+0  
 
0
1017
2
avatar+564 

Find the four  fourth roots of -3 + 4i and express the roots in polar coordinates. 

chilledz3non  May 29, 2014

Best Answer 

 #1
avatar+27229 
+5

If we have a general point (x, y) in Cartesian coordinates, the polar form is (R, Θ) where R = √(x2 + y2) and

Θ = tan-1(y/x). 

And if we are in the complex plane, the point z = x + iy is z = Re in polar coordinates.

Your point is z = -3 + 4i  so, in polar form it is z = Re where R = 5 and Θ = tan-1(4/-3) (note that this is an angle in the 2nd quadrant, because the x-value is negative while the y-value is positive). Because you can add any multiple of 2pi (360°) to Θ without changing the value of e, we can also write z = rei(Θ+2pi*k) where k is an integer.

To find the fourth root of a complex number in polar form we simply take the fourth root of r and divide the angle by 4.  That is: z1/4 = r1/4ei(θ+2pi*k)/4.  So  z1/4 = 51/4ei(tan-1(4/-3)+2pi*k). 

We can let k = 1, 2, 3 and 4 to get the four roots.

We can write this as z1/4 = re,  where r = 51/4 and θ = tan-1(4/-3) + 2pi*k.

The result of doing this is summarised in the image below:

fourthroots

Alan  May 29, 2014
 #1
avatar+27229 
+5
Best Answer

If we have a general point (x, y) in Cartesian coordinates, the polar form is (R, Θ) where R = √(x2 + y2) and

Θ = tan-1(y/x). 

And if we are in the complex plane, the point z = x + iy is z = Re in polar coordinates.

Your point is z = -3 + 4i  so, in polar form it is z = Re where R = 5 and Θ = tan-1(4/-3) (note that this is an angle in the 2nd quadrant, because the x-value is negative while the y-value is positive). Because you can add any multiple of 2pi (360°) to Θ without changing the value of e, we can also write z = rei(Θ+2pi*k) where k is an integer.

To find the fourth root of a complex number in polar form we simply take the fourth root of r and divide the angle by 4.  That is: z1/4 = r1/4ei(θ+2pi*k)/4.  So  z1/4 = 51/4ei(tan-1(4/-3)+2pi*k). 

We can let k = 1, 2, 3 and 4 to get the four roots.

We can write this as z1/4 = re,  where r = 51/4 and θ = tan-1(4/-3) + 2pi*k.

The result of doing this is summarised in the image below:

fourthroots

Alan  May 29, 2014
 #2
avatar+564 
0

Thank you Alan!

chilledz3non  May 29, 2014

17 Online Users

avatar
avatar
avatar

New Privacy Policy

We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive information about your use of our website.
For more information: our cookie policy and privacy policy.