The Real And Complex Number Systems

So, z 1 z 2 = ¯z 1¯z 2
(c) z¯z = |z| 2
Proof: Write z = x + iy and thus
z¯z = x 2 + y 2 = |z| 2 .
(d) z + ¯z =twice the real part of z
Proof: Write z = x + iy, then
z + ¯z = 2x,
twice the real part of z.
(e) (z − ¯z) /i =twice the imaginary part of z
Proof: Write z = x + iy, then
z − ¯z
i
= 2y,
twice the imaginary part of z.
1.30 Describe geometrically the set of complex numbers z which satisfies
each of the following conditions:
(a) |z| = 1
Solution: The unit circle centered at zero.
(b) |z| < 1
Solution: The open unit disk centered at zero.
(c) |z| ≤ 1
Solution: The closed unit disk centered at zero.
(d) z + ¯z = 1
Solution: Write z = x + iy, then z + ¯z = 1 means that x = 1/2. So, the
set is the line x = 1/2.
(e) z − ¯z = i
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