1. Complex conjugates
- If z=x+yi, then its conjugate is x-yi, denoted by z*.
- z and z* are reflections along the x-axis.
- \[ \left |z|\right =\sqrt{x^{2}+y^{2}}\], where z is called the modulus of z. (You can think of z as a vector in the argand diagram. Do you still remember the modulus of a vector?)
- \[ zz^{*}=z^{2} \]
The seven properties of the complex conjugates are very important. Please make use that you learn them by heart.
2. Division of complex numbers
- to calculate z1/z2, multiply it by z2*/z2* ( similar to how you rationalize the denominator when it is surds)
- While you are doing the computation, always remind yourself that \[i^{2}=-1\]
3. Polynomial equations with real coefficients.
- For a polynomial equations with real coefficients, if z is a root, then z* is also a root.
If you want to know more about this theorem, you can refer to http://en.wikipedia.org/wiki/Complex_conjugate_root_theorem
By now you should be able to do the whole tutorial 10a.
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