# Plotting complex solutions in Argand plane as arrows

I have:

Clear[z]
Solve[z^4 == 1 - Sqrt I, z] // ComplexExpand


Which produces these solutions:

{{z -> -(1/(2 2^(1/4))) - Sqrt/(2 2^(1/4)) +
I (-(1/(2 2^(1/4))) + Sqrt/(2 2^(1/4)))}, {z ->
1/(2 2^(1/4)) - Sqrt/(2 2^(1/4)) +
I (-(1/(2 2^(1/4))) - Sqrt/(2 2^(1/4)))}, {z -> -(1/(
2 2^(1/4))) + Sqrt/(2 2^(1/4)) +
I (1/(2 2^(1/4)) + Sqrt/(2 2^(1/4)))}, {z ->
1/(2 2^(1/4)) + Sqrt/(2 2^(1/4)) +
I (1/(2 2^(1/4)) - Sqrt/(2 2^(1/4)))}}


What would be the easiest way to convert these to a list of arrows to plot in the Argand plane. For example, the cube roots of -1:

Graphics[{
Circle[],
Blue, Thick,
Arrow[{{0, 0}, {-1, 0}}],
Arrow[{{0, 0}, {1/2, Sqrt/2}}],
Arrow[{{0, 0}, {1/2, -Sqrt/2}}]
}, Axes -> True, ImageSize -> Small]


Produces this image: But I am looking for a cute way to convert all the data in the first problem, $z^4=1-\sqrt3 i$, into a list of arrows.

sol = z /. Solve[z^4 == 1 - Sqrt I, z] // ComplexExpand coords = N @ ReIm @ sol (* N not needed, I just wanted a compact output *)


{{-1.14869, 0.307789}, {-0.307789, -1.14869}, {0.307789, 1.14869}, {1.14869, -0.307789}}

arr = Arrow[{{0, 0}, #}] & /@ coords


{Arrow[{{0, 0}, {-1.14869, 0.307789}}], Arrow[{{0, 0}, {-0.307789, -1.14869}}], Arrow[{{0, 0}, {0.307789, 1.14869}}], Arrow[{{0, 0}, {1.14869, -0.307789}}]}

For the circle:

{r = Abs[1 - Sqrt I]^(1/4), N@r}


{2^(1/4), 1.18921}

Graphics[{Circle[{0, 0}, r], Blue, Thick, arr}, Axes -> True,
AxesLabel -> {"Re z", "Im z"}] • This is a really well explained explanation. I'd like to thank you for all the intermediate steps. They greatly helped my understanding. – David Nov 14 '16 at 19:11
• I'm glad I could help :) – corey979 Nov 14 '16 at 21:30
• I just made a simplifying edit; if you do not like this revert the change or edit as you see fit. IMHO multiple baroque alternatives do not enhance this otherwise concise answer. – Mr.Wizard Nov 20 '16 at 19:02

Really just for fun to illustrate the geometric effect: $z^n=c$

f[c_, n_] :=
Table[ReIm@N[Abs[c]^(1/n) Exp[Arg[c] I/n + 2 Pi I j/n]], {j, 0, n - 1}]
Manipulate[
Module[{roots = f[Complex @@ p, n]},
Graphics[{Circle[], Red, Circle[{0, 0}, Norm@p], Arrow[{{0, 0}, p}],
Blue, Arrow[{{0, 0}, ##}] & /@ roots, Pink, Opacity[0.5],
Disk[{0, 0}, 0.5, {0, Arg[Complex @@ p]}],
LightBlue,
Disk[{0, 0}, Norm@p, {0, ArcTan @@ roots[]}]},
PlotLabel ->
Style[StringForm["\!$$\*SuperscriptBox[\(z$$, $$1$$]\)=2", n,
Complex @@ p], 20],
Axes -> True,
PlotRange -> Table[{-2, 2}, 2]
]],
{{p, {1, 0}}, {-2, -2}, {2, 2}}, {n, Range[2, 7]}]


Selected examples:   • Really nice example. I'm gonna have to study this one. Thanks ever so much for all of your responses. – David Nov 15 '16 at 6:58