How to Do Neutral Operations with Square Roots

Step-by-Step Guide

1. 1

   Step 1: Open a new workbook in Excel from the desktop

   Double click on Excel (either the green X on the dock or the app title in the folder) and select File New Workbook. ,, 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".,,, Select cells C2 and I4 together using the command key on a Mac and Format Cells Fill sky blue. , Select columns G and H and do Format Font Red.,, 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
2. 2

   Step 2: from the dock

   A5 should recalculate to
   1.55702560889077. ,,, Using the Command key on a Mac, select together cell C9 and I11 and Format Cells Fill sky blue., 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. , A12 should recalculate to
   51.2748509715808. ,,, 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 , 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. , D18 should recalculate to
   1.41438632639533. ,, 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 ,, Select Row 32 and do Insert Row(s).
   
   Select cell D32 and enter the formula, =PI() ,, 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. , 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. ,,,,, 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.,, 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: ,,,,,,,,,,,,,,,,,,,,,,, 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, For more art charts and graphs, you might also want to click on Category:
   Microsoft Excel Imagery, Category:
   Mathematics, Category:
   Spreadsheets or Category:
   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.
3. 3

   Step 3: or from within your Applications folder inside the Microsoft folder.

4. 4

   Step 4: In Preferences

5. 5

   Step 5: set R1C1 to unchecked or Off

6. 6

   Step 6: set Ribbon to checked or On

7. 7

   Step 7: 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.

8. 8

   Step 8: Click in the far upper left top corner above the 1 of row 1 and to the left of column A.

9. 9

   Step 9: Title the first worksheet

10. 10

    Step 10: "NeuOps sqrt(n)" and save the workbook as "Do Neu Ops w Sqrts" into an appropriate folder such as 'Microsoft Excel Imagery'

11. 11

    Step 11: 'Neu Ops' or 'LifeGuide Hub Articles'.

12. 12

    Step 12: Enter the Row 1 and Row 2 Headings: Enter to cell A1

13. 13

    Step 13: a Enter to cell A2

14. 14

    Step 14: By changing cell Enter to cell B1

15. 15

    Step 15: sqrt(a) Enter to cell C1

16. 16

    Step 16: sqrt(b) Enter to cell C2

17. 17

    Step 17: " = sqrt(a)/(sqrt(a)-1)" ...(w/o quotes and with a leading space) Enter to cell D1

18. 18

    Step 18: b = Enter to cell D2

19. 19

    Step 19: (sqrt(b))^2 Enter to cell E1

20. 20

    Step 20: c = Enter to cell E2

21. 21

    Step 21: sqrt(a) + sqrt(b) Enter to cell F1

22. 22

    Step 22: c = Enter to cell F2

23. 23

    Step 23: sqrt(a) * sqrt(b) Enter to cell G1

24. 24

    Step 24: Goal Enter to cell H1

25. 25

    Step 25: Goal Seek to 0

26. 26

    Step 26: 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

27. 27

    Step 27: sqrt(a) + sqrt(b) = sqrt(a) * sqrt(b) = c Enter to cell I2

28. 28

    Step 28: sqrt(a) + sqrt(b) -sqrt(b) = (sqrt(a) * sqrt(b)) - sqrt(b) Enter to cell I3

29. 29

    Step 29: sqrt(a) = sqrt(b) * (sqrt(a) - 1) (divide both sides by (sqrt(a) - 1)) to get: Enter to cell I4

30. 30

    Step 30: sqrt(a)/(sqrt(a) - 1) = sqrt(b) (notice this is the same result as in C1 and C2.

31. 31

    Step 31: Format the Goal Seeking columns red.

32. 32

    Step 32: Define the variable names for the worksheet: Select columns A:C Do Insert Name Create in Top Row

33. 33

    Step 33: OK. You should end up with 3 variable names: a

34. 34

    Step 34: sqrt_a and sqrt_b.

35. 35

    Step 35: Enter the Input cells and formulas for rows 3-5: Enter to cell A3 the formula

36. 36

    Step 36: =PI() Enter to cell A4 the formula

37. 37

    Step 37: =10 Enter to cell A5 the value

38. 38

    Step 38: as a preliminary guess.

39. 39

    Step 39: Do Goal Seeking to 2π: Enter to cell G5 the formula

40. 40

    Step 40: =2*PI() and Format Cells Fill yellow; Enter to cell H5 the formula

41. 41

    Step 41: =G5-F5 Select cell H5 and do Tools

42. 42

    Step 42: Goal Seek

43. 43

    Step 43: Set Cell H5

44. 44

    Step 44: To value 0

45. 45

    Step 45: By changing cell

46. 46

    Step 46: and click on cell A5.

47. 47

    Step 47: Select cell range A1:I5 and Edit Copy it and Edit Paste it to A8:I12

48. 48

    Step 48: Modify the algebraic solution: Enter to cell I8

49. 49

    Step 49: sqrt(a) - sqrt(b) = sqrt(a) * sqrt(b) Enter to cell I9

50. 50

    Step 50: sqrt(a) - sqrt(b) + sqrt(b) = (sqrt(a) * sqrt(b)) + sqrt(b) Enter to cell I10

51. 51

    Step 51: sqrt(a) = sqrt(b) * (sqrt(a) + 1) Enter to cell I11

52. 52

    Step 52: sqrt(a)/(sqrt(a) + 1) = sqrt(b)

53. 53

    Step 53: 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.

54. 54

    Step 54: Change the formulas appropriately: Enter to cell C10 the formula

55. 55

    Step 55: =sqrt_a/(sqrt_a+1) and Edit Fill Down to cell C12.

56. 56

    Step 56: Do Goal Seeking to 2π: Select cell H12 and do Tools

57. 57

    Step 57: Goal Seek

58. 58

    Step 58: Set Cell H12

59. 59

    Step 59: To value 0

60. 60

    Step 60: By changing cell

61. 61

    Step 61: and click on cell A12.

62. 62

    Step 62: Select cell range A1:I12 and Edit Copy it and Edit Paste it to A14:I25

63. 63

    Step 63: Modify the algebraic solution: Enter to cell I14

64. 64

    Step 64: sqrt(a) + sqrt(b) = sqrt(a) / sqrt(b) Enter to cell I15

65. 65

    Step 65: sqrt(a) + sqrt(b) - sqrt(a) = (sqrt(a) / sqrt(b)) - sqrt(a) Enter to cell I16

66. 66

    Step 66: sqrt(b) = sqrt(a) / (1/(sqrt(b)) - 1) Enter to cell I17

67. 67

    Step 67: sqrt(b)/ (1/(sqrt(b)) - 1) = sqrt(a)

68. 68

    Step 68: 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.

69. 69

    Step 69: Change the formulas appropriately: Enter to cell B16 the formula

70. 70

    Step 70: =sqrt_b/(1/(sqrt_b)-1) and Edit Fill Down to cell B18.

71. 71

    Step 71: Do Goal Seeking to -2π: Enter to cell G18 the formula

72. 72

    Step 72: =-2*PI() Select cell H18 and do Tools

73. 73

    Step 73: Goal Seek

74. 74

    Step 74: Set Cell H18

75. 75

    Step 75: To value 0

76. 76

    Step 76: By changing cell

77. 77

    Step 77: and click on cell D18.

78. 78

    Step 78: Modify the algebraic solution: Enter to cell I21

79. 79

    Step 79: sqrt(a) - sqrt(b) = sqrt(a) / sqrt(b) Enter to cell I22

80. 80

    Step 80: sqrt(a) - sqrt(b) - sqrt(a) = (sqrt(a) / sqrt(b)) - sqrt(a) Enter to cell I23

81. 81

    Step 81: " -sqrt(b) = sqrt(a) / (1/(sqrt(b)) - 1)"

82. 82

    Step 82: w/o quotes and w/ leading space Enter to cell I24

83. 83

    Step 83: " -sqrt(b)/ (1/(sqrt(b)) - 1) = sqrt(a)"

84. 84

    Step 84: w/o quotes and w/ leading space

85. 85

    Step 85: 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.

86. 86

    Step 86: Sort the range Select cell range A23:H25 and do Data Sort Options

87. 87

    Step 87: Uncheck My List has headers

88. 88

    Step 88: sort on column D

89. 89

    Step 89: smallest to largest.

90. 90

    Step 90: Extend the Range via a Fill Series command: Select D25:D33 and do Edit Fill Series

91. 91

    Step 91: Step Value = -1

92. 92

    Step 92: Copy cell range A23:F23 and Paste it to cell A35.

93. 93

    Step 93: Change the formulas appropriately: Enter to cell B23 the formula

94. 94

    Step 94: =-sqrt_b/(1/(sqrt_b)-1) and Edit Fill Down to cell B35.

95. 95

    Step 95: Do Goal Seeking to 2π: Enter to cell G23 the formula

96. 96

    Step 96: =2*PI() Select cell H23 and do Tools

97. 97

    Step 97: Goal Seek

98. 98

    Step 98: Set Cell H23

99. 99

    Step 99: To value 0

100. 100

     Step 100: By changing cell

101. 101

     Step 101: and click on cell D23.

102. 102

     Step 102: Enter to cell A36 the label

103. 103

     Step 103: and Format Cells Font Bold and Underlined;

104. 104

     Step 104: Enter to cell B36 the label

105. 105

     Step 105: and Format Cells Font Bold and Underlined;

106. 106

     Step 106: Enter to cell A37 the formula

107. 107

     Step 107: and Edit Fill down to cell range A37:A49;

108. 108

     Step 108: Enter to cell B37 the formula

109. 109

     Step 109: and Edit Fill down to cell range B37:B49;

110. 110

     Step 110: Select cell range A37:B49 and either Insert Chart or use the Chart Wizard or the Ribbon and select Charts

111. 111

     Step 111: Scatter

112. 112

     Step 112: Smooth Lined Scatter.

113. 113

     Step 113: Using Chart Layout

114. 114

     Step 114: choose Axis and select No Vertical Axis and No Horizontal Axis

115. 115

     Step 115: then Gridlines Horizontal none

116. 116

     Step 116: Legend none.

117. 117

     Step 117: 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

118. 118

     Step 118: or whatever appeals to you.

119. 119

     Step 119: Enter to D60

120. 120

     Step 120: sqrt(E)/sqrt(m) = sqrt(E) - sqrt(m)

121. 121

     Step 121: Enter to D61

122. 122

     Step 122: sqrt(E)/sqrt(m) - sqrt(E)= sqrt(E) - sqrt(m) -sqrt(E)

123. 123

     Step 123: Enter to D62

124. 124

     Step 124: sqrt(E)(1/sqrt(m) - 1) = -sqrt(m)

125. 125

     Step 125: Enter to D63

126. 126

     Step 126: sqrt(E) = -sqrt(m)/(1/sqrt(m) - 1)

127. 127

     Step 127: and multiply by -1/-1 on the right to get:

128. 128

     Step 128: Enter to D64

129. 129

     Step 129: sqrt(E) = sqrt(m)/(1 - 1/sqrt(m))

130. 130

     Step 130: Enter to A69

131. 131

     Step 131: Light Speed mps (where mps = miles per second)

132. 132

     Step 132: Enter to A70

133. 133

     Step 133: Miles/Second

134. 134

     Step 134: Enter to A71

135. 135

     Step 135: =299792458/1609.34400000087 (i.e.

136. 136

     Step 136: light speed in meters/second / meters/mile)

137. 137

     Step 137: giving you the result of 186

138. 138

     Step 138: 282.39705112 (lights peed in miles/sec.)

139. 139

     Step 139: Enter to B70

140. 140

     Step 140: Enter to C70

141. 141

     Step 141: Enter to D70

142. 142

     Step 142: sqrt(E)

143. 143

     Step 143: Enter to E70

144. 144

     Step 144: sqrt(m)

145. 145

     Step 145: Enter to F70

146. 146

     Step 146: sqrt(E)/sqrt(m)

147. 147

     Step 147: Enter to G70

148. 148

     Step 148: sqrt(E) - sqrt(m)

149. 149

     Step 149: Enter to H70

150. 150

     Step 150: Goal Seek to 0

151. 151

     Step 151: Enter to B71

152. 152

     Step 152: =D71^2

153. 153

     Step 153: Enter to C71

154. 154

     Step 154: 1 (as a preliminary guesstimate of mass m)

155. 155

     Step 155: Enter to D71 the formula

156. 156

     Step 156: =E71/(1-(1/E71))

157. 157

     Step 157: Enter to E71 the formula

158. 158

     Step 158: =SQRT(C71)

159. 159

     Step 159: Enter to F71 the formula

160. 160

     Step 160: =D71/E71

161. 161

     Step 161: Enter to G71 the formula

162. 162

     Step 162: =D71-E71

163. 163

     Step 163: Enter to H71 the formula

164. 164

     Step 164: =A71-F71

165. 165

     Step 165: Goal seek on cell H71 to value 0 by changing the mass in cell C71.

166. 166

     Step 166: 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

167. 167

     Step 167: Geometric and/or Trigonometric Art

168. 168

     Step 168: Charting/Diagramming and Algebraic Formulation.

Detailed Guide

About the Author