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The parallel operator (also known as reduced sum, parallel sum or parallel addition) \, (pronounced "parallel", following the parallel lines notation from geometry) is a mathematical function which is used as a shorthand in
electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
, but is also used in
kinetics Kinetics ( grc, κίνησις, , kinesis, ''movement'' or ''to move'') may refer to: Science and medicine * Kinetics (physics), the study of motion and its causes ** Rigid body kinetics, the study of the motion of rigid bodies * Chemical ki ...
, 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 A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', the ...
sum") and is defined by: :\begin \parallel: &&\overline \times \overline &\to \overline \\ &&(a, b) &\mapsto a \parallel b = \frac = \frac, \end with \overline = \mathbb\cup\ being the complex projective line (with corresponding rules). The operator gives half of the
harmonic mean In mathematics, the harmonic mean is one of several kinds of average, and in particular, one of the Pythagorean means. It is sometimes appropriate for situations when the average rate is desired. The harmonic mean can be expressed as the recipro ...
of two numbers ''a'' and ''b''. As a special case, for any number a \in \overline: :a \parallel a = \frac1 = \tfrac12a. Further, for all distinct numbers :\big, a \parallel b \big, > \tfrac12 \min\bigl(, a, , , b, \bigr), with \big, a \parallel b \big, representing the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
of a \parallel b, and \min(x, y) meaning the minimum (least element) among and . If a and b are distinct positive real numbers then \tfrac12 \min(a, b) < \big, a \parallel b \big, < \min(a, b). The concept has been extended from a
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers * Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
operation to matrices and further generalized.


Notation

The operator was originally introduced as reduced sum by Sundaram Seshu in 1956, studied as operator âˆ— 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 Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
network theory Network theory is the study of graphs as a representation of either symmetric relations or asymmetric relations between discrete objects. In computer science and network science, network theory is a part of graph theory: a network can be defi ...
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 âˆ¥ 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 ( âˆ¥ ) 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.


Rules

For
addition Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. ...
, the parallel operator follows the commutative law: :a \parallel b = b \parallel a and the associative law: : (a \parallel b) \parallel c = a \parallel (b \parallel c) = a \parallel b \parallel c = \frac = \frac. Multiplication is distributive over this operation: :k\bigl(a \parallel b\bigr) = (ka) \parallel (kb). Further, the parallel operator has \infty as neutral element. For any number a, :a \parallel \infty = \frac1 = a. For any non-zero number , the number is its inverse element: :a \parallel (-a) = \frac1 = \frac10 = \infty. However, \bigl(\overline, \bigr) is not an abelian group, as has no inverse element. For every non-zero , a \parallel 0 = 0. The quantity 0 \parallel (-0) = 0 \parallel 0 can either be left undefined (see indeterminate form) or defined to equal . (This is analogous to the way \bigl(\overline, \bigr) is not an abelian group because \infty has no additive inverse.) In the absence of parentheses, the parallel operator is defined as taking precedence over addition or subtraction, similar to multiplication.


Applications

In
electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
, the parallel operator can be used to calculate the total impedance of various serial and parallel electrical circuits. For instance, the total
resistance Resistance may refer to: Arts, entertainment, and media Comics * Either of two similarly named but otherwise unrelated comic book series, both published by Wildstorm: ** ''Resistance'' (comics), based on the video game of the same title ** ''T ...
of resistors connected in parallel is the reciprocal of the sum of the reciprocals of the individual
resistor A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active el ...
s. : :\frac = \frac + \frac + \cdots + \frac. Likewise for the total capacitance of serial capacitors. The same principle can be applied to various problems in other disciplines. For example, in geometric optics the thin lens approximation to the lens maker's equation. There is a
duality Duality may refer to: Mathematics * Duality (mathematics), a mathematical concept ** Dual (category theory), a formalization of mathematical duality ** Duality (optimization) ** Duality (order theory), a concept regarding binary relations ** Dual ...
between the usual (series) sum and the parallel sum.


Examples

Question: : Three resistors R_1 = 270\,\mathrm, R_2 = 180\,\mathrm and R_3 = 120\,\mathrm are connected in
parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster of IBM ...
. What is their resulting resistance? Answer: : R_1 \parallel R_2 \parallel R_3 = 270\,\mathrm \parallel 180\,\mathrm \parallel 120\,\mathrm = \frac \approx 56.84 \,\mathrm : The effectively resulting resistance is ca. 57 k Ω. 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 \parallel t_2 = 5\,\mathrm h \parallel 7\,\mathrm h = \frac \approx 2.92\,\mathrm h : They will finish in close to 3 hours.


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 The HP-42S RPN Scientific is a programmable RPN Scientific hand held calculator introduced by Hewlett Packard in 1988. It has advanced functions suitable for applications in mathematics, linear algebra, statistical analysis, computer science ...
since 2015, allowing to solve even cascaded problems with few keystrokes like .


Projective view

The parallel operator may be understood as a
homography In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, ...
on the projective line over a ring. The reciprocation operation is usually singular on null vectors, but with projective geometry the reciprocal is completed with "points at infinity". In fact, the translations from finite points are complemented by "translations at infinity" as valid projectivities. The parallel operator is the composition of two such translations at infinity.


Notes


References


Further reading

* * (10 pages) * * (33 pages) * *

(19 pages) * * * {{cite book , title=TLV3201, TLV3202: TLV320x 40-ns, microPOWER, Push-Pull Output Comparators , chapter=7.5 Electrical Characteristics: VCC = 5 V / 7.6 Electrical Characteristics: VCC = 2.7 V / 9.1.2.1 Inverting Comparator with Hysteresis , publisher=
Texas Instruments Incorporated Texas Instruments Incorporated (TI) is an American technology company headquartered in Dallas, Texas, that designs and manufactures semiconductors and various integrated circuits, which it sells to electronics designers and manufacturers globall ...
, publication-place=Dallas, Texas, USA , version=Revision B , id=SBOS561B , date=2022-06-03 , orig-date=2016, 2012 , pages=5, 6, 13–14 3, url=https://www.ti.com/lit/ds/symlink/tlv3201.pdf?ts=1660718632803 , access-date=2022-08-18 , url-status=live , archive-url=https://web.archive.org/web/20220817185705/https://www.ti.com/lit/ds/symlink/tlv3201.pdf?ts=1660718632803 , archive-date=2022-08-17 , quote-page=5 , quote=PARAMETER €¦TYP €¦UNIT €¦ INPUT IMPEDANCE €¦ Common mode €¦1013 ∥ 2 €¦Î© ∥ pF €¦ Differential €¦1013 ∥ 4 €¦Î© ∥ pF €¦} (37 pages) (NB. Unusual usage of ∥ for both values and units.)


External links

* https://github.com/microsoftarchive/edx-platform-1/blob/master/common/lib/calc/calc/calc.py Abstract algebra Elementary algebra Multiplication