Math 411: Honours Complex Variables - University of Alberta

Chapter 1
The Complex Numbers
Definition. The complex numbers—denoted by C—are R 2 equipped with the operations
for x,y,u,v ∈ R.
(x,y)+(u,v) := (x+u,y +v),
(x,y)(u,v) := (xu−yv,xv+yu)
Theorem 1.1 (C is a Field). The complex numbers are a field. Specifically, we have:
• (0,0) is the identity element of addition;
• −(x,y) = (−x,−y) for x,y ∈ R;
• (1,0) is the identity element of multiplication;
• (x,y) −1 �
= for x,y ∈ R with (x,y) �= (0,0).
�
x
x2 −y
+y2, x2 +y2 Proof (of the last claim only). Let x,y ∈ R besuch that (x,y) �= (0,0), andnote that
�
x
(x,y)
x2 −y
+y2, x2 +y2 � � 2 x
=
x2 −y2
−
+y2 x2 −xy
+y2, x2 xy
+
+y2 x2 +y2 �
� 2 2 x +y
=
x2 −xy +xy
+y2, x2 +y2 �
= (1,0).
Proposition 1.1. The set {(x,0) : x ∈ R} is a subfield of C, and the map
is an isomorphism onto its image.
θ: R → C, x ↦→ (x,0)
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