Problem Set 2 - MATH 4030 Matematical Analysis September...

MATH 4030 Matematical Analysis
September 18, 2017
Problem Set 2
1. Prove the distributive property for complex numbers.
2. Show that every nonzero complex number can be written
in one and only one way as the product of a positive real
number and a number of absolute value one.
3. Suppose that z = a + bi , w = u + vi and
a =
⎛
⎜
⎝
| w | + u
2
⎞
⎟
⎠
1 / 2
,
b =
⎛
⎜
⎝
| w | − u
2
⎞
⎟
⎠
1 / 2
.
Prove that z
2
= w if v ≥ 0 and that ( ¯ z)
2
= w if v ≤ 0.
Conclude that every complez number (with one excep-
tion!) has two complex square roots.