CP 9 Complex Numbers

Question 1

what is the denotation for complex numbers

Answer 1

ℂ

Question 2

given z=a+bi what are the real and imaginary parts

Answer 2

a=real part (RE)
b=imaginary part(Im)

Question 3

how would we add z=a+bi and x=c+di

Answer 3

z+x =
(a+c)+(b+d)i
- put the RE and the Im together

Question 4

how would we subtract z=a+bi and x=c+di

Answer 4

z-x =
(a-c)+(b-d)i

Question 5

how would we multiply z=a+bi and x=c+di

Answer 5

z·x =
(a+bi)·c + (a+bi)·di
- simplify from there

Question 6

Remember: i² =

Answer 6

-1

Question 7

given z=a+bi the complex conjugate(z‾) is

Answer 7

z=a+(-b)i
- the imaginary part gets opposite sign

Question 8

what is magnitude |z|

Answer 8

|z|= √a²+b²

Question 9

if z=3+4i what is the complex conjugate(z‾) and the magnitude(|z|)

Answer 9

z= 3+4i
z‾ = 3+(-4)i
|z|= 5

Question 10

what are the axis’ of the complex plane

Answer 10

real axis = horizontal
Im axis = vertical

Question 11

TF real #s are a subset of complex numbers

Answer 11

T

Question 12

what does |z| represent on the complex plane

Answer 12

The distance between 0 and z

Question 13

TF z·z‾ = |z|²

Answer 13

T

Question 14

how do we divide complex numbers
ex) 3+4i/5-6i

Answer 14

we multiply both the be conjugate of the denominator
ex)
(3+4i)x(5+6i) / (5-6i)x(5+6i)
then simplify

Question 15

what is a polynomial in terms of complex numbers

Answer 15

a real # mult. by a variable raised to the power of a natural #
P(x)=anxn

Question 16

what formula do we use to find the complex roots of polynomials

Answer 16

quadratic forumla
x=-b±√(b²-4ac)/2a

Question 17

TF when solving for the roots of a complex number we have to find for all solutions over ℂ

Answer 17

T

Question 18

how do we find the coordinates on a graph given angle θ where 0<θ<2π

Answer 18

-start at origin facing X-axis (the +ve axis)
- rotate counterclockwise by θ
- walk 1 distance in the direction we’re facing(unit)

then the point on the circle has coordinates (x,y) which equals (cos(θ),sin(θ))

Question 19

TF for every z∈ℂ, there exists r∈R with r≥0 and 0<θ<2π where
z=rcos(θ)+rsin(θ)

Answer 19

T

Question 20

TF the point on the circle has coordinates (x,y) which equals (cos(θ),sin(θ))

Answer 20

T