Question

For a positive integer n, the complex number z=r(cosθ+isinθ) has exactly how many solutions for θ in the interval [0, 2π)?

a) n
b) 1
c) 2
d) Infinite

Answer 1

For a positive integer n, the complex number z=r(cosθ+isinθ) has exactly n solutions for θ within the interval [0, 2π), as distinct multiples of 2π still fall within one revolution of the circle.

Step-by-step explanation:

For a positive integer n, the complex number z=r(cosθ+isinθ) has exactly n solutions for θ in the interval [0, 2π). This is because in polar form, the argument θ of the complex number can take on multiple values that differ by integral multiples of 2π, creating what are known as the complex number's arguments. These are essentially the angles at which the complex number's vector from the origin makes with the positive real axis. Given that θ is restricted to the interval [0, 2π) and n is a positive integer, each multiple of θ that differs by 2π will still fall within one revolution of the circle, thus providing exactly n distinct values of θ.