Math 411: Honours Complex Variables - University of Alberta

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