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Three Parts:Helpful guidanceThe tutorialTheory or Symmetrical Conjecture

This is a problem in Neutral Operations Theory, where addition is being neutralized versus multiplication. The problem is to find a+b+c+d = a*b*c*d = e. This article will guide you to the solution, step by step, showing you how it is done, to an accuracy of 14 decimal places.

EditPart 1 of 3: Helpful guidance

  1. 1
    Make use of helper articles when proceeding through this tutorial:
    • See the Related wikiHows below and the article How to Do the Sub Steps of Neutral Operations for a list of articles related to Excel, Geometric and/or Trigonometric Art, Charting/Diagramming and Algebraic Formulation relating to Neutral Operations.
    • For art charts and graphs, you might also want to click on Category:Microsoft Excel Imagery, Category:Mathematics or Category:Algebra to view many Excel worksheets and charts where trigonometry, geometry and algebra have been turned into art, or simply click on that category if it appears in the upper right white portion of this page.

EditPart 2 of 3: The tutorial

  1. New Sunflowers 2.png
    1
    Open a new Excel workbook and then list as text the following steps, adjust the column width to accomodate, and save the preliminary work.
  2. 2
    Let a+b = a*b; then do
    • a+b-b = ab-b, then do
    • a = b(a-1), then do
    • a/(a-1) = b and b has been isolated and defined in terms of a and 1.
  3. 3
    a may not = 1, or division by 0 will result in the denominator, in which case b = Infinity (just as it does for slope y/x of the y axis in the Cartesian Plane, where x=0 or x approaches 0 as closely as one pleases). b may also not equal 1, due to Addition and Multiplication both being subject to the Law of Commutation.
  4. 4
    Let a = 10; given that
    • a/(a-1) = b, b = 10/9
    • Test the original hypothesis: Does a+b = a*b = x?
    • a+b = 10+ 10/9 = 90/9 +10/9 = 100/9 = x
    • a*b = 10/1*10/9 = 100/9 = x √ check they are equal.
    • a^2 /(a-1) = 10^2 /9 = 100/9 = x √
    • b^2 / (b-1) = (10/9)^2 / (10/9 - 9/9) = 100/81 / 1/9 = 100/81 * 9/1 = 100/9 = x √
  5. 5
    If a+b = a*b = x =100/9 then to find c of a+b+c+d = a*b*c*d = e, let
    • 100/9 +c = 100/9 *c = y, then do
    • 100/9 +c-c =(100/9 *c)-c = y, then do
    • 100/9 = c(100/9 - 1), then do
    • 100/9 / (100/9 -1) = c = 1.0989010989011, then do
    • Test hypothesis: a+b+c = a*b*c = y?
    • a+b+c = 10 + 10/9 + 1.0989010989011 = y = 12.2100122100122
    • a*b*c = 10 * 10/9 * 1.0989010989011 = y = 12.2100122100122 √
  6. 6
    Because a+b+c = a*b*c = 12.2100122100122, to find d of a+b+c+d = a*b*d*d = e, let
    • 12.2100122100122 + d = 12.2100122100122 * d, then do
    • 12.2100122100122 + d-d =(12.2100122100122 *d) - d, then do
    • 12.2100122100122 = d*(12.2100122100122 - 1), then do
    • 12.2100122100122 / (12.2100122100122 - 1) = d = 1.08920596884871, then
    • Test the hypothesis: does a+b+c+d = a*b*c*d = e?
    • a+b+c+d = 10+10/9 +1.0989010989011 +1.08920596884871 = e=13.2992181788609
    • a*b*c*d = 10 *10/9 *1.0989010989011 * 1.08920596884871 = e=13.2992181788609√
    • Problem solved √
  7. 7
    In this way, Mother Nature may decide whether she wants to add a length or an area as she continues to grow, using the equivalent amount of resources. so long as the area has 1 unit of thickness.
    • Notice that Addition and Multiplication are both Commutative, and we might just as easily have started off by subtracting a from both sides instead of b. However, Subtraction is NOT commutative, e.g. 7-4 does not equal 4-7.
    • It is equally true for a+b=a*b that b/(b-1) = a. Also notice that if a+b = a*b = w, then a^2 /(a-1) = b^2 / (b-1) = w. This has the property of Symmetry, which is a very hot item in Mathematics these days!

EditPart 3 of 3: Theory or Symmetrical Conjecture

  • Note that there are Four Forces in Nature: the Strong Nuclear Force, the Weak Nuclear Force, Electromagnetism and Gravity -- this problem shows how to balance them additively and as a product simultaneously ... "perhaps", because the results of this Neutral Operation are related to each other by specific algebraic relations which may Not Hold True for these Physical Forces. But, within a "neutral zone" of some dimension, the neutrality might just hold true.
  • Neutral Operations can also balance them additively vs. a series of ratios, as in a continued fraction, i.e. a+b+c+d=a/b/c/d = e, where e is a "state" of neutrality. But it's just a theory being worked on at this point; it has no real professional scientific backing of any kind yet. Working this out is left as an Extra Credit Assignment for the reader. But yes, it's been solved; it is do-able.

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EditTips

  • It is also possible to do -a-b-c-d = -a*-b*-c*-d = e, and other such manipulations of neutral operations involving negative numbers and even complex/imaginary numbers. One may use transcendental numbers as well.
  • n.b. A new Part, The Quadratic Relation of Neutral Operations / Symmetry by Commutation, has been added to the article How to Convert a Quadratic Formula to Roots Form by Completing the Square

EditRelated wikiHows

How to
Do the Sub Steps of Neutral Operations

How to
Create an XL Worksheet for Continued Fractions

How to
Start Working with Continued Fractions

How to
Multiply and Divide Geometrically Like Mother Nature

How to
Find the Area of a Square Using the Length of its Diagonal

Article Info

Categories: Algebra | Mathematics

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