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Mathematics Dictionary
Dr. K. G. Shih
DeMovire's Theorem


  • Q01 | - DeMovire's Theorem : Rule 1
  • Q02 | - Expand : (1 + i)^2
  • Q03 | - Expand : (1 + i)^3
  • Q04 | - Expand : (1 + i)^5
  • Q05 | - References

    • Q01. DeMovire's Theorem

    Rule 1 Definition
    • Expression : x + i*y = r*(cos(A) + i*sin(A))
      • r = Sqr(x^2 + y^2)
      • A = arctan(y/x)
    • z = x + i*y is rectangular form
    • z = r*(cos(A) + i*sin(A)) is polor form
    • cis(A) = cos(A) + i*sin(A)
    • Rule 1
      • (cos(A) + i*sin(A))^n = cos(n*A) + i*sin(n*A)
    Example : Express z = 1 + i in polar form
    • r = Sqr(1^2 + 1^2) = Sqr(2)
    • A = arctan(1/1) = arctan(1) = 45 degrees
    • Hence Z = Sqr(2)*(cos(45) + i*sin(45) = Sqr(2)*cis(45)

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    Q02. Application : Expand (1 + i)^2

    Method 1 : Use binomial expansion
    • (1 + i)^2
    • = 1 + 2*1*i + i^2
    • = 1 + 2*i + (-1)
    • = 2*i
    Method 2 : Use DeMovire's theory 1
    • (1 + i) = Sqr(2)*(cos(45) + i*sin(45))
    • (1 + i)^2
      • = (Sqr(2)^2)*(cos(45) + i*sin(45))^2
      • = 2*(cos(2*45) + i*sin(2*45))
      • = 2*(cos(90) + i*sin(90))
      • = 2*(0 + i)
      • = 2*i
    Express (1 + i)^2 on complex coordiante
    • It is on y-axis with x unit = 0 and y unit = 2

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    Q03. Application : Expand (1 + i)^3

    Method 1 : Use binomial expansion
    • (1 + i)^3
    • = 1 + 3*(1^2)*i + 3(1)*(i^2) + i^3
    • = 1 + 3*i + 3*(1)*(-1) - i
    • = -2 + 2*i
    Method 2 : Use DeMovire's theory 1
    • (1 + i) = Sqr(2)*(cos(45) + i*sin(45))
    • (1 + i)^3
      • = (Sqr(2)^3)*(cos(45) + i*sin(45))^3
      • = 2*Sqr(2)*(cos(3*45) + i*sin(3*45))
      • = 2*Sqr(2)*(cos(135) + i*sin(135))
      • = 2*Sqr(2)*(-Sqr(2)/2 + i*sqr(2)/2)
      • = -2 + 2*i
    Express (1 + i)^3 on complex coordiante
    • It is in 2nd quadrant
    • It has x unit = -2 and y unit = 2

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    Q04. Application : Expand (1 + i)^5

    Method 1 : Use binomial expansion
    • (1 + i)^5
    • = 1 + 5*(1^4)*i + 10*(1^3)*(i^2) + 10*(1^2)*(i^3) + 5*(1)*(i^4) + i^5
    • = 1 + 5*i + 10*(1)*(-1) + 10*(1)*(-i) + 5*1*(1) + i
    • = 1 + 5*i -10 - 10*i + 5 + i
    • = -4 - 4*i
    Method 2 : Use DeMovire's theory 1
    • (1 + i) = Sqr(2)*(cos(45) + i*sin(45))
    • (1 + i)^5
      • = (Sqr(2)^5)*(cos(45) + i*sin(45))^5
      • = 4*Sqr(2)*(cos(5*45) + i*sin(5*45))
      • = 4*Sqr(2)*(cos(225) + i*sin(225))
      • = 4*Sqr(2)*(-Sqr(2)/2 - i*sqr(2)/2)
      • = -4 - 4*i
    Express (1 + i)^5 on complex coordiante
    • It is in 3rd quadrant
    • It has x unit = -4 and y unit = -4

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    Q5. References

    References :

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