# Parallel (operator)

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:^{[9]}^{[6]}^{[10]}^{[11]}

The operator gives half of the harmonic mean of two numbers *a* and *b*.^{[7]}^{[8]}

The concept has been extended from a scalar operation to matrices^{[13]}^{[14]}^{[15]}^{[16]}^{[17]} and further generalized.^{[18]}

The operator was originally introduced as **reduced sum** by Sundaram Seshu in 1956,^{[19]}^{[20]}^{[13]} studied as operator `∗`

by Kent E. Erickson in 1959,^{[21]}^{[22]}^{[13]} 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.^{[14]}^{[15]}^{[1]} While some authors continue to use this symbol up to the present,^{[7]}^{[8]} for example, Sujit Kumar Mitra used `∙`

as a symbol in 1970.^{[13]} In applied electronics, a `∥`

sign became more common as the operator's symbol around 1974.^{[23]}^{[24]}^{[25]}^{[26]}^{[27]}^{[nb 1]}^{[nb 2]} 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 `\|`

and `\parallel`

are often used (and rarely `\smallparallel`

is used) to denote the operator's symbol.

In the absence of parentheses, the parallel operator is defined as taking precedence over addition or subtraction, similar to multiplication..^{[1]}^{[28]}^{[9]}^{[10]}

In electrical engineering, the parallel operator can be used to calculate the total impedance of various serial and parallel electrical circuits.^{[nb 2]}

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

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.^{[7]}^{[8]}

Suggested already by Kent E. Erickson as a subroutine in digital computers in 1959,^{[21]} the parallel operator is implemented as a keyboard operator on the Reverse Polish Notation (RPN) scientific calculators WP 34S since 2008^{[29]}^{[30]}^{[31]} as well as on the WP 34C^{[32]} and WP 43S since 2015,^{[33]}^{[34]} allowing to solve even cascaded problems with few keystrokes like `270``↵ Enter``180``∥``120``∥`.