## Example 1.2.17 (b): Finding roots geometrically |

Find all 4 fourth-roots of |

With *-1 = 1 cis()* the formula gives us four roots:

z_{1}= cis(/4)z_{2}= cis(( + 2)/4) = cis(3/4)z_{3}= cis(( + 4)/4) = cis(5/4)z_{4}= cis(( + 6)/4) = cis(7/4)

Note that the next root in that pattern equals *z _{1}*
since

cis(/4) = cis(9/4)

But let's to do this geometrically, as before. We have *-1 = cis()*.

**Step 1: Draw the vector -1**

a = -1

**Step 2: Divide the angle by 4 (lengh = 1)**

z_{1}= cis(/4)

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

z_{1}= cis(/4)z_{2}= cis(3/4)z_{3}= cis(5/4)z_{3}= cis(7/4)