Parallel (operator): Difference between revisions

Revision as of 23:29, 26 March 2023

Revision as of 23:42, 26 March 2023

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</math>

where {{mvar|a}}, {{mvar|b}}, and <math>a \parallel b</math> are elements of the [[extended complex numbers]] <math>\overline{\mathbb{C}} = \mathbb{C}\cup\{ \infty\}.</math><ref name="Georg_1999"/><ref name=ACAH>{{Wikibooks -inline|Associative Composition Algebra/Homographies}}</ref>

The operator gives half of the [[harmonic mean]] of two numbers ''a'' and ''b''.<ref name="Ellerman_1995"/><ref name="Ellerman_2004"/>

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:<math>a \parallel a = \frac1{2/a} = \tfrac12a.</math>

Further, for all distinct numbers {{nowrapnobr|<math>a \neq b</math>:}}

:<math>\big| a \parallel b \big| > \tfrac12 \min\bigl(|a|, |b|\bigr),</math>

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The identity element <math>\infty</math> is its own inverse, <math>\infty \parallel \infty = \infty.</math>

Every element <math>a \neq \infty</math> of <math>\widetilde{\C}</math> has a ''[[multiplicative inverse]]'' {{nowrapnobr|<math>a^{-1} = 1/a</math>:}}

:<math>a\cdot\frac1a = 1.</math>

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k (a \parallel b) = \frac{k}{\dfrac1a + \dfrac1b} = \frac{1}{\dfrac1{ka} + \dfrac1{kb}} = ka \parallel kb.

</math>

=== Repeated parallel ===

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=== Parallel Functions ===

A ''parallel function'' is one which commutes with the parallel operation:{{citation neededcn|date=January 2023}}

:<math>

f\left(a\parallel b\right) = f(a)\parallel f(b)

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=== Factoring parallel polynomials ===

As with a [[polynomial]] under addition, a parallel polynomial with coefficients <math>a_k</math> in <math display=inline>\widetilde\C</math> (with {{nowrapnobr|<math>a_0 \neq \infty</math>)}} can be [[factorization of polynomials|factored]] into a product of monomials:

:<math>\begin{align}

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\end{align}</math>

for some roots <math>r_k</math> (possibly repeated) in <math display=inline>\widetilde\C.</math>

Analogous to polynomials under addition, the polynomial equation

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[[File:Hp30bwp34s.jpg|thumb|150px|[[WP&nbsp;34S]] with parallel operator (<kbd>∥</kbd>) on the {{keypress|g|÷}} key.]]

Suggested already by Kent E. Erickson as a subroutine in digital computers in 1959,<ref name="Erickson_1959"/> the parallel operator is implemented as a keyboard operator on the [[Reverse Polish Notation]] (RPN) scientific calculators [[WP&nbsp;34S]] since 2008<ref name="Bonin_2012"/><ref name="Bonin_2015"/><ref name="Bonin_2016"/> as well as on the [[WP&nbsp;34C]]<ref name="Dowrick_2015"/> and [[WP&nbsp;43S]] since 2015,<ref name="Bonin_2019_OG"/><ref name="Bonin_2019_RG"/> allowing to solve even cascaded problems with few keystrokes like {{keypress|270}}{{keypress|ENTER}}{{keypress|180}}{{keypress|∥}}{{keypress|120}}{{keypress|∥}}.

==Projective view==