Complex Math

RV'SMath, Jul 26, 2016

NOTE: (IT IS BEST TO USE A 23" MONITOR TO VIEW THIS MATERIAL, THANK YOU) 12/02/2022 - Today I have fixed the triangle in the Dynamic Coordinates Work Sheet so it can take on neqitive vales and return to its shape in the first quadrant in the ln(x) section 06/12/2020 I deleted "equations in x" because I made a mistake and had each number f(x) off by 1 place. It is an easy fix, but I took it down until I correct the mistake. 04/16/20 - On "Integral Ln x" by accident I left out the "number 2" in the denominator in line 2 on the right hand side, I believe, when showing 1 = xt. 12-30-2018 - Correction - I found that in the "Vector Addition" worksheet there is a "Dimension Leader Line" that needs to be - moved - so it's "dimension" is equal to " 1." It's labeled; " i " and should be relocated so the corresponding math above is understandable and applicable to the drawing itself. 09-14-2018 - After inspecting my ' time clocks ' they seem to be off about 1 sec. from being at 20 sec. per cycle. 11-18-2017 This book is made for observation purposes only or something only to consider as relationships of numbers. Applied Math 03-01-2017 Today I made some changes to two worksheets involving the " a " vectors in the (1,-1) position. Their lengths were supposed to be governed by the slider " a " dimension which I had mistakenly made the " Square root of " a " " which is now corrected. It is easy to hastily make these mistakes while trying to build this idea. Sorry for any confusion this may have caused. 09-28-2016 - I made some changes to some worksheets. When I find that I made mistakes I try to fix them. 08-17-2016 - In drawing this relationship of Triangle in this manner it must be said that the 1st quadrant has technically a 2:1 scale as opposed to below the axis which is 1to1; consequently when drawing the vectors it will look different than what I had drawn on a blank page. Theoretically though, my point to drawing this triangle in this manner was in fact to set all elements equal to each other and then determine their dimension or relationship; the math is proportional to itself so as long as you keep the relationships of "a" you will get the same results. I really did not notice that this would create such a big problem! I did a lot of different versions of this relationship and that is why it is called "Independent Axis" so long as you keep the relationships proportional to each other which "scale" may not be so apparent to the viewer but is common amongst drafters which I incorporated drafting into the idea of math. Again this math was not supposed to replace common math ideas but is experimental. You can see the relationship in my Complex Trigonometry Worksheet. Today's Date 07-01-2016 My Approximation for Pi ≈ 1.35 + 1.41 + .24 + √ 8/ 20 = 3.14142135.... - Also see the "Box Function" in Chapter Ln(x). The Null Set is important too. The applications Chapters are Ch. 1 and Ch.13 through Ch. 18 - Show this relationship at work. The other Chapters primarily show geometry which takes time to look at. 2 - 29 - 2016 ( See i - pairs where I show the Null Set:{-1,1,0,i,i²} graphically. ) - I was looking at the 'Step Function' again and that thing is pretty cool: See 'Four Dimensional Space.' 2 - 21 - 2016: I updated the worksheet - Complex Geometry - Time Motion to show the velocity vector in action. 2 - 19 - 2016 I Expanded my Complex Trigonometry Grid to Have reference equations as ' Units' for the Grid- which give location reference on the Grid itself. 2 - 6 - 2016; This math is ' experimental ' geometrical relationships, that I hope supports my Hypothesis in the work sheet; ' Independent Axis. ' (This math is not meant to be a contradiction to traditional mathematics; My math is applied mathematics with specific ideas that I use for the basis of the geometries.) On the other hand. (I do hope; that both my books on applied math - can be applied to traditional mathematics, with the hope of helping solve some traditional math problems?) 1 - 31 - 2016: I updated Complex Trigonometry Worksheet - Incorporated a "Time" Grid. a) I Removed my "Time Equation": After looking at the functioning of the equation; it needs to be reconfigured; It must be an Increasing Function! 1 - 30 - 2016: http://www.space.com/31765-ancient-babylonians-tracked-jupiter-with-math.html - See Story and Compare to Complex Trigonometry - I added two versions of worksheets, using this relationship, and i - geometry. 1 - 31 - 2016: ( I made a correction to Worksheet; Time(i) space time of "a") 1- 17 - 2016 - I removed the " Complex Trigonometry " work sheet - I need to look at that more closely. 1 - 18 -2016 - ( I moved this to its' own chapter - I clarified the Geometry and the Problem I stated above ) 1 - 11 - 2016 i geometry dimensions ; a = 1 - Shows how the Geometry fits in perfectly with a circle of diameter 4. 10 - 18 - 2015 - Prime Numbers - The Complete Solution Generates prime number values - I listed some in left bottom column of page http://news.yahoo.com/math-could-reveal-hidden-sources-chaos-125023192.html - ( Some really good geometry is shown in " Center Line e(x) " ) http://www.livescience.com/17455-butterfly-effect-weather-prediction.html - Compare my diagrams to this?, The Circles.

Table of Contents

  1. Independent Axis
    1. Independence of Axis
    2. Proof of the" Time Vector " and " Hypothesis " Complex Math
    3. Complex algebra - Dot Product
    4. Graphical Theory
    5. Graphical Theory 2
    6. i - pairs
    7. Bernoulli Distribution - Defining 'p'
    8. The Determination of x²
    9. Showing x² = dy/dx
    10. Showing Calculated Value of Pi
    11. Pi at a = 5000
    12. Vector Equations
    13. Summation(x)
    14. Showing i-squared = - 1
    15. square box
    16. cylinder
    17. square roots of X
    18. Showing the Derivative of (x) Squared = a
    19. Imaginary Planes
  2. Fine Tuning Time Space
    1. SubSpace - Closed adition ; Closed Mulltiplication; and Zero
    2. How to do Addition and Subtraction
    3. Research
    4. i = pi * d^2 - d
    5. How to show the Distance between two points.
  3. Four Dimensional Space
    1. Four Dimensions of Space
    2. Step Function
    3. Pi Distribution
    4. Pi Distribution Random Data Graph
    5. Squarring the Circle
    6. 54 (D)
    7. Cubic Box - x^2 + x +1 = 0
    8. Limits of Integration
    9. Equation of a Line in 'a'
    10. Ratio of a Circle
    11. Descriptive Geometry
  4. Graphs
    1. Graphics Negative Area i^2
    2. Negative Area Defining i
    3. The 3D Space Circles, Ellipses, Lines
    4. i distribution (rv(a))
    5. x^2 - x
    6. Relationship of Numbers to the Circle
  5. Multiplication
    1. Direction of Flow - Equilibium Conditions
    2. Block Multiplication : The Three Types
    3. Force Triangle Results Complex Math
    4. Projection of Lines
    5. Riemann Summ/ i - pairs
    6. Complex Mechanics Of Materials
    7. Dynamic Analysis - Descriptive Geometry
    8. (Multiplication 2)
    9. n - Factorial
    10. Gamma Function
    11. Multiplication
  6. The Definition of i
    1. Defining the Space i
    2. The Expansion of the Cube
    3. i(th) areas
    4. Time(i) Space Time of " a "
    5. Details of Fisher's Formula Long Version
    6. Fisher's Exact Test Two - Changes
    7. Square root of a circle
    8. a +bi
    9. Circumference of a Circle using " i "
  7. Construction of the i distribution
    1. Converting Imaginary Numbers 2
    2. Integrating Gamma by a
    3. Inverse Gamma Imaginary Function
    4. matrix 2
    5. Circle Geometry(i)
    6. Defining i^2
  8. Transformations
    1. Imaginary Hypotenuse i
    2. Boltzman Transport Equation
    3. Properties of : a, i, i^2 , x^2
    4. Pi Factored and Transformed = 2
    5. The Geometry of i - A Complex Relationship
    6. sin 2 inverse
    7. y parabolic
    8. The Sphere
    9. The SemiCircle
    10. Solving the roots of x^2 + 1 = 0
  9. Geometry of i
    1. Full Geometry of i - Proof 2
    2. Coefficients of i - The Construction ....
    3. Imaginary Pi(i) - Construction i
    4. RV(i) Coordinate System - Evaluator
    5. Proof 3 - Complex(Pi) Trigonometry Geometry Solution
    6. i geometry dimensions; a = 1
    7. Complex Geometry ' i '
    8. Speed of Light Constant: Time
    9. vector addition
    10. 2 i
    11. Kepler's Ellipse : on the Complex Number Line
    12. x- geometry
    13. Inverse Trig functions
    14. i[squared] and i
    15. Imaginary Parabola
  10. Commentary
    1. Imaginary Number Line(i) - Time Space
    2. Implicit Differential Equations (i) revisited
    3. 2^x - 3 Dimensional Number Line
    4. How to Calculate Pi Using 2^x
    5. Geometry - Calculations - Relationships
    6. Prime Number 13
    7. The Inside of the funciton :(a)
    8. The Factoring Machine Speed of Light
    9. i= - 1/i
    10. Parabolas in tx
  11. Art
    1. Roof Design(i)
    2. Measuring Functions - Ln(x)
    3. GeoGraphing i Mathematical Design
    4. Dynamic Geometry(i)
    5. Equation of a line in "a"
    6. Circular Motion of a Line
    7. Cubic Triangle
  12. Ln(x)
    1. Area Diagram - Ln x
    2. Box Function - Fermat's Equation
    3. Partial Beta Function - Turbulance
    4. i / i^2 - Hyperbola
    5. Dam Project
    6. Integral ln x
    7. Dynamic Coordinates(i)
  13. Time Clock
    1. Orbit(i) Around the Planets
    2. " The i Clock "
    3. 3 D - i Clock
    4. Orbit(i) Around Planets 2
    5. Imaginary Numbers Orbits
  14. Probabilities
    1. i - probabilities (part two) - reference points
    2. Evenly divided Integers of ' a '
    3. Normal vs RVsigma - Proof 6 - varifacation(i)
    4. Center Line e(x)
    5. Rectangler Probabilities - Sums of Areas
    6. Normal Circular Probabilities
    7. Triangular Normal Probabilities
    8. Direct Link: Area/Variance = Constant
    9. Independent Distributions
    10. Independent Distributions 2
  15. Analysis of the Normal Curve
    1. Proof : i = i²
    2. Analysis of the Normal Curve
    3. Ln(4.47x) Further Developement
    4. Moment About the Origin
  16. Prime Numbers
    1. Primes 1 - 43
    2. All Components of (i)
    3. q = e^x ln(x) - ratio of areas
    4. Prime Area Equation: Complete Solution
    5. Number Line
    6. Prime Number Factors
    7. a + b = c
    8. Differential Equations
  17. Null Set
    1. Proof the Null Set ϵ { i², i , ž }
    2. Null Space: Full Proof
    3. Undefined - Dividing by Zero
    4. Compostie Function(i); The Null Set
    5. Proof: 1 = t(1 + t²) = a
    6. 1 = t(1 + t²) = a: Graph
    7. Showing - Geometry(i) - equivelance
    8. Counting Numbers(i)
    9. tau = 1
  18. Complex Trigonometry
    1. Complex Trigonemtric Formulas
    2. Solving ' a ' geometry - expotential equations
    3. i - geometry probabilities
    4. Version 1 - Area of Trapezoid
    5. Version 2 - Area of a Trapezoid
    6. Exponential Log Numbers
    7. Time Motion - Increasing Velocity
    8. Complex Trigonometry
  19. The Four Subspaces
    1. The Four Subspaces of a Matrix
    2. Regression Analysis Y = XB + e
    3. Y = X B + e Matrix Form
    4. Design Experiment - Data Analysis
  20. Lines

1.

Independent Axis

Ch 1 The Idea and Rules behind Dimensions

  • 1. Independence of Axis
  • 2. Proof of the" Time Vector " and " Hypothesis " Complex Math
  • 3. Complex algebra - Dot Product
  • 4. Graphical Theory
  • 5. Graphical Theory 2
  • 6. i - pairs
  • 7. Bernoulli Distribution - Defining 'p'
  • 8. The Determination of x²
  • 9. Showing x² = dy/dx
  • 10. Showing Calculated Value of Pi
  • 11. Pi at a = 5000
  • 12. Vector Equations
  • 13. Summation(x)
  • 14. Showing i-squared = - 1
  • 15. square box
  • 16. cylinder
  • 17. square roots of X
  • 18. Showing the Derivative of (x) Squared = a
  • 19. Imaginary Planes

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1.11.

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1.12.

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1.13.

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1.14.

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1.15.

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1.16.

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1.17.

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1.18.

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1.19.

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2.

Fine Tuning Time Space

  • 1. SubSpace - Closed adition ; Closed Mulltiplication; and Zero
  • 2. How to do Addition and Subtraction
  • 3. Research
  • 4. i = pi * d^2 - d
  • 5. How to show the Distance between two points.

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3.

Four Dimensional Space

  • 1. Four Dimensions of Space
  • 2. Step Function
  • 3. Pi Distribution
  • 4. Pi Distribution Random Data Graph
  • 5. Squarring the Circle
  • 6. 54 (D)
  • 7. Cubic Box - x^2 + x +1 = 0
  • 8. Limits of Integration
  • 9. Equation of a Line in 'a'
  • 10. Ratio of a Circle
  • 11. Descriptive Geometry

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4.

Graphs

  • 1. Graphics Negative Area i^2
  • 2. Negative Area Defining i
  • 3. The 3D Space Circles, Ellipses, Lines
  • 4. i distribution (rv(a))
  • 5. x^2 - x
  • 6. Relationship of Numbers to the Circle

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5.

Multiplication

  • 1. Direction of Flow - Equilibium Conditions
  • 2. Block Multiplication : The Three Types
  • 3. Force Triangle Results Complex Math
  • 4. Projection of Lines
  • 5. Riemann Summ/ i - pairs
  • 6. Complex Mechanics Of Materials
  • 7. Dynamic Analysis - Descriptive Geometry
  • 8. (Multiplication 2)
  • 9. n - Factorial
  • 10. Gamma Function
  • 11. Multiplication

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6.

The Definition of i

  • 1. Defining the Space i
  • 2. The Expansion of the Cube
  • 3. i(th) areas
  • 4. Time(i) Space Time of " a "
  • 5. Details of Fisher's Formula Long Version
  • 6. Fisher's Exact Test Two - Changes
  • 7. Square root of a circle
  • 8. a +bi
  • 9. Circumference of a Circle using " i "

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7.

Construction of the i distribution

  • 1. Converting Imaginary Numbers 2
  • 2. Integrating Gamma by a
  • 3. Inverse Gamma Imaginary Function
  • 4. matrix 2
  • 5. Circle Geometry(i)
  • 6. Defining i^2

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8.

Transformations

  • 1. Imaginary Hypotenuse i
  • 2. Boltzman Transport Equation
  • 3. Properties of : a, i, i^2 , x^2
  • 4. Pi Factored and Transformed = 2
  • 5. The Geometry of i - A Complex Relationship
  • 6. sin 2 inverse
  • 7. y parabolic
  • 8. The Sphere
  • 9. The SemiCircle
  • 10. Solving the roots of x^2 + 1 = 0

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9.

Geometry of i

  • 1. Full Geometry of i - Proof 2
  • 2. Coefficients of i - The Construction ....
  • 3. Imaginary Pi(i) - Construction i
  • 4. RV(i) Coordinate System - Evaluator
  • 5. Proof 3 - Complex(Pi) Trigonometry Geometry Solution
  • 6. i geometry dimensions; a = 1
  • 7. Complex Geometry ' i '
  • 8. Speed of Light Constant: Time
  • 9. vector addition
  • 10. 2 i
  • 11. Kepler's Ellipse : on the Complex Number Line
  • 12. x- geometry
  • 13. Inverse Trig functions
  • 14. i[squared] and i
  • 15. Imaginary Parabola

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10.

Commentary

  • 1. Imaginary Number Line(i) - Time Space
  • 2. Implicit Differential Equations (i) revisited
  • 3. 2^x - 3 Dimensional Number Line
  • 4. How to Calculate Pi Using 2^x
  • 5. Geometry - Calculations - Relationships
  • 6. Prime Number 13
  • 7. The Inside of the funciton :(a)
  • 8. The Factoring Machine Speed of Light
  • 9. i= - 1/i
  • 10. Parabolas in tx

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11.

Art

  • 1. Roof Design(i)
  • 2. Measuring Functions - Ln(x)
  • 3. GeoGraphing i Mathematical Design
  • 4. Dynamic Geometry(i)
  • 5. Equation of a line in "a"
  • 6. Circular Motion of a Line
  • 7. Cubic Triangle

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12.

Ln(x)

  • 1. Area Diagram - Ln x
  • 2. Box Function - Fermat's Equation
  • 3. Partial Beta Function - Turbulance
  • 4. i / i^2 - Hyperbola
  • 5. Dam Project
  • 6. Integral ln x
  • 7. Dynamic Coordinates(i)

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13.

Time Clock

  • 1. Orbit(i) Around the Planets
  • 2. " The i Clock "
  • 3. 3 D - i Clock
  • 4. Orbit(i) Around Planets 2
  • 5. Imaginary Numbers Orbits

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14.

Probabilities

  • 1. i - probabilities (part two) - reference points
  • 2. Evenly divided Integers of ' a '
  • 3. Normal vs RVsigma - Proof 6 - varifacation(i)
  • 4. Center Line e(x)
  • 5. Rectangler Probabilities - Sums of Areas
  • 6. Normal Circular Probabilities
  • 7. Triangular Normal Probabilities
  • 8. Direct Link: Area/Variance = Constant
  • 9. Independent Distributions
  • 10. Independent Distributions 2

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15.

Analysis of the Normal Curve

This is a complex geometric analysis of the Normal Curve using RV-sigma as the leading parameter while adhering to i - geometry; Linking RV-sigma and i - geometry by using e^(-.69) = .5 = i / 2. This layout conforms to the geometry while using Standard Deviation to govern the range of motion. The main occurrence here is the uniqueness of the set of recurring numbers: 1, 1.16, 1.35 , 1.64 , 1.79 , 1.57 , 1.41 , 1.42 , 3.14 , 1.08 ..... etc.

  • 1. Proof : i = i²
  • 2. Analysis of the Normal Curve
  • 3. Ln(4.47x) Further Developement
  • 4. Moment About the Origin

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16.

Prime Numbers

I referenced this story and believe I have shown a contradiction to the theory by showing a relationship between a , b , c in the expression a + b = c By Davide Castelvecchi and Nature magazine | October 8, 2015 Math Mystery: Shinichi Mochizuki and the Impenetrable Proof Primal importance The abc conjecture refers to numerical expressions of the type a + b = c. The statement, which comes in several slightly different versions, concerns the prime numbers that divide each of the quantities a, b and c. Every whole number, or integer, can be expressed in an essentially unique way as a product of prime numbers—those that cannot be further factored out into smaller whole numbers: for example, 15 = 3 × 5 or 84 = 2 × 2 × 3 × 7. In principle, the prime factors of a and b have no connection to those of their sum, c. But the abc conjecture links them together. It presumes, roughly, that if a lot of small primes divide a and b then only a few, large ones divide c

  • 1. Primes 1 - 43
  • 2. All Components of (i)
  • 3. q = e^x ln(x) - ratio of areas
  • 4. Prime Area Equation: Complete Solution
  • 5. Number Line
  • 6. Prime Number Factors
  • 7. a + b = c
  • 8. Differential Equations

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17.

Null Set

Here I prove that the Null Set Contains Imaginary Numbers and Integers = { i^2 , i , Z }

  • 1. Proof the Null Set ϵ { i², i , ž }
  • 2. Null Space: Full Proof
  • 3. Undefined - Dividing by Zero
  • 4. Compostie Function(i); The Null Set
  • 5. Proof: 1 = t(1 + t²) = a
  • 6. 1 = t(1 + t²) = a: Graph
  • 7. Showing - Geometry(i) - equivelance
  • 8. Counting Numbers(i)
  • 9. tau = 1

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18.

Complex Trigonometry

Here I clarify the Geometry associated with the Geometric shape, using variables " a " & " i "

  • 1. Complex Trigonemtric Formulas
  • 2. Solving ' a ' geometry - expotential equations
  • 3. i - geometry probabilities
  • 4. Version 1 - Area of Trapezoid
  • 5. Version 2 - Area of a Trapezoid
  • 6. Exponential Log Numbers
  • 7. Time Motion - Increasing Velocity
  • 8. Complex Trigonometry

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19.

The Four Subspaces

  • 1. The Four Subspaces of a Matrix
  • 2. Regression Analysis Y = XB + e
  • 3. Y = X B + e Matrix Form
  • 4. Design Experiment - Data Analysis

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20.

Lines

Various relationships of lines to a circle


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