+0

# Finding the fourth root?

0
562
2
+564

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

chilledz3non  May 29, 2014

#1
+26399
+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:

Alan  May 29, 2014
Sort:

#1
+26399
+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:

Alan  May 29, 2014
#2
+564
0

Thank you Alan!

chilledz3non  May 29, 2014

### 7 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details