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I need to draw $z^4 +16 = 0$ on the complex numbers plane.

By solving $z^4 +16 = 0$ I get:

$z = 2 (-1)^{3/4}$ or $z = -2 (-1)^{3/4}$ or $z = -2 (-1)^{1/4}$ or $z = 2 (-1)^{1/4}$

However, the suggested solution by my teacher is:

enter image description here

Where you can see that the solutions that he found are:

$z = \sqrt{2} + i\sqrt{2}$ or $z = \sqrt{2} - i\sqrt{2}$ or $z = - \sqrt{2} + i\sqrt{2}$ or $z = - \sqrt{2} - i\sqrt{2}$

How did my teacher find these solutions?

Am I missing something here? Maybe I don't need to solve the equation in order to draw it on the plane of complex numbers?

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Use polar coordinates. – lhf Dec 16 '11 at 20:09
Tried but I do something obviously wrong. I get $r = 16$ and $θ = 0$. – Dimme Dec 16 '11 at 20:12
Your solutions are correct, but in an inconvenient form. Try substituting -1 = e^(i*pi). – JGWeissman Dec 16 '11 at 20:20
Note that writing things like $(-1)^{3/4}$ and $(-1)^{1/4}$ is not very useful, since there are four possible solutions for each; in fact, you are listing 16 complex numbers, each of which occurs four times in your list. – Arturo Magidin Dec 16 '11 at 20:22
up vote 2 down vote accepted

One easy solution is to use polar coordinates. Write $z=re^{i\theta}$. Then $r^4 e^{i4\theta}=z^4 =-16 = 16 e^{i\pi}$ implies $r=2$ and $4\theta \equiv \pi \bmod 2\pi$. This gives $\theta = \pi/4 + k \pi/2$, which is the picture you posted.

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How did you get that $z^4 = -16$ implies $r=2$ and $4θ=\pi \mod {2\pi}$? – Dimme Dec 16 '11 at 21:02
@Dimme, I've added a clarification. Recall that two complex numbers are equal iff they have the same modulus and the same arguments, but arguments are only defined mod $2\pi$. – lhf Dec 16 '11 at 21:21
Thanks got it now! – Dimme Dec 16 '11 at 21:25

The solutions that your teacher found are the $4$ solutions to $z^4=16$. Here we will find these roots algebraically.

As you point out, the solutions should be of the form $z=2\cdot (-1)^{1/4}$, where "$(-1)^{1/4}$" stands for any fourth root of $-1$. Thus, we need to find the solutions to $$z^4=-1,$$ or equivalently, the solutions to $z^4+1=0$. Over the complex numbers, $-1$ is a square, namely $i^2=-1$, so $$z^4+1 = z^4-(-1) = z^4-i^2$$ and therefore $z^4+1$ factors as $(z^2-i)(z^2+i)$. In order to factor this expression completely, you need to find the square roots of $i$ and the square roots of $-i$.

Let us find the square roots of $i$, i.e., we are looking for a complex number $a+bi$ such that $(a+bi)^2=i$. Since $(a+bi)^2=a^2-b^2+2abi$, we conclude that $a^2-b^2=0$ (so $a=\pm b$) and $1=2ab=\pm 2b^2$. Thus, $a=b=\pm \frac{\sqrt{2}}{2}$. In other words, the square roots of $i$ are $$z_1=\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i,\quad \text{and} \quad z_2=-\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}i.$$ You can similarly find that the square roots of $-i$ are given by $$z_3=\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}i,\quad \text{and} \quad z_4=-\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i.$$ Therefore, it follows from our previous discussion that the roots of $z^4+16=0$ are $2z_1$, $2z_2$, $2z_3$ and $2z_4$, which are the solutions your teacher found.

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Let's concentrate on the geometry. We look at $w^4+1=0$, and multiply by $2$ later.
Instead of looking at $w^4+1=0$, look at $$(w^4+1)(w^4-1)=0,\qquad\qquad(\ast)$$ that is, $$w^8=1.$$ The solutions of this are the vertices of a regular octagon inscribed in the unit circle. The "first" vertex is $1$, the second is $1$ rotated counterclockwise by $45$ degrees, and so on. Draw the circle. Put in, in pencil, the vertices of this octagon.

Every solution of $w^4+1=0$ is among these vertices. But also among these vertices are, by $(\ast)$, the solutions of $w^4-1=0$. These are the $4$ vertices of the square inscribed in the unit circle, with first vertex at $1$, the next at $1$ rotated $90$ degrees, and so on. Erase these $4$ vertices.

What's left is the $4$ solutions of $w^4+1=0$. The "first" one is $1$ rotated through $45$ degrees, that is, $\cos(\pi/4)+i\sin(\pi/4)$.

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