Complex Numbers

Question 1

(x1+iy1)+(x2+iy2)=

Answer 1

(x1+x2)+i(y1+y2)

Question 2

(x1+iy2)-(x2+iy2)=

Answer 2

(x1-x2)+i(y1-y2)

Question 3

(x1+iy1)(x2+iy2)=

Answer 3

(x1x2-y1y2)+i(x1y2+x2y1)

Question 4

z=x+iy
ż=

Answer 4

x-iy

Question 5

żz=|z|^2=

Answer 5

x^2 + y^2

Question 6

z1/z2=

Answer 6

(z1xż2)/(z2xż2)

Question 7

|z|=r=

Answer 7

sqrt(x^2 + y^2)
(Modulus/Absolute Value)

Question 8

When x>0 find arg(p)

Answer 8

tan(p)=y/x

Question 9

When x<0 find arg(p)

Answer 9

tan^-1(y/x) + pi = p

Question 10

Trigonometric/Polar form
z=

Answer 10

r(cosp + isinp)

Question 11

De moivre formula

Answer 11

(cosp+isinp)^n = cos(np)+isin(np)

Question 12

Take z=cosp+isinp and w=cosa+isina
zw=
arg(zw)=

Answer 12

cos(p+a)+isin(p+a)
arg(z)+arg(w)

Question 13

Take z=cosp+isinp and W=R(cosa+isina)
zW=

Answer 13

R(cos(p+a)+isin(p+a))

Question 14

arg(z^n)=

Answer 14

narg(z)

Question 15

Z=R1(cosp+isinp) and W=R2(cosa+isina)
ZW=
|ZW|= |Z||W|=
arg(ZW)=

Answer 15

ZW=R1R2(cos(p+a)+isin(p+a))
=R1R2
arg(Z)+arg(W)

Question 16

|Z^n|=|Z|^n
arg(Z^n)=

Answer 16

n x arg(Z)

Question 17

on the unit circle only
z^-1 =

Answer 17

ż

Question 18

z=r(cosp+isinp)
z^n=

Answer 18

r^n (cosnp+isinnp)

Question 19

z^-1=

Answer 19

(1/r^2)(ż)

Question 20

Z=R1(cosp+isinp) and W=R2(cosa+isina)
Z/W=

Answer 20

(R1/R2)(cosp-a+isinp-a)