CP 9 Complex Numbers Flashcards
what is the denotation for complex numbers
ℂ
given z=a+bi what are the real and imaginary parts
a=real part (RE)
b=imaginary part(Im)
how would we add z=a+bi and x=c+di
z+x =
(a+c)+(b+d)i
- put the RE and the Im together
how would we subtract z=a+bi and x=c+di
z-x =
(a-c)+(b-d)i
how would we multiply z=a+bi and x=c+di
z·x =
(a+bi)·c + (a+bi)·di
- simplify from there
Remember: i² =
-1
given z=a+bi the complex conjugate(z‾) is
z=a+(-b)i
- the imaginary part gets opposite sign
what is magnitude |z|
|z|= √a²+b²
if z=3+4i what is the complex conjugate(z‾) and the magnitude(|z|)
z= 3+4i
z‾ = 3+(-4)i
|z|= 5
what are the axis’ of the complex plane
real axis = horizontal
Im axis = vertical
TF real #s are a subset of complex numbers
T
what does |z| represent on the complex plane
The distance between 0 and z
TF z·z‾ = |z|²
T
how do we divide complex numbers
ex) 3+4i/5-6i
we multiply both the be conjugate of the denominator
ex)
(3+4i)x(5+6i) / (5-6i)x(5+6i)
then simplify
what is a polynomial in terms of complex numbers
a real # mult. by a variable raised to the power of a natural #
P(x)=anxn
what formula do we use to find the complex roots of polynomials
quadratic forumla
x=-b±√(b²-4ac)/2a
TF when solving for the roots of a complex number we have to find for all solutions over ℂ
T
how do we find the coordinates on a graph given angle θ where 0<θ<2π
-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))
TF for every z∈ℂ, there exists r∈R with r≥0 and 0<θ<2π where
z=rcos(θ)+rsin(θ)
T
TF the point on the circle has coordinates (x,y) which equals (cos(θ),sin(θ))
T
