ( | a | , | b | ) , {\displaystyle {\big |}\,a\parallel b\,{\big |}>{\tfrac {1}{2}}\min {\bigl (}|a|,|b|{\bigr )},}
with | a ∥ b | {\displaystyle {\big |}\,a\parallel b\,{\big |}} representing the absolute value of a ∥ b {\displaystyle a\parallel b} , and min ( x , y ) {\displaystyle \min(x,y)} meaning the minimum (least element) among x and y. If a {\displaystyle a} and b {\displaystyle b} are distinct positive real numbers then 1 2 min ( a , b ) < | a ∥ b | < min ( a , b ) . {\displaystyle {\tfrac {1}{2}}\min(a,b)<{\big |}\,a\parallel b\,{\big |}<\min(a,b).}
The concept has been extended from a scalar operation to matrices and further generalized.
Notation The operator was originally introduced as reduced sum by Sundaram Seshu in 1956, studied as operator ∗ by Kent E. Erickson in 1959, and popularized by Richard James Duffin and William Niles Anderson, Jr. as parallel addition or parallel sum operator : in mathematics and network theory since 1966. While some authors continue to use this symbol up to the present, for example, Sujit Kumar Mitra used ∙ as a symbol in 1970. In applied electronics, a ∥ sign became more common as the operator's symbol around 1974. This was often written as doubled vertical line (||) available in most character sets (sometimes italicized as //), but now can be represented using Unicode character U+2225 ( ∥ ) for "parallel to". In LaTeX and related markup languages, the macros \| and \parallel are often used (and rarely \smallparallel is used) to denote the operator's symbol.
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