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

chilledz3non
May 29, 2014

#1**+5 **

If we have a general point (x, y) in Cartesian coordinates, the polar form is (R, Θ) where R = √(x^{2} + y^{2}) and

Θ = tan^{-1}(y/x).

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

Your point is z = -3 + 4i so, in polar form it is z = Re^{iΘ} 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^{iΘ}, we can also write z = re^{i(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: z^{1/4} = r^{1/4}e^{i(θ+2pi*k)/4}. So z^{1/4} = 5^{1/4}e^{i(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^{iθ}, where r = 5^{1/4} and θ = tan^{-1}(4/-3) + 2pi*k.

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

Alan
May 29, 2014

#1**+5 **

Best Answer

If we have a general point (x, y) in Cartesian coordinates, the polar form is (R, Θ) where R = √(x^{2} + y^{2}) and

Θ = tan^{-1}(y/x).

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

Your point is z = -3 + 4i so, in polar form it is z = Re^{iΘ} 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^{iΘ}, we can also write z = re^{i(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: z^{1/4} = r^{1/4}e^{i(θ+2pi*k)/4}. So z^{1/4} = 5^{1/4}e^{i(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^{iθ}, where r = 5^{1/4} and θ = tan^{-1}(4/-3) + 2pi*k.

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

Alan
May 29, 2014