Well done Christopher Tynan, St. Bees School, for this very nice
  solution and also Andrei Lazanu, School No. 205, Bucharest,
  Romania and Jesse Rasowsky of Bethlehem Central High School and
  Matthew Hartley.<br />
  <br />
  Below the complex numbers $3-i$ and $2-10i$ are plotted, as well
  as their corresponding image when multiplied by -1. <img alt=
  "the complex numbers are plotted" src=
  "/content/03/07/15plus1/sol_img_01.gif" /> <br />
  <br />
  Our conjecture is that multiplying a point on the Argand diagram
  by -1 is equivalent to a 180 degree rotation about the origin.
  <br />
  <br />
  Let $(a,b)$ on the Argand diagram represent the complex number
  $a+ib$, where $i$ is the square root of -1. So, multiplying
  $a+ib$ by -1, we get $-a-ib$, which is represented by the point
  $(-a,-b)$ in the Argand diagram. However, the point $(-a,-b)$ is
  a rotation of the point $(a,b)$ by 180 degrees about the origin.
  QED. <br />
  <br />
  Using the same numbers for the second part, our conjecture here
  is that multiplying a complex number by $i$ gives a rotation of
  90 degrees anti-clockwise on the Argand diagram. <br />
  <img alt="Graph Plot" src=
  "/content/03/07/15plus1/sol_img_02.gif" /> <br />
  Again, assuming $(a,b)$ is a complex number plotted on the Argand
  diagram, by multiplying $a+ib$ by $i$, we get the complex number
  $-b+ai$, which is represented by the point $(-b,a)$ on the Argand
  diagram. This is equivalent to a rotation of 90 degrees
  anti-clockwise about the origin. QED. <br />
  <br />
  Assuming we were to multiply the number by $i$ twice, this would
  give a rotation of 180 degrees (as shown in the first part
  because $i^2=-1$). Multiplying by $i$ thrice would be equivalent
  to a rotation of 270 degrees anti-clockwise, or 90 degrees
  clockwise and four time maps the number onto itself (essentially
  multiplying by 1). Multiplying by $i$ $n$ times is equivalent to
  rotating the point 90 degrees anti-clockwise $n$ times.
  Therefore; if $n$ is 0 mod 4, it has no effect; if $n$ is 1 mod
  4, it's equivalent to an anti-clockwise rotation of 90 degrees;
  if $n$ is 2 mod 4, it's equivalent to a rotation of 180 degrees;
  if $n$ is 3 mod 4, it's equivalent to a rotation of 90 degrees
  clockwise.<br />
  <br />
  NB. All these rotations are rotations of the initial point (a,b)
  about the origin as the centre of rotation.<br />