Picture of a plane with the 3 axis of rotation

If you have solved quadratic equations you have met complex numbers. For example if you solve the equation

x^{2} - 4 x + 13 = 0

you get the solutions

x = 2 \pm \sqrt{-} 9 = 2 \pm 3 i

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?