3.4.E: Problems on Complex Numbers (Exercises)
Complete the proof of Theorem 1 (associativity, distributivity, etc.).
Verify that the "real points" in \(C\) form an ordered field.
Prove that \(z \overline{z}=|z|^{2} .\) Deduce that \(z^{-1}=\overline{z} /|z|^{2}\) if \(z \neq 0 .^{4}\)
Prove that
\[
\overline{z+z^{\prime}}=\overline{z}+\overline{z^{\prime}} \text { and } \overline{z z^{\prime}}=\overline{z} \cdot \overline{z^{\prime}}
\]
Hence show by induction that
\(\overline{z^{n}}=(\overline{z})^{n}, n=1,2, \ldots,\) and \(\quad \overline{\sum_{k=1}^{n} a_{k} z^{k}}=\sum_{k=1}^{n} \overline{a}_{k} \overline{z}^{k}\)
Define
\[
e^{\theta i}=\cos \theta+i \sin \theta.
\]
Describe \(e^{\theta i}\) geometrically. Is \(\left|e^{\theta i}\right|=1 ?\)
Compute
(a) \(\frac{1+2 i}{3-i}\);
(b) \((1+2 i)(3-i) ;\) and
(c) \(\frac{x+1+i}{x+1-i}, x \in E^{1}\).
Do it in two ways: (i) using definitions only and the notation \((x, y)\) for \(x+y i ;\) and \((\text { ii) using all laws valid in a field. }\)
Solve the equation \((2,-1)(x, y)=(3,2)\) for \(x\) and \(y\) in \(E^{1}\).
Let
\[
\begin{aligned} z &=r(\cos \theta+i \sin \theta) \\ z^{\prime} &=r^{\prime}\left(\cos \theta^{\prime}+i \sin \theta^{\prime}\right), \text { and } \\ z^{\prime \prime} &=r^{\prime \prime}\left(\cos \theta^{\prime \prime}+i \sin \theta^{\prime \prime}\right) \end{aligned}
\]
as in Corollary \(2 .\) Prove that \(z=z^{\prime} z^{\prime \prime}\) if
\[
r=|z|=r^{\prime} r^{\prime \prime}, \text { i.e., }\left|z^{\prime} z^{\prime \prime}\right|=\left|z^{\prime}\right|\left|z^{\prime \prime}\right|, \text { and } \theta=\theta^{\prime}+\theta^{\prime \prime}.
\]
Discuss the following statement: To multiply \(z^{\prime}\) by \(z^{\prime \prime}\) means to rotate \(\overrightarrow{0 z^{\prime}}\) counterclockwise by the angle \(\theta^{\prime \prime}\) and to multiply it by the scalar \(r^{\prime \prime}=\) \(\left|z^{\prime \prime}\right| .\) Consider the cases \(z^{\prime \prime}=i\) and \(z^{\prime \prime}=-1\).
[Hint: Remove brackets in
\[
r(\cos \theta+i \sin \theta)=r^{\prime}\left(\cos \theta^{\prime}+i \sin \theta^{\prime}\right) \cdot r^{\prime \prime}\left(\cos \theta^{\prime \prime}+i \sin \theta^{\prime \prime}\right)
\]
and apply the laws of trigonometry.]
By induction, extend Problem 7 to products of \(n\) complex numbers, and derive de Moivre's formula, namely, if \(z=r(\cos \theta+i \sin \theta),\) then
\[
z^{n}=r^{n}(\cos (n \theta)+i \sin (n \theta)).
\]
Use it to find, for \(n=1,2, \ldots\)
\[
(a) i^{n}; \hskip 12pt (b) (1 + i)^{n}; \hskip 12pt (c) \frac{1}{(1 + i)^{n}}.
\]
From Problem \(8,\) prove that for every complex number \(z \neq 0,\) there are exactly \(n\) complex numbers \(w\) such that
\[
w^{n}=z;
\]
they are called the \(n\) th roots of \(z\)
[Hint: If
\[
z=r(\cos \theta+i \sin \theta) \text { and } w=r^{\prime}\left(\cos \theta^{\prime}+i \sin \theta^{\prime}\right),
\]
the equation \(w^{n}=z\) yields, by Problem 8
\[
\left(r^{\prime}\right)^{n}=r \text { and } n \theta^{\prime}=\theta,
\]
and conversely.
While this determines \(r^{\prime}\) uniquely, \(\theta\) may be replaced by \(\theta+2 k \pi\) without affecting \(z .\) Thus
\[
\theta^{\prime}=\frac{\theta+2 k \pi}{n}, \quad k=1,2, \ldots
\]
Distinct points \(w\) result only from \(k=0,1, \ldots, n-1\) (then they repeat cyclically).
Thus \(n\) values of \(w\) are obtained.]
Use Problem 9 to find in \(C\)
\[
\text { (a) all cube roots of } 1 ; \quad \text { (b) all fourth roots of } 1
\]
Describe all \(n\) th roots of 1 geometrically.