Brief Notes on the Eighth-Root of [ 1 ]

by Frank C. Fung - 1st published in March, 2008.

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Key Findings and Introduction :

• During a brief study of { Complex Numbers } , the question arose :
  • as to whether [ i ] and [ -i ] can be freely interchanged , without adverse effects .

• On further investigations , the Finiding here is simply this :
  • If the { eighth-roots of [ 1 ] } are :
    • 

    then each of the following 3 sets , i.e. the sets :

    • , , ;

      would each exhibit a different and distinct { combinatorics-symmetry pattern } .

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The Investigation :

• We start-off our investigation with the { Fermat Little Theorem } :
  • 

    where [ P ] is an { odd prime number } and [ a ] is a { positive integer } less than [ P ] .

  We then bring-in an extension thereof previously discussed in { Approach to the Characteristics Of Numbers } , namely :

  • For the { odd prime number } [ P ] , we set up the value [ Q ] such that :
    • 

    and the equation :

    • 

      has solutions , other than [ 1 ] , if and only if [ R ] is divisible into [ Q ] .

• For the { primer number } [ 233 ] , with { 233 = 8 x 29 + 1 } , the equation :
  • 

    has solutions other than [ 1 ] .

  We then bring-in this { tree-diagram } on-the-left , below , to give the 8 { roots } :

  

  • And the { tree-diagram } on-the-right , above , then gives the 8 { eighth-roots } of [ 1 ] , with :
    • 

• Let us now bring-in this equation for the diagram on-the-left :
  • 

    and look at an equation of similar format , modelled therefrom , for the diagram on-the-right :

  • 

  We then have :

  • 

    yielding the relation :

  • 

• We shall now make a arbitrary choice here , namely , we set :
  • 

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The { Counter-Multiplicative Conjugate } Relation :

• Let us now define , for the value [ w ] :
  • the { Additive Conjugate } [ x ] , as the value satisfying the relation :
    • 

  • the { Multiplicative Conjugate } [ y ] , as the value satisfying the relation :
    • 

  • the { Counter-Multiplicative Conjugate } [ z ] , as the value satisfying the relation :
    • 

  We then bring-in this diagram for the 3 sets , i.e. the sets :

  • , , ;

    whereby :

    • each { horizontal pair } in the diagram below would depict a { additive conjugate } relation ,
    • each { vertical pair } in the diagram below would depict a { muliplicative conjugate } relations .

    

    • And each { cross-diagonal pair } here would depict a { counter-multiplicative conjugate } relation .

    Let us now bring-in this set of 3 diagrams from { An Approach to the I-CHING } ,

    

    • which also appeared in { Epilog III } of { An Approach to the Four Body Problem } .

    CLEARLY , each of the 3 sets shows a different and distinct { combinatorics-symmetry pattern } .

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The { Tetrahedron Diagram } :

• Let us now put the four (4) elements in the set :
  • 

    at the 4 vertices of an { equilateral tetrahedron } , as per this diagram below :

  

  We see here that each pair of opposite edges would then depict a different { conjugate relation } :

  • And there are only three (3) such relations , namley :
    • the { additive conjugate } relation ,
    • the { multiplicative conjugate } relation , and
    • the { counter-multiplicative conjugate } relation .

    Seems likely that all three are always all-important .

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The { Green Cross } :

• We now bring-in this diagram , which shows the positions of the 4 elements in the set :
  • 

    in the { imaginary plane } , on the right-hand-side .

  

• The question then arises as to whether :
  • using the { R-B Cross } on-the-left above as the basis for { complex analysis }

    is always superior to using the { Green Cross } on-the-right ?

  May be not . And we shall leave this for the reader's further evaluation .

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• Riemann Zeta Function anyone ?
  • Surely you must be joking .

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( Three Body Problem / Characteristics of Numbers / Matrix and Linear Algebra / I-Ching / Triangle / Odd-On-Odd Format / Multiplication Table for Modulo P / Symmetry and Combinatorics / Solid Object with 14 Faces / Four Body Problem )

Original dated 2008-3-13 *** Updated 2008-11-01 / 2009-10-26