Let $n$ be a postive integer and let $$Z = \big|Z\big|\big(cos(\theta) + sin(\theta)i\big)$$ be a complex number where $0 \lt \theta \le 2\pi.$ This app displays the $n$-th roots of $Z$ which can be expressed as $$R_{k} = \big|Z\big|^{\frac{1}{n}}\big[\cos\big(\frac{\theta+2\pi k}{n}\big) + \sin\big( \frac{\theta+2\pi k}{n} \big)i\big]$$ where $k = 0, 1, ..., n-1$. The root with the smallest positive angle will be called the basic primitive root
of $Z$. From the formula above, it is the root $$R_{0} = \big|Z\big|^{\frac{1}{n}}\big[cos\big(\frac{\theta}{n}\big) + sin\big(\frac{\theta}{n}\big)i\big]$$ and is displayed as a magenta vector. The remaining $n-1$ roots are displayed as cyan vectors equally spaced around a yellow circle of radius $\big|Z\big|^{\frac{1}{n}}$ with an angle of $\frac{2\pi}{n}$ separating one root from the next.

Instructions

1. Enter $Z$ (drawn in green) by
   • Selecting 0.5, 1, or 2 from the |Z| menu.
   • Selecting 15, 30, 45, …, or 360 as its angle $\theta$ from the Z's Angle menu.
2. Enter the roots to be found by selecting Square Roots, Cube Roots, …, or Sixth Roots from the Roots menu.
3. Click the Start button.

An animation will begin that provides a visualization of the generation of all $n$ of $Z$'s roots beginning with $Z$'s basic primitive root $R_{0}$.

Details:

• You can change the angle mode (degrees or radians) and the animation rate by using the Angle Mode, the Angle Curves and the Animation Rate menus, respectively.

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