IB Analysis Unit 6: Complex Numbers

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.