Complex Numbers

Question 1

(z + w)*

Answer 1

z* + w*

Question 2

(zw)*

Answer 2

zw

Question 3

|z*|

Answer 3

|z|

Question 4

z**

Answer 4

z

Question 5

Triangle Inequality

Answer 5

||z| - |w|| <= |z + w| <= |z| + |w|

Question 6

If 1 root is known, how to find others

Answer 6

1. If p(z0) = 0 then p(z0*) is also a root.
2. If p(z0) = 0 then p(z) = (z - z0)q(z)
3. An integer root needs to divide a0

Question 7

If z1 = r1(cosØ1 + isinØ1) and z2 = r2(cosØ2 + isinØ2) what is z1z2?

Answer 7

cos(Ø1 + Ø2) + isin(Ø1 + Ø2)

Question 8

(r(cosØ + isinØ))^n, n in Z

Answer 8

r^n(cos(nØ) + isin(nØ))

Question 9

e^(iØ)

Answer 9

cosØ + isinØ

Question 10

(e^(iØ))^n

Answer 10

e^(inØ)