Do Neutral Operations with Square Roots

You will learn how to algebraically solve sqrt(a) + sqrt(b) = sqrt(a) * sqrt(b)=c, sqrt(a) - sqrt(b) = sqrt(a) * sqrt(b)=c, sqrt(a) + sqrt(b) = sqrt(a) / sqrt(b)=c and sqrt(a) - sqrt(b) = sqrt(a) / sqrt(b)=c (whereby two operators are held neutral to each other between constants, rendering a "tipping point") and to graph the final a and b as x and y. You will also learn goal seeking to 2π for each above equation. This article has applicability to the Special Theory of Relativity because, just as E = mc^2, c = sqrt(E) / sqrt(m), which can be set equal in Neutral Operations to either sqrt(E)-sqrt(m) or sqrt(E)+sqrt(m), among other possibilities. Bear in mind all that is being proposed concerns a certain velocity, not a physical event. Were it to be put forth that E = mc^2 = m+c^2 and m+c^2 is the moment of tangence of mass and light speed squared, then a physical phenomenon has been shown to be realized by experimentation.

Steps

√a + √b = √a * √b = c

  1. Open a new workbook in Excel from the desktop, from the dock, or from within your Applications folder inside the Microsoft folder. Double click on Excel (either the green X on the dock or the app title in the folder) and select File New Workbook.
  2. In Preferences, set R1C1 to unchecked or Off, set Ribbon to checked or On, set Show Formula Bar to checked or On and set Calculation to .000 000 000 000 001 (without interim spaces) and 100 iterations and Calculate Automatically.
  3. Click in the far upper left top corner above the 1 of row 1 and to the left of column A. Doing so will select the entire worksheet. Format Cells Number Number to decimal places 15, show comma. Format Cells Alignment Center. Format column width 1`.64".
  4. Title the first worksheet, "NeuOps sqrt(n)" and save the workbook as "Do Neu Ops w Sqrts" into an appropriate folder such as 'Microsoft Excel Imagery', 'Neu Ops' or 'wikiHow Articles'.
  5. Enter the Row 1 and Row 2 Headings:
    • Enter to cell A1, a
    • Enter to cell A2, By changing cell
    • Enter to cell B1, sqrt(a)
    • Enter to cell C1, sqrt(b)
    • Enter to cell C2, " = sqrt(a)/(sqrt(a)-1)" ...(w/o quotes and with a leading space)
    • Enter to cell D1, b =
    • Enter to cell D2, (sqrt(b))^2
    • Enter to cell E1, c =
    • Enter to cell E2, sqrt(a) + sqrt(b)
    • Enter to cell F1, c =
    • Enter to cell F2, sqrt(a) * sqrt(b)
    • Enter to cell G1, Goal
    • Enter to cell H1, Goal Seek to 0
  6. Enter the algebraic solution to the problem sqrt(a)+sqrt(b)=sqrt(a)*sqrt(b) in column I:
    • Select column I and Format Cells Font Bold and Format Column Set Column Width 3.82";
    • Enter to cell I1, sqrt(a) + sqrt(b) = sqrt(a) * sqrt(b) = c
    • Enter to cell I2, sqrt(a) + sqrt(b) -sqrt(b) = (sqrt(a) * sqrt(b)) - sqrt(b)
    • Enter to cell I3, sqrt(a) = sqrt(b) * (sqrt(a) - 1) (divide both sides by (sqrt(a) - 1)) to get:
    • Enter to cell I4, sqrt(a)/(sqrt(a) - 1) = sqrt(b) (notice this is the same result as in C1 and C2. Select cells C2 and I4 together using the command key on a Mac and Format Cells Fill sky blue.
  7. Format the Goal Seeking columns red. Select columns G and H and do Format Font Red.
  8. Define the variable names for the worksheet:
    • Select columns A:C
    • Do Insert Name Create in Top Row, OK. You should end up with 3 variable names: a, sqrt_a and sqrt_b.
  9. Enter the Input cells and formulas for rows 3-5:
    • Enter to cell A3 the formula, =PI()
    • Enter to cell A4 the formula, =10
    • Enter to cell A5 the value, 1, as a preliminary guess. Do not put an equals sign before it.
    • Select cells A3:A5 and Format Cells Fill canary yellow as these are Input Cells. Select cell A5 and Format Cell Font Red as this is a Goal Seeking "By changing cell" cell value.
    • Enter to cell B3 the formula, =SQRT(a)
    • Enter to cell C3 the formula, =sqrt_a/(sqrt_a-1)
    • Enter to cell D3 the formula, =sqrt_b^2
    • Enter to cell E3 the formula, =sqrt_a+sqrt_b
    • Enter to cell F3 the formula, =sqrt_a*sqrt_b; (these last 2 should equate)
    • Select cell range B3:F3 and copy it to B4:F5.
    • Check that E3=F3 and E4=F4;
  10. Do Goal Seeking to 2π:
    • Enter to cell G5 the formula, =2*PI() and Format Cells Fill yellow;
    • Enter to cell H5 the formula, =G5-F5
    • Select cell H5 and do Tools, Goal Seek, Set Cell H5, To value 0, By changing cell, and click on cell A5. A5 should recalculate to 1.55702560889077.

√a - √b = √a * √b = c

  1. Select cell range A1:I5 and Edit Copy it and Edit Paste it to A8:I12
  2. Modify the algebraic solution:
    • Enter to cell I8, sqrt(a) - sqrt(b) = sqrt(a) * sqrt(b)
    • Enter to cell I9, sqrt(a) - sqrt(b) + sqrt(b) = (sqrt(a) * sqrt(b)) + sqrt(b)
    • Enter to cell I10, sqrt(a) = sqrt(b) * (sqrt(a) + 1)
    • Enter to cell I11, sqrt(a)/(sqrt(a) + 1) = sqrt(b)
  3. Edit in the formula bar cell I11 and copy just the part that reads sqrt(a)/(sqrt(a) + 1) and Edit Paste it to cell C9. Using the Command key on a Mac, select together cell C9 and I11 and Format Cells Fill sky blue.
  4. Change the formulas appropriately:
    • Enter to cell C10 the formula, =sqrt_a/(sqrt_a+1) and Edit Fill Down to cell C12.
    • Enter to cell E9 the label, sqrt_a-sqrt_b
    • Enter to cell E10 the formula, =sqrt_a-sqrt_b and Edit Fill Down to cell E12.
  5. Do Goal Seeking to 2π:
    • Select cell H12 and do Tools, Goal Seek, Set Cell H12, To value 0, By changing cell, and click on cell A12. A12 should recalculate to 51.2748509715808.

√a + √b = √a / √b = c

  1. Select cell range A1:I12 and Edit Copy it and Edit Paste it to A14:I25
  2. Modify the algebraic solution:
    • Enter to cell I14, sqrt(a) + sqrt(b) = sqrt(a) / sqrt(b)
    • Enter to cell I15, sqrt(a) + sqrt(b) - sqrt(a) = (sqrt(a) / sqrt(b)) - sqrt(a)
    • Enter to cell I16, sqrt(b) = sqrt(a) / (1/(sqrt(b)) - 1)
    • Enter to cell I17, sqrt(b)/ (1/(sqrt(b)) - 1) = sqrt(a)
  3. Edit in the formula bar cell I17 and copy just the part that reads sqrt(b)/ (1/(sqrt(b)) - 1) and Edit Paste it to cell B15. Using the Command key on a Mac, select together cell B15 and I17 and Format Cells Fill sky blue. Select cell C15 and do Edit Clear All.
    • Copy cell range A2:A5 and Paste it to cell range D15:D18
  4. Change the formulas appropriately:
    • Enter to cell B16 the formula, =sqrt_b/(1/(sqrt_b)-1) and Edit Fill Down to cell B18.
    • Enter to cell A16 the formula, =sqrt_a^2 and Edit Fill Down to cell A18. Format cell range A16:A18 Format Cells Fill None.
    • Enter to cell C16 the formula, =sqrt(D16) and Edit Fill Down to cell C18.
    • Enter to cell E15 the label, sqrt_a+sqrt_b
    • Enter to cell E16 the formula, =sqrt_a+sqrt_b and Edit Fill Down to cell E18.
    • Enter to cell F15 the label, sqrt(a)/sqrt(b)
    • Enter to cell F16 the formula, =sqrt_a/sqrt_b and Edit Fill Down to cell F18.
  5. Do Goal Seeking to -2π:
    • Enter to cell G18 the formula, =-2*PI()
    • Select cell H18 and do Tools, Goal Seek, Set Cell H18, To value 0, By changing cell, and click on cell D18. D18 should recalculate to 1.41438632639533.

√a - √b = √a / √b = c

  1. Modify the algebraic solution:
    • Enter to cell I21, sqrt(a) - sqrt(b) = sqrt(a) / sqrt(b)
    • Enter to cell I22, sqrt(a) - sqrt(b) - sqrt(a) = (sqrt(a) / sqrt(b)) - sqrt(a)
    • Enter to cell I23, " -sqrt(b) = sqrt(a) / (1/(sqrt(b)) - 1)", w/o quotes and w/ leading space
    • Enter to cell I24, " -sqrt(b)/ (1/(sqrt(b)) - 1) = sqrt(a)", w/o quotes and w/ leading space
  2. Edit in the formula bar cell I24 and copy just the part that reads " -sqrt(b)/ (1/(sqrt(b)) - 1)" (w/o quotes) and Edit Paste it to cell B22. Using the Command key on a Mac, select together cell B22 and I24 and Format Cells Fill sky blue. Select cell C22 and do Edit Clear All.
    • Copy cell range A2:A5 and Paste it to cell range D22:D25
  3. Sort the range
    • Select cell range A23:H25 and do Data Sort Options [Sort Top to Bottom], Uncheck My List has headers, sort on column D, smallest to largest.
  4. Extend the Range via a Fill Series command:
    • Select D25:D33 and do Edit Fill Series, Step Value = -1, OK.
    • Select Row 32 and do Insert Row(s).
    • Select cell D32 and enter the formula, =PI()
  5. Copy cell range A23:F23 and Paste it to cell A35.
  6. Change the formulas appropriately:
    • Enter to cell B23 the formula, =-sqrt_b/(1/(sqrt_b)-1) and Edit Fill Down to cell B35.
    • Enter to cell A23 the formula, =sqrt_a^2 and Edit Fill Down to cell A35. Format cell range A23:A35 Format Cells Fill None.
    • Enter to cell C23 the formula, =sqrt(D16) and Edit Fill Down to cell C35.
    • Enter to cell E22 the label, sqrt_a-sqrt_b
    • Enter to cell E23 the formula, =sqrt_a-sqrt_b and Edit Fill Down to cell E35.
    • Enter to cell F22 the label, sqrt(a)/sqrt(b)
    • Enter to cell F23 the formula, =sqrt_a/sqrt_b and Edit Fill Down to cell F35.
  7. Do Goal Seeking to 2π:
    • Enter to cell G23 the formula, =2*PI()
    • Select cell H23 and do Tools, Goal Seek, Set Cell H23, To value 0, By changing cell, and click on cell D23. D23 should recalculate to 1.41438632639533. Copy cell D23 and paste it to cell D35.
    • Column E should equal column F from top to bottom at this point; if not, go back and reason through where the error was made. No one's infallible.

Create the chart of √a - √b = √a / √b

  1. Enter to cell A36 the label, x = a, and Format Cells Font Bold and Underlined;
  2. Enter to cell B36 the label, y = b, and Format Cells Font Bold and Underlined;
  3. Enter to cell A37 the formula, =A23, and Edit Fill down to cell range A37:A49;
  4. Enter to cell B37 the formula, =D23, and Edit Fill down to cell range B37:B49;
  5. Select cell range A37:B49 and either Insert Chart or use the Chart Wizard or the Ribbon and select Charts, All, Scatter, Smooth Lined Scatter. A new chart will appear atop your data. Move it to the right and position its upper left corner at about G26, and expand it rightwards and downwards by pulling on the bottom righthand corner when the cursor is over it and turns into a double headed arrow to about cell I54.
  6. Using Chart Layout, choose Axis and select No Vertical Axis and No Horizontal Axis, then Gridlines Horizontal none, Legend none.
  7. Double click in the Plot Area and do Fill medium purple; double click in the Chart Area and do Fill black; double click on the series line or plot line and do Line Weight 2.75 and Solid Color red, or whatever appeals to you. Make sure the series in the formula bar reads as follows: =SERIES(,'NeuOps sqrt(n)'!$A$37:$A$49,'NeuOps sqrt(n)'!$B$37:$B$49,1) and here is what the chart should look like:

c = sqrt(E)/sqrt(m) = sqrt(E) - sqrt(m)

  1. Enter to D60, sqrt(E)/sqrt(m) = sqrt(E) - sqrt(m)
  2. Enter to D61, sqrt(E)/sqrt(m) - sqrt(E)= sqrt(E) - sqrt(m) -sqrt(E)
  3. Enter to D62, sqrt(E)(1/sqrt(m) - 1) = -sqrt(m)
  4. Enter to D63, sqrt(E) = -sqrt(m)/(1/sqrt(m) - 1), and multiply by -1/-1 on the right to get:
  5. Enter to D64, sqrt(E) = sqrt(m)/(1 - 1/sqrt(m))
  6. Enter to A69, Light Speed mps (where mps = miles per second)
  7. Enter to A70, Miles/Second
  8. Enter to A71, =299792458/1609.34400000087 (i.e., light speed in meters/second / meters/mile), giving you the result of 186,282.39705112 (lights peed in miles/sec.)
  9. Enter to B70, E
  10. Enter to C70, m
  11. Enter to D70, sqrt(E)
  12. Enter to E70, sqrt(m)
  13. Enter to F70, sqrt(E)/sqrt(m)
  14. Enter to G70, sqrt(E) - sqrt(m)
  15. Enter to H70, Goal Seek to 0
  16. Enter to B71, =D71^2
  17. Enter to C71, 1 (as a preliminary guesstimate of mass m)
  18. Enter to D71 the formula, =E71/(1-(1/E71))
  19. Enter to E71 the formula, =SQRT(C71)
  20. Enter to F71 the formula, =D71/E71
  21. Enter to G71 the formula, =D71-E71
  22. Enter to H71 the formula, =A71-F71
  23. Goal seek on cell H71 to value 0 by changing the mass in cell C71. Your # of iterations in Set Preferences, Calculation should probably be about 500 with an accuracy to 15 decimal places. The answer you may well arrive at will not equal zero but be close enough, it will be 6.24 448 875 896 633E-06 (to 20 decimal places). That is the speed of light, c, in A71 = 186282.39705112, while the answer in F71 for sqrt(E)/sqrt(m) is almost equal to it, at 186282.397044876 miles/second. The mass you should get is 1.000 010 736 473 96

Helpful Guidance

  1. Make use of helper articles when proceeding through this tutorial:
    • See 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.
    • For more art charts and graphs, you might also want to click on Microsoft Excel Imagery, 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.

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