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

Printable View

- April 9th 2005, 12:48 PMEliberto[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 - April 9th 2005, 02:09 PMpaultwang
Hints:

Polar representation: Length * exp( Theta * i )

Euler's Formula: exp( Theta * i ) = cos(Theta) + sin(Theta)*i - April 9th 2005, 11:49 PMtheprofQuote:

Originally Posted by**Eliberto**

http://theprof.altervista.org/complex0.png

bye - April 11th 2005, 04:23 AMticbol
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)]. - July 31st 2008, 08:05 AMSMcCavanaughok