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In this exercise you will get some introductory experiences with the 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 Re$(-5-7i)$ and Im$(-5-7i)$?

D

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

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 $i\,$?

A

Find by the use of elementary computations the rectangular form for 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 Re$\displaystyle{\left(\frac{1}{a+ib}\right)}\,$ and Im$\displaystyle{\left(\frac{1}{a+ib}\right)}\,\,$.

A

Show that $\,\overline{\overline{z}}=z\,,$ and

$\,\overline{z_1\cdot z_2}=\overline{z_1}\cdot\overline{z_2}\,.$

B

Let $z_0=a+ib\neq 0$ be a given complex number. Which complex number corresponds to the mirror image of $z_0$ in

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

  2. the real axis,

  3. the imaginary axis,

  4. the angle bisector in the first quadrant?

Write the result partly by using $a$, $b$ and $i$ and partly by using $z_0$, $\overline{z_0}$ and $i$. Draw all of it in one figure.

B

  1. $-z_{0}\,$,

  2. $\overline{z_0}\,$,

  3. $-\overline{z_0}\,$,

  4. $i\, \overline{z_0}$.

By the absolute value $\,\left|z\right|\,$ of a complex number $z$ we understand the length of the position vector for $z$ in the complex number plane.

A

Given a complex number in 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 by 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.

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

A

Sketch $\,M\,.$

B

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

In this exercise we consider two rational numbers $a=\frac{41}{42}$ and $b=\frac{98}{99}$

A

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

B

Find 3 rational numbers that lie between $a$ and $b$.

C

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

%### Et reelt tal som ikke er rationalt (advanced)\label{U1LD7} 


%####### begin:question
%Vis at $\,\sqrt{5}\notin \mathbb Q\,.$\\\\
%Vink: Skriv $\,\sqrt{5}\,$ som en brøk $\,\frac pq,\,\,q \neq 0$. Tæller og nævner kan, som alle hele tal, på entydig måde (bortset fra evt. rækkefølge) skrives som et produkt af primtal (aritmetikkens fundamentalsætning).
%####### end:question

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

  1. Only one of the statements $\,a<b,$ $\,b<a$ or $\,a=b\,$ 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 by a few examples.

B

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