Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
Chapter 1<br />
The <strong>Complex</strong> Numbers<br />
Definition. The complex numbers—denoted by C—are R 2 equipped with the operations<br />
for x,y,u,v ∈ R.<br />
(x,y)+(u,v) := (x+u,y +v),<br />
(x,y)(u,v) := (xu−yv,xv+yu)<br />
Theorem 1.1 (C is a Field). The complex numbers are a field. Specifically, we have:<br />
• (0,0) is the identity element <strong>of</strong> addition;<br />
• −(x,y) = (−x,−y) for x,y ∈ R;<br />
• (1,0) is the identity element <strong>of</strong> multiplication;<br />
• (x,y) −1 �<br />
= for x,y ∈ R with (x,y) �= (0,0).<br />
�<br />
x<br />
x2 −y<br />
+y2, x2 +y2 Pro<strong>of</strong> (<strong>of</strong> the last claim only). Let x,y ∈ R besuch that (x,y) �= (0,0), andnote that<br />
�<br />
x<br />
(x,y)<br />
x2 −y<br />
+y2, x2 +y2 � � 2 x<br />
=<br />
x2 −y2<br />
−<br />
+y2 x2 −xy<br />
+y2, x2 xy<br />
+<br />
+y2 x2 +y2 �<br />
� 2 2 x +y<br />
=<br />
x2 −xy +xy<br />
+y2, x2 +y2 �<br />
= (1,0).<br />
Proposition 1.1. The set {(x,0) : x ∈ R} is a subfield <strong>of</strong> C, and the map<br />
is an isomorphism onto its image.<br />
θ: R → C, x ↦→ (x,0)<br />
5