## Proposition 1.2.16: Finding Roots |

For any positive integer |

As is often the case in mathematics, the proof is easy if you happen to
know the correct answer by working backwords. In our case we start by
saying: let *a = r cis(t)* and define

Then, according to our theorem on muliplying geometrically we have:z =

for all *k = 0, 1, 2, ... n-1* because of the periodicity of
*cos*.

[ x ]

It is more enlightning to try to understand how this theorem works
geometrically. For simplicity take *a = cis(t)*, i.e.
*|a|=1*:

To find anThat gives you the first root.n-th root of a vector with anglet, divide that angle byn

Divide the unit circle intonequally spaced pieces, starting at the first anglet/n

That will be your *n*-roots total. As an example, let's find
the three third-roots of *i*, i.e. we want to find all solutions
to *z ^{3} = i*.

**Step 1: Draw the vector i**

**Step 2: Divide the angle by 3 for your first root**

**Step 3: Draw 3 equally spaced segments, starting at the first root**