At this point it is worth taking some time out to get really familiar with twizzle arithmetic. Make your own drawings, or use the twizzle addition or twizzle multiplication animations to get some practice.

You should begin to see how it's quite easy to add without a picture if the twizzle is written like 2+3i or 3-4i .

When you multiply you'll find the cis notation easier - e.g. use 3cis(30) or 2cis(10) .

Check that you agree with each of these results:

\begin{eqnarray} 2 cis (30) \times 3 cis (60) = 6i \\ (1 cis (120))^3 = 1 \\ i^4 = 1 \\ (1 cis (72))^5 = 1 \\ (2 cis (60))^3 = -8 \end{eqnarray}

1cis(120) is a cube root of 1. Can you think of any others? How many are there?

i is a fourth root of 1. Can you think of any others? How many are there?

1cis(72) is a fifth root of 1. Can you think of any others? How many are there?

Are you beginning to see a pattern?

Tackle Twizzle Wind Up and Twizzle Twists and you might begin to see how general this is and why this pattern is appearing.