The Real And Complex Number Systems

Proof: Write z = x + iy, then z − ¯z = i means that y = 1/2. So, the set
is the line y = 1/2.
(f) z + ¯z = |z| 2
Proof: Write z = x + iy, then 2x = x 2 + y 2 ⇔ (x − 1) 2 + y 2 = 1. So, the
set is the unit circle centered at (1, 0) .
1.31 Given three complex numbers z 1 , z 2 , z 3 such that |z 1 | = |z 2 | = |z 3 | =
1 and z 1 + z 2 + z 3 = 0. Show that these numbers are vertices of an equilateral
triangle inscribed in the unit circle with center at the origin.
Proof: It is clear that three numbers are vertices of triangle inscribed in
the unit circle with center at the origin. It remains to show that |z 1 − z 2 | =
|z 2 − z 3 | = |z 3 − z 1 | . In addition, it suffices to show that
Note that
which is equivalent to
which is equivalent to
|z 1 − z 2 | = |z 2 − z 3 | .
|2z 1 + z 3 | = |2z 3 + z 1 | by z 1 + z 2 + z 3 = 0
|2z 1 + z 3 | 2 = |2z 3 + z 1 | 2
(2z 1 + z 3 ) (2¯z 1 + ¯z 3 ) = (2z 3 + z 1 ) (2¯z 3 + ¯z 1 )
which is equivalent to
|z 1 | = |z 3 | .
1.32 If a and b are complex numbers, prove that:
(a) |a − b| 2 ≤ ( 1 + |a| 2) ( 1 + |b| 2)
Proof: Consider
(
1 + |a|
2 ) ( 1 + |b| 2) − |a − b| 2 = (1 + āa) ( 1 + ¯bb ) − (a − b) ( ā − ¯b )
= (1 + āb) ( 1 + a¯b )
= |1 + āb| 2 ≥ 0,
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