Complex Numbers

z1 = r1(cos(θ) + i sin(θ)) and z2 = r2(cos(φ) + i sin(φ)).

What is the term for z1z2 ?

z1z2 = r1r2(cos(θ + φ) + i sin(θ + φ)).

That is, the argument of z1z2 is arg(z1z2) = θ + φ and the modulus is |z1z2| = r1r2.

What are the positive and negative power rules for complex numbers in polar form?

z^n = r^n (cos(nθ) + i sin(nθ))

z ^−n = (r ^ −n) (cos(nθ) − i sin(nθ))

Using the principle arguement what can you add a multiple of to the arguement which will not change the value of the polar form?

2k(pi)

Say we have z^n how many solutions would be have?

n solutions

To find the nth roots of a value in polar form where z= r^(1/n) (cos(%)+isin(%)) what does % equal to find the n distinct roots?

%= (x+2k(pi))/n

What is the “Shorthand” version of polar form?

r(e^iθ)

How would you apply de moivres theorem to z1 = r1(e^iθ) and z2 = r2(e^iφ),

z1z2 = (r1r2)(e^i(θ+φ))

Find the 4th roots of 6 + 6i in short hand terms.