Solve a+b+c+d=abcd=e in Neutral Operations

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. It is possible that this problem is related to the Four Fundamental Properties of Mass, Distance, Duration and Electric Charge, and.or the Four Fundamental Forces of the Strong Nuclear Force, Weak Nuclear Force, Electromagnetism and Gravity. (But electric charge is not measured directly, it is calculated from other measurements, and its units defined through electric current.)

Steps

The tutorial

  1. Open a new Excel workbook and then list as text the following steps, adjust the column width to accommodate, and save the preliminary work.
  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. Realize 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. Let a = 10;
    • Given that a/(a-1) = b, b = 10/9
    • Test the original hypothesis: Does a+b = a*b = x?
    • Let a+b = 10+ 10/9 = 90/9 +10/9 = 100/9 = x
    • Let a*b = 10/1*10/9 = 100/9 = x √ check they are equal.
    • Let a^2 /(a-1) = 10^2 /9 = 100/9 = x √
    • Let b^2 / (b-1) = (10/9)^2 / (10/9 - 9/9) = 100/81 / 1/9 = 100/81 * 9/1 = 100/9 = x √
  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
    • Test hypothesis: a+b+c = a*b*c = y?
    • Let a+b+c = 10 + 10/9 + 1.0989010989011 = y = 12.2100122100122
    • Let a*b*c = 10 * 10/9 * 1.0989010989011 = y = 12.2100122100122 √
  6. Find that because a+b+c = a*b*c =12.2100122100122, to find d of a+b+c+d =a*b*c*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?
    • Let a+b+c+d = 10+10/9 +1.0989010989011 +1.08920596884871 = e=13.2992181788609
    • Let a*b*c*d = 10 *10/9 *1.0989010989011 * 1.08920596884871 = e=13.2992181788609√
    • Problem solved √
  7. Realize that 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.
    • Find 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!

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.
  • You can also balance Neutral Operations 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.

Helpful Guidance

  1. Make use of helper articles when proceeding through this tutorial:
    • See the Related Articles 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 more art charts and graphs, you might also want to click on Algebra, Mathematics, Spreadsheets or Graphics to view many Excel worksheets and charts where Trigonometry, Geometry and Calculus have been turned into Art, or simply click on the category as appears in the upper right white portion of this page, or at the bottom left of the page.

Tips

  • 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

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