For those that remember their Group and Ring/Field Theory, the complex plane is actually a Field with the operations (+,*) defined as within the Complex Number page of this chapter.[br][br]Here are some propositions that demonstrate the properties of a Field within the context of complex numbers:[br] for all [img]https://latex.codecogs.com/gif.latex?%28x%2Cy%29%2C%28a%2Cb%29%2C%28c%2Cd%29%5Cepsilon%20%5Cmathbb%7BC%7D[/img][br]1) [img]https://latex.codecogs.com/gif.latex?%28x%2Cy%29+%28a%2Cb%29%3D%28x+a%2Cy+b%29[/img][br]2) [img]https://latex.codecogs.com/gif.latex?%28%28x%2Cy%29+%28a%2Cb%29%29+%28c%2Cd%29%3D%28x%2Cy%29+%28%28a%2Cb%29+%28c%2Cd%29%29[/img][br]3) [img]https://latex.codecogs.com/gif.latex?%28x%2Cy%29+%28a%2Cb%29%3D%28a%2Cb%29+%28x%2Cy%29[/img][br]4) [img]https://latex.codecogs.com/gif.latex?%28x%2Cy%29+%280%2C0%29%3D%28x%2Cy%29[/img][br]5) [img]https://latex.codecogs.com/gif.latex?%28x%2Cy%29+%28-x%2C-y%29%3D%280%2C0%29[/img][br]6) [img]https://latex.codecogs.com/gif.latex?%28x%2Cy%29*%28%28a%2Cb%29+%28c%2Cd%29%29%3D%28x%2Cy%29%28a%2Cb%29+%28x%2Cy%29%28c%2Cd%29[/img][br]7) [img]https://latex.codecogs.com/gif.latex?%28x%2Cy%29*%28a%2Cb%29%3D%28xa-yb%2Cxb+ya%29[/img][br]8) [img]https://latex.codecogs.com/gif.latex?%28x%2Cy%29%28%28a%2Cb%29%28c%2Cd%29%29%3D%28x%2Cy%29%28a%2Cb%29%28c%2Cd%29[/img][br]9) [img]https://latex.codecogs.com/gif.latex?%28x%2Cy%29%28a%2Cb%29%3D%28a%2Cb%29%28x%2Cy%29[/img][br]10) [img]https://latex.codecogs.com/gif.latex?%28x%2Cy%29%281%2C0%29%3D%28x%2Cy%29[/img][br]11) [img]https://latex.codecogs.com/gif.latex?%28x%2Cy%29%28x/%28x%5E2+y%5E2%29%2C-y/%28x%5E2+y%5E2%29%29%3D%281%2C0%29[/img][br][br]So we have an abelian group with respect to (+) with unit element (0,0) and another abelian group with respect to operation (*) with unit element (1,0)