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'Twizzles Venture Forth' printed from http://nrich.maths.org/
Remember that i is the
name of the twizzle that has a number arrow equal to the unit arrow
rotated through 90 degrees.
In other words i = 1 cis
The twizzle multiplication rule is neatly expressed by saying
'Multiply the number arrow lengths, and add the angles'.
So $i^2 = i \times i= 1 cis 90 \times 1 cis 90 = 1 cis 180 = -1.$
Now you should have enough information to multiply out (z-i)(z+i) .