How can i use the roots function to show that all the roots of a complex number lie on a circle.

The question gives the following information: Given z=1+3i , what is z^1/4 where arg(z) is in the range: pi/2 < arg(z) < pi I am looking how i would go about using the roots function to solve the problem and how i would find the other roots to generate a figure which shows that all the roots lie on a circle in the complex plane. My code so far without using the roots function:z = 1+3i; Im = imag(z); Re = real(z); Mag = abs(z); theta = atan(Im/Re); %more data of powers n = 1/4; k = 1:4; arg = (theta*n + (2*k*pi)*n) x = cosd(arg); y = 1i*sind(arg); r = Mag.^(n); zn = r.*(x) + r.*(y); zn = zn' Any advice would be greatly appreciated.

Answer 1

[John D'Errico]

Well, you did make an effort. Something we see too little of from students lately. :)

You want to find the 4 roots of a complex number.

z = 1+3i;

Just use roots, on the polynomial

x^4 - z == 0

Clearly, any solutions to that equation must be roots of z. That takes no more than

format long g
r = roots([1 0 0 0 z])
r =
        -1.18702520404887 +     0.607659917216594i
       -0.607659917216594 -      1.18702520404886i
        0.607659917216592 +      1.18702520404886i
         1.18702520404886 -     0.607659917216592i

We can view those numbers in polar form. The radial component would be:

abs(r)
ans =
        1.33352143216333
        1.33352143216332
        1.33352143216332
        1.33352143216332

That is consistent with this prediction:

nthroot(abs(z),4)
ans =
          1.33352143216332

Are they on a circle? Of course.

plot(real(r),imag(r),'bo')
axis equal
grid on
r1 = abs(r(1));
t = linspace(0,2*pi,100);
hold on
plot(r1*cos(t),r1*sin(t),'r-')

Note that axis equal was important there, otherwise the plot would look elliptical.

If you want to convince yourself the points are equally spaced around the circle, try this:

(angle(r) - min(angle(r)))/pi
ans =
                     1.5
                       0
                       1
                     0.5

So the differences between each point in angle is a nice simple multiple of pi.

One caveat to remember: this exposition using roots shows only that the roots of THIS PARTICULAR complex number lie on a circle. While we know it to be true in general, this is not a proof of the general case.

[Todd Trainor]

Thank you soo much that has helped massively!!, One last thing, how would i go about plotting lines from the origin of the circle to each of the 4 plots, essentially plotting a large cross in the circle? Thanks again in advance :)

[John D'Errico]

As long as hold is still on,

plot([zeros(1,4);real(r).'],[zeros(1,4);imag(r).'],'g-')

[Todd Trainor]

Brilliant! Thank you very much sir. Just to be clear are these roots for z=1+3j or z^1/4 ?

[John D'Errico]

Why would I have calculated the 4th root of z^(1/4)??? I think you are getting confused. Start over.

Surely z^(1/4) is the 4th root of z, which is what you were looking for?

I calculated the solutions to the problem x^4-z==0. If x is such a solution, then it MUST be one of the 4th roots of z, since

(z^(1/4))^4 = z

[free5721]

If trying to find the 4th roots of z, the expression "roots([1 0 0 0 z])" as shown in the example should be "roots([1 0 0 0 -z])"