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Complex Rotations

Age 16 to 18 Challenge Level:

Picture of a plane with the three axes of rotation
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?