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Complex numbers
a number with both a real and imaginary part, often displayed in the form
a+bi
Complex conjugate
the conjugate of a complex number. Switch the sign in between a and b to it’s inverse.
equal
complex numbers are ______ if the real parts are equal and the imaginary parts are equal
argand diagram
diagrams drawn to represent complex number on a graph. One axis is for imaginary numbers, one is for real numbers. You can graph an a+bi imaginary number using this.
modulus
represented by r or |z|, represents the size of a complex number
√a2+b2
argument
represents the angle of a complex number (z), represented by θ
tan-1(b/a) when z is in quadrant 1
tan-1(b/a)+π when z is in quadrant 2, 3
tan-1(b/a)+2π when z is in quadrant 4
polar form
what form of a complex number is this?
z=r(cosθ+i sinθ)
muliply, add
when you multiply complex numbers, you ________ the moduli and ______ their arguments
DeMoivre’s Theorem
given z=r cisθ, the nth power of z=
zn=rn cis nθ
dilate
multiplying a complex number, z, by a positive real number, k, will ______ the vector z by a factor of k in the plane
180
multiplying a complex number, z, by -1 is equivalent to rotating z by _____ degrees in the complex plane
90
multiplying a complex number, z, by i, (an imaginary number) is equivalent to rotating z by ___ degrees counterclockwise in the complex plane.
rotate
multiplying z by k cis θ will _______ z by θ counterclockwise and scale the modulus of z by k in the plane
roots of unity
xn=1 where the equation has exactly n solutions on the unit circle that are equidistant from each other
generic complex roots
solutions to xn=z where z is a complex number. There will be n solutions.
complex conjugate theorem
a theorem that states that if a+bi is a root of a polynomial then a-bi is also a root. Corresponding factors would be (x-(a+bi)) and (x-(a-bi))
fundamental theorem of algebra
a polynomial of degree n has n complex factors and corresponding roots.
Note
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