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Twizzle Wind Up

A loopy exploration of z^2+1=0 (z squared plus one) with an eye on winding numbers. Try not to get dizzy!

Twizzles Venture Forth

Age 14 to 16 Challenge Level:

In this animation, the 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) . You can check that by multiplying the red and green twizzles using Twizzle Arithmetic . You need to know that i is the name we give the twizzle which has a number arrow equal to the unit arrow but rotated through 90 degrees.

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There are lots of things to investigate with this animation, and lots of things to think about...

Which values of the blue twizzle make the grey twizzle equal to zero?

When you set the blue twizzle to one of these values, what happens to the red and green twizzles?

When the grey twizzle is zero, (z-i)(z+i)=0 . This should suggest to you the two values of z for which this is true. Multiply out the expression (z-i)(z+i) and so write the equation in a simpler form.

Explain why you can't solve this equation using ordinary numbers.

If you get stuck, look at the hints. Before you progress to Twizzle Wind Up, look at the notes.