Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
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6 CHAPTER 1. THE COMPLEX NUMBERS<br />
Proposition 1.1 is <strong>of</strong>ten worded as:<br />
R “is” a subfield <strong>of</strong> C.<br />
Set 1 := (1,0) and i := (0,1). Then, for any z = (x,y) ∈ C, we have<br />
We write<br />
and<br />
z = (x,0)+(0,y) = (1,0)(x,0)+(0,1)(y,0) = x+iy.<br />
Rez := x = “the real part <strong>of</strong> z”<br />
Imz := y = “the imaginary part <strong>of</strong> z”.<br />
The complex number i is called the imaginary unit and satisfies<br />
i 2 = (0,1) 2 = (−1,0) = −1.<br />
Unlike R, the set C = {(x,y) : x ∈ R,y ∈ R} is not ordered; there is no notion <strong>of</strong><br />
positive and negative (greater than or less than) on the complex plane. For example,<br />
if i were positive or zero, then i 2 = −1 would have to be positive or zero. If i were<br />
negative, then −i would be positive, which would imply that (−i) 2 = i 2 = −1 is<br />
positive. It is thus not possible to divide the complex numbers into <strong>of</strong> negative, zero,<br />
and positive numbers.<br />
The frequently appearing notation √ −1 for i is misleading and should be avoided,<br />
becausetherule √ xy = √ x √ y (whichonemightanticipate)doesnotholdfornegative<br />
x and y, as the following contradiction illustrates:<br />
1 = √ 1 = � (−1)(−1) = √ −1 √ −1 = i 2 = −1.<br />
Furthermore, bydefinition √ x ≥ 0, butonecannotwritei ≥ 0, sinceCisnotordered.<br />
Definition. For z = x+iy ∈ C, its complex conjugate is defined as ¯z = x−iy.<br />
Proposition 1.2. For z,w ∈ C, the following hold true:<br />
(i) Rez = 1<br />
1<br />
(z + ¯z) and Imz = (z − ¯z);<br />
2 2i<br />
(ii) z +w = ¯z + ¯w;<br />
(iii) zw = ¯z¯w;<br />
(iv) z −1 = ¯z −1 if z �= 0.