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## 'Complex Rotations' printed from http://nrich.maths.org/

If you have solved quadratic equations you have met complex
numbers. For example if you solve the equation $$x^2-4x+13=0$$ you
get the solutions $x=2\pm\sqrt{-9}=2\pm3i$ where
$i=\sqrt{-1}$.

The complex number $a+ib$ is represented in the plane by the point
with coordinates $(a,b)$. This is called an Argand diagram. Make
your own choice of some complex numbers, and mark them on a graph
with lines joining the points to the origin. Now multiply your
numbers by $-1$ and join their images to the origin. Make and prove
a conjecture about the geometric effect of multiplying complex
numbers by $-1$.

Again make a choice of some complex numbers and multiply each one
by $i$. Draw the complex numbers and their images on a graph and
make and prove a conjecture about the effect of multiplying complex
numbers by $i$.

What happens if you multiply a complex number by $i$ twice, three
times, four times, ..., $n$ times?