Parallel (operator)

The parallel operator (also known as reduced sum, parallel sum or parallel addition) formula_1 (pronounced "parallel", following the parallel lines notation from geometry) is a mathematical function which is used as a shorthand in electrical engineering, but is also used in kinetics, fluid mechanics and financial mathematics.
Overview.
The parallel operator represents the reciprocal value of a sum of reciprocal values (sometimes also referred to as "reciprocal formula") and is defined by:
with formula_3 being the extended complex numbers (with corresponding rules). The latter form is sometimes also referred to as "product over sum".
The operator gives half of the harmonic mean of two numbers "a" and "b".
As a special case, for formula_4:
Further, for all formula_6:
with formula_8 representing the absolute value of formula_9.
With formula_10 and formula_11 being positive real numbers follows formula_12.
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 codice_1 by Kent E. Erickson in 1959, and popularized by Richard James Duffin and William Niles Anderson, Jr. as parallel addition or parallel sum operator codice_2 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 codice_3 as a symbol in 1970. In applied electronics, a codice_4 sign became more common as the operator's symbol later on. This was often written as doubled vertical line () available in most character sets, but now can be represented using Unicode character U+2225 ( ∥ ) for "parallel to". In LaTeX and related markup languages, the macros codice_5 and codice_6 are often used to denote the operator's symbol.
Rules.
For addition, the parallel operator follows the commutative law:
and the associative law:
Multiplication is distributive over this operation.
Further, the parallel operator has formula_15 as neutral element and, for formula_16 the number formula_17 as inverse element. However, formula_18 is not an abelian group, as formula_19 for every nonzero , and formula_20 is not well defined (indeterminate form).
In the absence 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.
Likewise for the total capacitance of serial capacitors.
The same principle can be applied to various problems in other disciplines.
There is a duality between the usual (series) sum and the parallel sum.
Examples.
Question:
Answer:
Question:
Answer:
Implementation.
Suggested already by Kent E. Erickson as a subroutine in digital computers in 1959, the parallel operator is implemented as a keyboard operator on the Reverse Polish Notation (RPN) scientific calculators WP 34S since 2008 as well as on the WP 34C and WP 43S since 2015, allowing to solve even cascaded problems with few keystrokes like .