\\\\( \nonumber \newcommand{\bevisslut}{$\blacksquare$} \newenvironment{matr}[1]{\hspace{-.8mm}\begin{bmatrix}\hspace{-1mm}\begin{array}{#1}}{\end{array}\hspace{-1mm}\end{bmatrix}\hspace{-.8mm}} \newcommand{\transp}{\hspace{-.6mm}^{\top}} \newcommand{\maengde}[2]{\left\lbrace \hspace{-1mm} \begin{array}{c|c} #1 & #2 \end{array} \hspace{-1mm} \right\rbrace} \newenvironment{eqnalign}[1]{\begin{equation}\begin{array}{#1}}{\end{array}\end{equation}} \newcommand{\eqnl}{} \newcommand{\matind}[3]{{_\mathrm{#1}\mathbf{#2}_\mathrm{#3}}} \newcommand{\vekind}[2]{{_\mathrm{#1}\mathbf{#2}}} \newcommand{\jac}[2]{{\mathrm{Jacobi}_\mathbf{#1} (#2)}} \newcommand{\diver}[2]{{\mathrm{div}\mathbf{#1} (#2)}} \newcommand{\rot}[1]{{\mathbf{rot}\mathbf{(#1)}}} \newcommand{\am}{\mathrm{am}} \newcommand{\gm}{\mathrm{gm}} \newcommand{\E}{\mathrm{E}} \newcommand{\Span}{\mathrm{span}} \newcommand{\mU}{\mathbf{U}} \newcommand{\mA}{\mathbf{A}} \newcommand{\mB}{\mathbf{B}} \newcommand{\mC}{\mathbf{C}} \newcommand{\mD}{\mathbf{D}} \newcommand{\mE}{\mathbf{E}} \newcommand{\mF}{\mathbf{F}} \newcommand{\mK}{\mathbf{K}} \newcommand{\mI}{\mathbf{I}} \newcommand{\mM}{\mathbf{M}} \newcommand{\mN}{\mathbf{N}} \newcommand{\mQ}{\mathbf{Q}} \newcommand{\mT}{\mathbf{T}} \newcommand{\mV}{\mathbf{V}} \newcommand{\mW}{\mathbf{W}} \newcommand{\mX}{\mathbf{X}} \newcommand{\ma}{\mathbf{a}} \newcommand{\mb}{\mathbf{b}} \newcommand{\mc}{\mathbf{c}} \newcommand{\md}{\mathbf{d}} \newcommand{\me}{\mathbf{e}} \newcommand{\mn}{\mathbf{n}} \newcommand{\mr}{\mathbf{r}} \newcommand{\mv}{\mathbf{v}} \newcommand{\mw}{\mathbf{w}} \newcommand{\mx}{\mathbf{x}} \newcommand{\mxb}{\mathbf{x_{bet}}} \newcommand{\my}{\mathbf{y}} \newcommand{\mz}{\mathbf{z}} \newcommand{\reel}{\mathbb{R}} \newcommand{\mL}{\bm{\Lambda}} \newcommand{\mnul}{\mathbf{0}} \newcommand{\trap}[1]{\mathrm{trap}(#1)} \newcommand{\Det}{\operatorname{Det}} \newcommand{\adj}{\operatorname{adj}} \newcommand{\Ar}{\operatorname{Areal}} \newcommand{\Vol}{\operatorname{Vol}} \newcommand{\Rum}{\operatorname{Rum}} \newcommand{\diag}{\operatorname{\bf{diag}}} \newcommand{\bidiag}{\operatorname{\bf{bidiag}}} \newcommand{\spanVec}[1]{\mathrm{span}{#1}} \newcommand{\Div}{\operatorname{Div}} \newcommand{\Rot}{\operatorname{\mathbf{Rot}}} \newcommand{\Jac}{\operatorname{Jacobi}} \newcommand{\Tan}{\operatorname{Tan}} \newcommand{\Ort}{\operatorname{Ort}} \newcommand{\Flux}{\operatorname{Flux}} \newcommand{\Cmass}{\operatorname{Cm}} \newcommand{\Imom}{\operatorname{Im}} \newcommand{\Pmom}{\operatorname{Pm}} \newcommand{\IS}{\operatorname{I}} \newcommand{\IIS}{\operatorname{II}} \newcommand{\IIIS}{\operatorname{III}} \newcommand{\Le}{\operatorname{L}} \newcommand{\app}{\operatorname{app}} \newcommand{\M}{\operatorname{M}} \newcommand{\re}{\mathrm{Re}} \newcommand{\im}{\mathrm{Im}} \newcommand{\compl}{\mathbb{C}} \newcommand{\e}{\mathrm{e}} \\\\)

The following exercises are designed to generate deep learning of the topics and syllabus of the day. They typically lead with procedural exercises settling your skills within standard methods and gradually become more challenging training your depth and logical thinking. It is recommended that you work with the exercises with your study group and solve them manually (i.e. on paper, and later we will introduce a mathematical software) by giving them your best shot before peeking at hints or answers. Store your solutions for future reference and test rehearsals. Enjoy.

Exercise 1: The Number i

In this exercise you will get some introductory experience with the simple complex number $\,i\,.$

A

What is $i^2$, $i^3$, $i^4$, $i^5$, $(-i)^2$, $(-i)^3$, $(-i)^4$ and $(-i)^{-5}\,$?

B

What is the real part and the imaginary part of $-5-i7\,$?

C

What is $\mathrm{Re}(-5-7i)$ and $\mathrm{Im}(-5-7i)$?

D

Write the complex numbers $\,7i-5\,$, $\,i(7i-5)\,$ and $\,i(7i-5)i\,$ on rectangular form.

Exercise 2: The Complex Number Plane

A

Consider the following ten numbers: $-2,\,0,\,i,\,2-i,\,1+2i,\,1,\,-2+3i,\,-5i,\,3\,$ and $\,-1-2i\,.$ Which are complex, which are real, and which are purely imaginary? Draw the ten numbers in the complex number plane.

B

Given the number $z=4+i\,$.

  1. Draw the four numbers $\,z\,,\,iz\,,\,i^2z\,$ og $\,i^3z\,$ in the complex number plane.

  2. What happens geometrically when a number is multiplied by $i\,$?

  3. And divided by $i\,$?

Exercise 3: Basic Computations

A

Find by hand the rectangular form of the following complex numbers.

  1. $(5+i)(1+9i)$

  2. $i+i^2+i^3+i^4$

  3. $\displaystyle{\frac{1}{1+3i}+\frac{1}{(1+3i)^2}}$

  4. $\displaystyle{\frac{1}{(1+i)^4}}$

  5. $\displaystyle{\frac{5+i}{2-2i}}$

  6. $\displaystyle{\frac{3i}{4}}\,$ and $\displaystyle{\frac{i2}{4}}$

B

Given two real numbers $a$ and $b\,$.

  1. Why is the number $\,\,\displaystyle{\frac{1}{a+ib}}\,\,$ not on rectangular form?

  2. Compute $\mathrm{Re}\displaystyle{\left(\frac{1}{a+ib}\right)}\,$ and $\mathrm{Im}\displaystyle{\left(\frac{1}{a+ib}\right)}\,\,$.

Exercise 4: Complex Conjugation

A

Show that $\,\overline{\overline z}=z\,$ and that $\,\overline{z_1\cdot z_2}=\overline{z_1}\cdot\overline{z_2}\,.$

B

Let $z_0=a+ib\neq 0$ be a given complex number. Which complex numbers correspond to the mirror images of $z_0$ about

  1. the point $(0,0)$,

  2. the real axis,

  3. the imaginary axis,

  4. and the angle bisector in the first quadrant?

Write the result first in terms of $a$, $b$ and $i$ and then in terms of $z_0$, $\overline{z_0}$ and $i$. Draw all of them in one figure.

Exercise 5: Absolute value

By the absolute value $\,\left|z\right|\,$ of a complex number $\,z\,$ we are referring to the length of $z$’s position vector in the complex number plane.

A

Given a complex number on rectangular form $\,z=a+ib\,.$ Determine $\,\left|z\right|\,.$

B

Investigate the geometrical meaning of $\,\left|z_1-z_2\right|\,$ for two arbitrary complex numbers $\,z_1\,$ and $\,z_2$. Illustrate with examples.

C

A set of points in the complex number plane is given by

$$\big\{z \in {\Bbb C}\,\,\big| \,\,|z-1|\, = \, 3\big\}\,.$$

Give a geometric description of the set of points.

Exercise 6: The Real Criterion

In the complex number plane we consider the set $\,M=\left\{z\,\,\big|\,\,|z-1+2i|\leq 3\,\right\}\,.$

A

Sketch $\,M\,.$

B

Determine the subset of $\,M\,$ that is real.

Exercise 7: The Magnitude of Rational Numbers (Advanced)

In this exercise we are considering the two rational numbers $\,a=\frac{41}{42}\,$ and $\,b=\frac{98}{99}\,$.

A

Which of the two numbers $a$ and $b$ is largest?

B

Present three rational numbers that lie between $a$ and $b$.

C

How many rational numbers are there between $a$ and $b$?

Exercise 8: Ordering of Complex Numbers (Advanced)

For the real numbers we have the well-known less than order relation $\,<\,$ that for all $\,a,b\,$ and $\,c\,$ fulfills the following:

  1. Only one of the statements $\,a<b,$ $\,b<a$ and $\,a=b\,$ is true.

  2. If $\,a<b\,$ and $\,b<c\,$ then $\,a<c\,.$

  3. If $\,a<b\,$ then $\,a+c<b+c\,.$

  4. If $\,a<b\,$ and $\,0<c\,$ then $\,ac<bc\,.$

A

Test the four statements with a few examples.

B

Show that the order relation $\,<\,$ from the real numbers cannot be extended to apply to all complex numbers.