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Math Help - [SOLVED] Complex Numbers

  1. #1
    Eliberto
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    Question [SOLVED] Complex Numbers

    Express the complex numbers 1+i and 1-i in the form r(cos x+ i sin x), where r is the modulus and x the argument of the complex number. Find the modulus and argument of (1+i)^5/(1-i)^7.

    Eli
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  2. #2
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    Hints:

    Polar representation: Length * exp( Theta * i )

    Euler's Formula: exp( Theta * i ) = cos(Theta) + sin(Theta)*i
    Attached Thumbnails Attached Thumbnails [SOLVED] Complex Numbers-realimag1.png  
    Last edited by paultwang; April 9th 2005 at 01:13 PM.
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  3. #3
    Junior Member theprof's Avatar
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    Quote Originally Posted by Eliberto
    Express the complex numbers 1+i and 1-i in the form r(cos x+ i sin x), where r is the modulus and x the argument of the complex number. Find the modulus and argument of (1+i)^5/(1-i)^7.

    Eli
    look at this
    http://theprof.altervista.org/complex0.png

    bye
    Attached Thumbnails Attached Thumbnails [SOLVED] Complex Numbers-work.gif  
    Last edited by MathGuru; April 10th 2005 at 05:43 PM.
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  4. #4
    MHF Contributor
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    Your question has two parts:
    a) the expressing of 1+i into r(cos x +i sin x), etc.
    b) the modulus and argument of [(1+i)^5]/[(1-i)^7]

    ----------------

    Part a).

    The usual way is by (a +bi).

    1 +i
    Here a=1 and b=1 also.

    r = sqrt(a^2 +b^2) = sqrt(1^2 +1^2) = sqrt(2) ....***
    x = arctan(b/a) = arctan(1/1) = arctan(1) = 45degrees = pi/4 radians ....***

    But I want to show another way.

    1+i = r(cos x +i sin x)
    1 +i = r(cosX +i*sinX)
    1 +i = r*cosX +r*i*sinX

    The real parts are
    1 and r*cosX
    So,
    1 = r*cosX
    r = 1/cosX ....(1)

    The imaginary parts are
    i and r*i*sinX
    So,
    i = r*i*sinX
    Divide both sides by i,
    1 = r*sinX
    r = 1/sinX ....(2)

    r from (1) equals r from (2),
    1/cosX = 1/sinX
    Multiply both sides by sinX,
    sinX/cosX = 1
    tanX = 1
    x = arctan(1) = 45 degrees = pi/4 ....***

    So,
    r = 1/cosX ....(1)
    r = 1/cos(pi/4) = 1/(1/sqrt(2)) = sqrt(2) ....***

    Therefore,
    1+i = r(cosX +i*sinX)
    1+i = sqrt(2)*[cos(pi/4) +i*sin(pi/4)] .....answer.

    Following the same process, you will find that
    1-i = sqrt(2)*[cos(pi/4) -i*sin(pi/4)] .....answer.

    ----------------------

    Part b).

    DeMoivre's theorem says:
    z^n = (r^n)*[cos(n*X) +i*sin(n*X)] ----***
    where
    z is any complex number r(cosX +i*sinX)
    n is any positive integer

    So,
    (1+i)^5 = [(sqrt(2))^5]*[cos(5*pi/4) +i*sin(5*pi/4)] ....(3)
    (1-i)^7 = [(sqrt(2))^7]*[cos(7*pi/4) -1*sin(7*pi/4)] ....(4)

    (3)/(4) = {[(sqrt(2))^5]*[cos(5pi/4) +i*sin(5pi/4)]} / {[(sqrt(2))^7]*[cos(7pi/4) -i*sin(7pi/4)]}

    (3)/(4) = {cos(5pi/4) +i*sin(5pi/4)} / {[(sqrt(2))^2]*[cos(7pi/4) -i*sin(7pi/4)]}

    Angle (5pi/4) is in the 3rd quadrant, where cosine is negative, and sine is negative also.
    Angle (7pi/4) is in the 4th quadrant, where cosine is positive, while sine is negative.

    (3)/(4) = {-1/sqrt(2) +i*(-1/sqrt(2))} / {2*[1/sqrt(2) -i(-1/sqrt(2)]}
    (3)/(4) = {(-1/sqrt(2) *(1+i)} / {2* 1/sqrt(2) *(1+i)]
    (3)/(4) = -1/2

    Whoaa....

    So, (1+i)^5 / (1-i)^7 = -1/2.

    It's time to use the complex plane, or an Argand diagram.
    In the complex plane, -1/2 is a point on the real axis. This point is on the negative or left side of the real axis. It is 1/2 unit to the left of the origin (0,0). It is 180 degrees, or pi radians, from the positive rightside of the real axis. Hence,
    >>>the real part is -1/2
    >>>the imaginary part is zero
    >>>the point is at X=pi.

    Therefore, the modulus is 1/2, and the argument is pi radians. ....answer.
    Or,
    (1+i)^5 / (1-i)^7 = (1/2)*[cos(pi) +i*sin(pi)].
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  5. #5
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    ok

    Quote Originally Posted by Eliberto View Post
    Express the complex numbers 1+i and 1-i in the form r(cos x+ i sin x), where r is the modulus and x the argument of the complex number. Find the modulus and argument of (1+i)^5/(1-i)^7.

    Eli
    Great post. I'm just not completely sure that I'm understanding the depth to this. Can anybody explain this out a little bit more?
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