We're going to start with the same animation we looked at in

Twizzles Venture Forth . If you haven't done so yet, it would
be worth getting in some twizzle arithmetic practice using the

notes for that problem as a guide.

In this animation, you can change the value of the

blue twizzle

z .

The red twizzle takes the value

(z-i) .

The green twizzle takes the value

(z+i) .

The grey twizzle takes the value

(z-i)(z+i) .

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The equation (z-i)(z+i) =
0 has two solutions; z=i and z=-i .

Switch on 'Show Tracings' and draw a small loop around each of
these solutions with the blue z twizzle. Observe carefully what
happens to the paths traced by the red, green, and grey twizzles
and how the paths relate to each twizzle's zero spot.

Now click on 'Show Tracings' twice - once to clear the paths, and
once to switch tracing back on, and draw a small loop around the
other solution. Again, observe carefully the paths of the other
twizzles.

Click 'Show Tracings' twice again to clear the paths.

This time, draw a loop that goes around both solutions (so both the
solutions are inside your loop). Again, observe what paths the
other twizzles trace out. How many times to they turn around their
spots?

### Here's the question!

When you trace out a loop with the blue z twizzle, what determines how many
times the grey (z-i)(z+i)
twizzle winds around its zero spot?

### Generalise!

You might like to experiment some more with the next slightly
more general animation. You can change the value of the beige
a twizzle as well as the
value of the blue z
twizzle. Roll over the twizzles to see how they are
calculated.

Something special happens when the beige a twizzle is zero. What is it?:

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