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Mathematics Dictionary
Dr. K. G. Shih
Demoivre's Theory
Subjects


  • AL 18 00 | - Outlines
  • AL 18 01 | - Imaginary number and complex numbers
  • AL 18 02 | - Solve X^3 - 1 = 0 geometrically
  • AL 18 03 | - Solve X^4 - 1 = 0 geometrically
  • AL 18 04 | - Solve X^5 - 1 = 0 geometrically
  • AL 18 05 | - Solve x^3 + 1 = 0 by construction
  • AL 18 06 | - Solve x^4 + 1 = 0 by construction
  • AL 18 07 | - Solve x^5 + 1 = 0 by construction
  • AL 18 08 | - Solve x^3 + x^2 + x + 1 = 0
  • AL 18 09 | - Solve x^4 + x^3 + x^2 + x + 1 = 0 by construction
  • AL 18 10 | - Solve x^4 - x^3 + x^2 - x + 1 = 0 by construction
  • AL 18 11 | - Solve x^4 - i = 0 by construction
  • AL 18 12 | - Solve x^4 + i = 0
  • AL 18 13 | -
  • AL 18 14 | - Quiz for complex numbersNew
  • AL 18 15 | - Answer to quiz
  • AL 18 16 | - Reference
  • AL 18 17 | - Summary and formula
    • Answers


    AL 18 01. Imaginary number and complex numbers

    Outlines
    • Complex number expression
      • z = a + b*i is complex number.
      • a is sames x in rectangular coordinate
      • b is sames y in rectangular coordinate
    • What is conjugate complex number ?
      • z1 = a + b*i is conjugate of z2 = a - b*i
      • Properties 1 : z1 + z2 = real
      • Properties 2 : z1*z2 = real = a^2 + b^2
    More defintion and examples     Apllication of DeMoivre Theory
    • Solve x^n - 1 = 0
      • Draw a large unit circle. On circle draw n point A, B, C, ...
      • Angles between A, B, C, ... are Ang = 360/n
      • 1st angle AOX = 0/n = 0
      • 2nd angle BOX = 1*Ang + 1st angle
      • 3rd angle COX = 2*Ang + 1st angle
      • ...
  • Solve x^n + 1 = 0
    • Draw a large unit circle. On circle draw n point A, B, C, ...
    • Angles between A, B, C, ... are Ang = 360/n
    • 1st angle AOX = 180/n
    • 2nd angle BOX = 1*Ang + 1st angle
    • 3rd angle COX = 2*Ang + 1st angle
    • ...
  • Solve x^n - i = 0
    • Draw a large unit circle. On circle draw n point A, B, C, ...
    • Angles between A, B, C, ... are Ang = 360/n
    • 1st angle AOX = 90/n = 0
    • 2nd angle BOX = 1*Ang + 1st angle
    • 3rd angle COX = 2*Ang + 1st angle
    • ...
  • Solve x^n + i = 0
    • Draw a large unit circle. On circle draw n point A, B, C, ...
    • Angles between A, B, C, ... are Ang = 360/n
    • 1st angle AOX = 270/n = 0
    • 2nd angle BOX = 1*Ang + 1st angle
    • 3rd angle COX = 2*Ang + 1st angle
    • ...
    Go to Begin

    AL 18 02. Solve X^3 - 1 = 0 geometrically

    Method
    • 1. Draw coordiante OX and OY
    • 2. Draw unit circle with center at O
    • 3. Draw 3 points A, B and C on unit circle
    • 4. Angles between OA, OB and OC are 360/3 = 120
      • Angle AOX = 0
      • Angle BOX = 1*120 = 120
      • Angle COX = 2*120 = 240
    Solution : The roots are R0, R1 and R2
    • Find R0 using complex coordiante at A : R0 = x + y*i
      • At A : x = 1 and y = 0
      • Hence R0 = 1 + 0*i = 1
    • Find R1 using complex coordiante at B : R1 = x + y*i
      • At B : x = -0.5 and y = 0.866 by measurement
      • Hence R1 = -0.5 + 0.866*i
    • Find R2 using complex coordiante at C : R2 = x + y*i
      • At C : x = -0.5 and y = -0.866 by measurement
      • Hence R2 = -0.5 - 0.866*i
    Note
    • Note 1 : R2 is conjugate of R1, Hence we can get R2 using R1
    • Note 2 : Using right angle triangle wirh 30 degrees, we can find R1 = -1/2 + i*Sqr(3)/2

    Go to Begin

    AL 18 03. Solve X^4 - 1 = 0 geometrically

    Method
    • 1. Draw coordiante OX and OY
    • 2. Draw unit circle with center at O
    • 3. Draw 3 points A, B C and D on unit circle
    • 4. Angles between OA, OB, OC and OD are 360/4 = 90
      • Angle AOX = 0
      • Angle BOX = 1*90 = 90
      • Angle COX = 2*90 = 180
      • Angle DOX = 3*90 = 270
    Solution : The roots are R0, R1, R3 and R4
    • Find R0 using complex coordiante at A : R0 = x + y*i
      • At A : x = 1 and y = 0
      • Hence R0 = 1 + 0*i = 1
    • Find R1 using complex coordiante at B : R1 = x + y*i
      • At B : x = 0.0 and y = 1 by measurement
      • Hence R1 = 0 + 1*i = i
    • Find R2 using complex coordiante at C : R2 = x + y*i
      • At C : x = -1 and y = 0 by measurement
      • Hence R2 = -1 - 0*i = -1
    • Find R3 using complex coordiante at D : R3 = x + y*i
      • At D : x = 0 and y = -1 by measurement
      • Hence R3 = 0 - 1*i = -i
    Note
    • Note 1 : R1 is conjugate of R3 and R1*R3 = -i^2 = 1
    • Note 2 : Diagram in AL 18 04)
    Go to Begin

    Q04.Solve X^5 - 1 = 0 geometrically

    Method
    • 1. Draw coordiante OX and OY
    • 2. Draw unit circle with center at O
    • 3. Draw 3 points A, B C, D and E on unit circle
    • 4. Angles between OA, OB, OC, OD and OE are 360/5 = 72
      • Angle AOX = 0
      • Angle BOX = 1*72 = 072
      • Angle COX = 2*72 = 144
      • Angle DOX = 3*72 = 216
      • Angle EOX = 4*72 = 288
    Solution : The roots are R0, R1, R3 and R4
    • Find R0 using complex coordiante at A : R0 = x + y*i
      • At A : x = 1 and y = 0
      • Hence R0 = 1 + 0*i = 1
    • Find R1 using complex coordiante at B : R1 = x + y*i
      • At B : x = 0.309 and y = 0.951 by measurement
      • Hence R1 = 0.309 + 0.951*i
    • Find R2 using complex coordiante at C : R2 = x + y*i
      • At C : x = -0.309 and y = 0.951 by measurement
      • Hence R2 = -0.309 + 0.951*i
    • Find R3 using complex coordiante at D : R3 = x + y*i
      • At D : x = -0.309 and y = -0.951 by measurement
      • Hence R3 = -0.309 - 0.951*i
    • Find R4 using complex coordiante at E : R3 = x + y*i
      • At D : x = 0.309 and y = -0.951 by measurement
      • Hence R3 = 0.309 - 0.951*i
    Diagram
    • Subject | 21 01 : Diagrams.
    • Enter 21 01
    • Open application program
      • Select run at current location (No download)
      • Select yes to run
      • Click 01 in upper box
      • click 09 in lower box : Solve x^n - 1 = 0
      • Give data : 5 (n = 5)
    • On the diagram : Five points on circle are the solutions
    Note
    • Note 1 : R1 is conjugate of R4 and R1*R4 = 1
    • Note 2 : R2 is conjugate of R3 and R2*R3 = 1

    Go to Begin

    AL 18 05. Solve x^3 + 1 = 0

    Method
    • 1. Draw coordiante OX and OY
    • 2. Draw unit circle with center at O
    • 3. Draw 3 points A, B and C on unit circle
    • 4. Angles between OA, OB and OC are 360/3 = 120
    • 5. Angle AOX = 180/3 = 60
      • Angle AOX = 60
      • Angle BOX = 60 + 1*120 = 180
      • Angle COX = 60 + 2*120 = 300
    Solution : The roots are R0, R1 and R2
    • Find R0 using complex coordiante at A : R0 = x + y*i
      • At A : x = 0.5 and y = 0.866
      • Hence R0 = 0.5 + 0.866
    • Find R1 using complex coordiante at B : R1 = x + y*i
      • At B : x = -1 and y = 0 by measurement
      • Hence R1 = -1 + 0*i = -1
    • Find R2 using complex coordiante at C : R2 = x + y*i
      • At C : x = 0.5 and y = -0.866 by measurement
      • Hence R2 = 0.5 - 0.866*i
    Notes
    • 1. R2 is conjugate of R1, Hence we can get R2 using R1
    • 2. Using right angle triangle wirh 30 degrees, we can find R0 = 1/2 + i*Sqr(3)/2
    Example : Solve x^2 - x + 1 = 0
      Method 1 : Using above method
      • R0 and R2 are solutions
      • Since x^2 + x + 1 is a factor of (x^3 - 1).
      Method 2 : Using quadratic formula
      • x1 = (1 + Sqr(1^2 - 4*1*1))/2 = (1 + i*Sqr(3))/2.
      • x2 = (1 - Sqr(1^2 - 4*1*1))/2 = (1 - i*Sqr(3))/2.

    Go to Begin

    AL 18 06. Solve x^4 + 1 = 0.

    Method
    • 1. Draw coordiante OX and OY
    • 2. Draw unit circle with center at O
    • 3. Draw 3 points A, B C and D on unit circle
    • 4. Angles between OA, OB, OC and OD are 360/4 = 90
    • 5. Angle AOX = 180/4 = 45
      • Angle AOX = 45
      • Angle BOX = 45 + 1*90 = 135
      • Angle COX = 45 + 2*90 = 225
      • Angle DOX = 45 + 3*90 = 315
    Solution : The roots are R0, R1, R3 and R4
    • Find R0 using complex coordiante at A : R0 = x + y*i
      • At A : x = y = Sqr(2)/2
      • Hence R0 = Sqr(2)/2 + i*Sqr(2)/2
    • Find R1 using complex coordiante at B : R1 = x + y*i
      • At B : x = -Sqr(2)/2 and y = Sqr(2)/2 by measurement
      • Hence R1 = -Sqr(2)/2 + i*Sqr(2)/2
    • Find R2 using complex coordiante at C : R2 = x + y*i
      • At C : x = -Sqr(2)/2 and y = -Sqr(2)/2 by measurement
      • Hence R2 = -Sqr(2)/2 - i*Sqr(2)/2
    • Find R3 using complex coordiante at D : R3 = x + y*i
      • At D : x = Sqr(2)/2 and y = -Sqr(2)/2 by measurement
      • Hence R3 = Sqr(2)/2 - i*Sqr(2)/2

    Go to Begin

    AL 18 07. Solve x^5 + 1 = 0

    Method
    • 1. Draw coordiante OX and OY
    • 2. Draw unit circle with center at O
    • 3. Draw 3 points A, B C, D and E on unit circle
    • 4. Angles between OA, OB, OC, OD and OE are 360/5 = 72
    • Angle AOX = 180/5
      • Angle AOX = 36
      • Angle BOX = 36 + 1*72 = 108
      • Angle COX = 36 + 2*72 = 180
      • Angle DOX = 36 + 3*72 = 252
      • Angle EOX = 36 + 4*72 = 324
    Solution : The roots are R0, R1, R3 and R4
    • Find R0 using complex coordiante at A : R0 = x + y*i
      • At A : x = 0.809 and y = 0.588
      • Hence R0 = 0.809 + 0.588*i
    • Find R1 using complex coordiante at B : R1 = x + y*i
      • At B : x = 0.-809 and y = 0.588 by measurement
      • Hence R1 = -0.809 + 0.588*i
    • Find R2 using complex coordiante at C : R2 = x + y*i
      • At C : x = -1 and y = 0 by measurement
      • Hence R2 = -1
    • Find R3 using complex coordiante at D : R3 = x + y*i
      • At D : x = -0.809 and y = -0.588 by measurement
      • Hence R3 = -0.809 - 0.588*i
    • Find R4 using complex coordiante at E : R3 = x + y*i
      • At D : x = 0.809 and y = -0.588 by measurement
      • Hence R3 = 0.809 - 0.588*i
    Notes
    • Note 1 : R0 is conjugate of R4 and R0*R4 = 1
    • Note 2 : R1 is conjugate of R3 and R1*R3 = 1

    Go to Begin

    AL 18 08. Solve x^3 + x^2 + x + 1 = 0

    Solution
    • Since x^4 - 1 = (x - 1)*(x^3 + x^2 + x + 1) = 0
    • Hence x^3 + x^2 + x + 1 = 0 has 3 roots in x^4 - 1 = 0
    • Hence x = -1, x = i and x = -i are the solutions of x^3 + x^2 + x + 1 = 0
    Prove that x = i is solution of x^3 + x^2 + x + 1 = 0
    • Since i^3 + i^2 + i + 1 = (-i) + (-1) + i + 1 = 0
    • Hence x = i is a solution

    Go to Begin

    AL 18 09. Solve x^4 + x^3 + x^2 + x + 1 = 0 by construction

    Using solution of x^5 - 1 = 0 in AL 18 05
    • Since (x^5 - 1) = (x - 1)*(x^4 + x^3 + x^2 + x + 1).
    • Hence we can use the solutions of x^5 - 1 = 0 in AL18 05.

    Go to Begin

    AL 18 10. Solve x^4 - x^3 + x^2 - x + 1 = 0

    Using solution of x^5 + 1 = 0 in AL 18 07
    • Since (x^5 + 1) = (x + 1)*(x^4 - x^3 + x^2 - x + 1).
    • Hence we can use the solutions of x^5 + 1 = 0 in AL18 07.

    Go to Begin

    AL 18 11. Solve x^4 - i = 0

    Construction
    • Draw unit circle. Draw 4 points A,B,C,D on circle
    • Angles between points = 360/4 and 1st angle is 90/4 = 22.5
    • Draw angle AOX = 22.5 degrees
    • Draw angle BOX = 1*90 + 22.5 = 115.5degrees
    • Draw angle COX = 2*90 + 22.5 = 205.5
    • Draw angle DOX = 3*90 + 22.5 = 295.5
    Solution : The four roots can be obtained by measurements
    • R1 = Complex coordinate of A = a + p*i
      • a = 0.924
      • p = 0.383
    • R2 = Complex coordinate of A = b + q*i
      • b = -0.431
      • q = +0.903
    • R3 = Complex coordinate of A = c + r*i
      • c = -0.903
      • r = -0.431
    • R4 = Complex coordinate of A = d + s*i
      • d = +0.431
      • s = -0.267
    Prove that 0.924 + 0.383*i is a solution of x^4 - i = 0
    • x^4 = (0.924 + 0.383*i)^4
    • = (0.924^4)+ 4*(0.924^3)*(0.383)*i+ 6*(0.924^2)*(0.383^2)*(i^2)
    • + (0.924)*(0.383^3)*(i^3) + (0.383^4)*(i^4)
    • = 0.729 + (4*0.789*0.383)*i + (6*0.854*0.147)*(1) + ...
    • = 0.00 + 0.99*i
    • Hence it is a solution
    Exercise
    • Draw a large unit circle
    • Find a,b,c,d and p,q,r,s by measurement

    Go to Begin

    AL 18 12. Solve x^4 + i = 0

    Construction
    • Draw unit circle. Draw 4 points A,B,C,D on circle
    • Angles between points = 360/4 and 1st angle is 270/4 = 67.5
    • Draw angle AOX = 22.5 degrees
    • Draw angle BOX = 1*90 + 67.5 = 157.5degrees
    • Draw angle COX = 2*90 + 67.5 = 247.5
    • Draw angle DOX = 3*90 + 67.5 = 337.5
    Solution : The four roots can be obtained by measurements
    • R1 = Complex coordinate of A = a + p*i
      • a = 0.3826
      • p = 0.9239
    • R2 = Complex coordinate of A = b + q*i
      • b = -
      • q = +
    • R3 = Complex coordinate of A = c + r*i
      • c = -
      • r = -
    • R4 = Complex coordinate of A = d + s*i
      • d = +
      • s = -
    Prove that 0.3826 + 0.9293*i is a solution of x^4 + i = 0
    • (0.3826 + 0.9293*i)
    • = 0.00 - 0.99*i
    • Hence it is a solution

    Go to Begin

    AL 18 13.


    Go to Begin

    AL 18 14. Quiz
    • 1. Rationalize z = 1/(1/2 - i*Sqr(3)/2)
    • 2. What are conjugate complex numbers ?
    • 3. Solve x^2 - i = 0 by construction
    • 4. Solve x^3 - i = 0 by construction

    Go to Begin

    AL 18 15. Answer to Quiz
    • 1. Rationalize z = 1/(1/2 - i*Sqr(3)/2).
      • Multiply numeritor and denominator by (1/2 + i*Sqr(3)/2).
      • z = (1/2+i*Sqr(3)/2)/((1/2-i*Sqr(3)/2)*(1/2+i*Sqr(3)/2)).
      • z = (1/2+i*Sqr(3)/2)/(1/4 - (i^2)*3/4).
      • z = (1/2+i*Sqr(3)/2)/(1/4 +3/4).
      • z = (1/2-i*Sqr(3)/2).
    • 2. What are conjugate complex numbers.
      • Sum of complex numbers = real.
      • Product of complex numbers = real.
    • 3. Solve x^2 - i = 0 by construction
      • Draw large unit circle
      • Draw two point A and B on circle
      • Angle between A and B is 360/2 = 180
      • Let angle AOX = 90/2 = 45
      • Let angle BOX = 180 + 45 = 225
      • Hence 1st root = a + p*i = +0.7071 + 0.7071*i
      • Hence 2nd root = b + q*i = -0.7071 - 0.7071*i
    • 4. Solve x^2 + i = 0 by construction
      • Draw large unit circle
      • Draw two point A and B on circle
      • Angle between A and B is 360/2 = 180
      • Let angle AOX = 270/2 = 135
      • Let angle BOX = 180 + 135 = 315
      • Hence 1st root = a + p*i = -0.7071 + 0.7071*i
      • Hence 2nd root = b + q*i = +0.7071 - 0.7071*i
    Go to Begin

    AL 18 16. Reference


    Go to Begin

    AL 18 17. Formula

    1. Factors
    • x^2 - 1 = (x - 1)*(x + 1)
    • x^4 - 1 = (x - 1)*(x + 1)*(x^2 + 1) = x^3 + x^2 + x + 1
    • x^3 + 1 = (x + 1)*(x^2 - x + 1)
    • x^5 + 1 = (x + 1)*(x^4 - x^3 + x^2 - x + 1)
    • x^7 + 1 = (x + 1)*(x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)
    • x^3 - 1 = (x - 1)*(x^2 + x + 1)
    • x^5 - 1 = (x - 1)*(x^4 + x^3 + x^2 + x + 1)
    • x^7 - 1 = (x - 1)*(x^6 ) x^5 + x^4 + x^3 + x^2 + x + 1)

    Go to Begin

    AL 18 00. Summary and formula

    1. Imaginary number and complex number
    • Imaginary number i = Sqr(-1)
      • i^2 = -1
      • i^3 = -i
      • i^4 = +1
      • i^5 = +i
    • Complex Number
      • Rectangular form z = x + i*y.
    • Solve x^n - 1 = 0
      • n is positive integer
      • The solutions can be used for x^(n-1) + x^(n-2) + .... + x + 1 = 0
    • Solve x^n + 1 = 0
      • n is positive integer
      • The solutions can be used for x^(n-1) - x^(n-2) + .... - x + 1 = 0 if n is even
    2. Solve x^n - 1 = 0 by construction
    • Draw large unit circle
    • Draw n points A, B, C, .... on circle
    • Angles between points are Ang = 360/n
    • 1st angle AOX = 0/n
    • 2nd angle BOX = 1*Ang + 1st angle
    • ....
    3. Solve x^n + 1 = 0
    • Same as above but 1st angle = 180/n
    3. Solve x^n + i = 0
    • Same as above but 1st angle = 90/n

    Go to Begin
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