EU01S-OPG

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

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

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

What is Re $(-5-7i)$ and Im $(-5-7i)$ ?

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

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.

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

Draw the four numbers $\,z\,,\,iz\,,\,i^2z\,$ og $\,i^3z\,$ in the complex number plane.
What happens geometrically when a number is multiplied by $i\,$ ?
And divided by $i\,$ ?

Find by the use of elementary computations the rectangular form for the following complex numbers .

$(5+i)(1+9i)$
$i+i^2+i^3+i^4$
$\displaystyle{\frac{1}{1+3i}+\frac{1}{(1+3i)^2}}$
$\displaystyle{\frac{1}{(1+i)^4}}$
$\displaystyle{\frac{5+i}{2-2i}}$
$\displaystyle{\frac{3i}{4}}\,$ and $\displaystyle{\frac{i2}{4}}$

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

Why is the number $\,\,\displaystyle{\frac{1}{a+ib}}\,\,$ not on rectangular form?
Compute Re $\displaystyle{\left(\frac{1}{a+ib}\right)}\,$ and Im $\displaystyle{\left(\frac{1}{a+ib}\right)}\,\,$ .

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

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

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

the point $(0,0)$ ,
the real axis,
the imaginary axis,
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.

$-z_{0}\,$ ,
$\overline{z_0}\,$ ,
$-\overline{z_0}\,$ ,
$i\, \overline{z_0}$ .

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.

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

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

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 $\,M=\left{z\,|\,\,|z-1+2i|\leq 3\,\right}\,.$

Sketch $\,M\,.$

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

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

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

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

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

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


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%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:

Only one of the statements $\,a<b,$ $\,b<a$ or $\,a=b\,$ is true.
If $\,a<b\,$ and $\,b<c\,$ then $\,a<c\,.$
If $\,a<b\,$ then $\,a+c<b+c\,.$
If $\,a<b\,$ and $\,0<c\,$ then $\,ac<bc\,.$

Test the four statements by a few examples.

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