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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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
Hints:
Polar representation: Length * exp( Theta * i )
Euler's Formula: exp( Theta * i ) = cos(Theta) + sin(Theta)*i
look at thisQuote:
Originally Posted by Eliberto
http://theprof.altervista.org/complex0.png
bye
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]
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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)].
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?Quote:
Originally Posted by Eliberto