Complex numbers question: (1 Viewer)

[ashokkumar]

a) Find the least positive integer k such that cos(4pi/9) + isin(4pi/9) is a solution of z^k =1

b) Show that if the complex number w is a solution of z^n =1, then so is w^m, where m and n are arbitrary integers.

 

[Trebla]

a) Find the least positive integer k such that cos(4pi/9) + isin(4pi/9) is a solution of z^k =1

b) Show that if the complex number w is a solution of z^n =1, then so is w^m, where m and n are arbitrary integers.

(cos 4π/9 + i sin 4π/9)^k = (cos 4πk/9 + i sin 4πk/9) by DeMoivre's Theorem
For this to equal 1, we would like 4πk/9 = 4π
(for positive integer k and to find the least value of k, note that the choices 0 and 2π are valid but do not yield positive integers of k)
=> k = 9

b) zⁿ = 1
Exponentiate both sides to power of some integer m
(zⁿ)^m = 1^m
=> z^mn = 1 (note the solution set has not changed)
Since z = w satisfies this equation then
w^mn = 1
=> (w^m)ⁿ = 1
This means that z = w^m satisifies zⁿ = 1