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 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mn>3</mn><mo>-</mo><mi>i</mi></mrow></math> and 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mn>2</mn><mo>-</mo><mn>10</mn><mi>i</mi></mrow></math> are
plotted, as well as their corresponding image
when multiplied by -1.

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<img src="mcprob1/sol_img_01.gif" alt=
"the complex numbers are plotted" /> <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 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></math> on the Argand diagram represent the
complex number 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>a</mi><mo>+</mo><mi>ib</mi></mrow></math>, where 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>i</mi></mrow></math> is the square
root of -1. So, multiplying 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>a</mi><mo>+</mo><mi>ib</mi></mrow></math> by -1, we get

<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mo>-</mo><mi>a</mi><mo>-</mo><mi>ib</mi></mrow></math>, which is represented by the point

<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mo stretchy="false">(</mo><mo>-</mo><mi>a</mi><mo>,</mo><mo>-</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></math> in the Argand diagram. However, the
point 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mo stretchy="false">(</mo><mo>-</mo><mi>a</mi><mo>,</mo><mo>-</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></math> is a rotation of the point

<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></math> 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 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>i</mi></mrow></math> gives a rotation of 90 degrees
anti-clockwise on the Argand diagram.

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<br />
<img src="mcprob1/sol_img_02.gif" alt="Graph Plot" /> <br />





 
Again, assuming 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></math> is a complex number
plotted on the Argand diagram, by multiplying

<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>a</mi><mo>+</mo><mi>ib</mi></mrow></math> by 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>i</mi></mrow></math>, we get the complex number 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mo>-</mo><mi>b</mi><mo>+</mo><mi>ai</mi></mrow></math>,
which is represented by the point 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mo stretchy="false">(</mo><mo>-</mo><mi>b</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></math> 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 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>i</mi></mrow></math>
twice, this would give a rotation of 180 degrees
(as shown in the first part because 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<msup><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow>
</msup>
<mo>=</mo><mo>-</mo><mn>1</mn></mrow></math>).
Multiplying by 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>i</mi></mrow></math> 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 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>i</mi></mrow></math> 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>n</mi></mrow></math> times is equivalent to
rotating the point 90 degrees anti-clockwise 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>n</mi></mrow></math>
times. Therefore; if 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>n</mi></mrow></math> is 0 mod 4, it has no
effect; if 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>n</mi></mrow></math> is 1 mod 4, it's equivalent to an
anti-clockwise rotation of 90 degrees; if 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>n</mi></mrow></math> is
2 mod 4, it's equivalent to a rotation of 180
degrees; if 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>n</mi></mrow></math> 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.

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