Question

how do you find the argument, given the modulus is 2???

Answer 1

complex numbers?

Answer 2

Yes the question allows you to determine that they are the cube roots of -8, so the modulus is 2. I'm not sure how to find the argument given this information

Answer 3

ok we can do this but of course you cannot simply ask "what it the arguement if the modulus is r" because it could be literally anything, since there are an infinite number of complex numbers whose modulus is r, namely any on the circle of radius 2 with center at the origin

Answer 4

you want the three cube roots of -8 right? in other words you are trying to solve
\[z^2=-8\] for z

Answer 5

right of course, my prof just said in the solution "using simple geometry you can determine that the argument is 60"

Answer 6

we can do this several ways, first lets do it the algebra way
\[z^3=-8\]
\[z^3+8=0\]
\[(z+2)(z^2-2z+4)=0\] and the first answer we get is \(z=-2\) and the other two we get from using the quadratic formula for \(z^2-2z+4=0\)

Answer 7

now we do it the demoivre way

Answer 8

you already know one real root by common sense, the one real root is -2

Answer 9

right ok makes sense! i did the quadratic formula and got 2+/- sq root -12 all over 2

Answer 10

that is supposed to be a picture of -2 in the complex plane. since you seem to be working in degrees, you divide the circle up in to 3 equal parts, and therefore you angles are 60 degrees, 180 degrees, and 300 degrees

Answer 11

right! ok that makes sense.. so that's what he means by the argument is 60?

Answer 12

your solutions are \(z=2(\cos(60)+i\sin(60))=1+\sqrt{3}i\)
\[z=2(\cos(180)+i\sin(180))=-2\]
\[z=2(\cos(300)+i\sin(300))=1-\sqrt{3}i\]

Answer 13

he is asking for the first cube root of -8

Answer 14

-8 in the complex plane is on the real axis like this

Answer 15

then angle for -8 is 180, i.e. \(-8=8(\cos(180)+i\sin(180))\) clearly since it is on the axis.
now since you are asked for the cubed root, divide the angle by 3 to get 60

Answer 16

so the "arguement" for the first cubed root is 60 and we get the answers above

Answer 17

ok! thank you! I was wondering if you could help me with another part of the same question... we were asked to find the eigenvalues which were 1+/- sqroot3i and the matrix was 4 -3
4 -2 can you help me find the eigenvectors?

Answer 18

that's a 2x2 matrix by the way

Answer 19

???