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(section:uge3S)=
# Extra (Optional) Exercises -- Long Day

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These exercises are optional extra material for students who are done with the exercises of the day but have the time for and the interest in more.

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## Extra Exercise 1:  Ordering the Complex Numbers

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

1. Only one of the claims $\,a<b,$ $\,b<a$ 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 $\,ac<bc\,.$

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### Question a

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Test these four claims on some examples.

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### Question b

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Show that the order relation $\,<\,$ from the real numbers *cannot* be extended to apply to the complex numbers. More precisely, show that no order relation $\,<\,$ exists on $\mathbb C$ that extends the order relation $\,<\,$ from the real numbers while satisfying the four above bullets for all $\,a,b\,$ og $\,c\,$ i $\mathbb C$.  

```{hint}
:class: dropdown
Attempt a proof by contradiction. That is, assume that the extension does exist and then try reaching a contradiction.
```

```{hint}
:class: dropdown
If the extension exists then either $0<i$ or $i<0$ holds due to bullet 1. Try reaching a contradiction in each case.
```

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## Extra Exercise 2: Yet another Equation with Modulus

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In Exercise 9 we showed that all complex numbers $z$ that are solutions to the equation $|z-1|=|2z-3|$ form a circle in the complex plane. We are now given complex numbers $z_0,z_1$ and $z_2$. Show that the solution set to the equation $|z-z_0|=|z_2\cdot z-z_1|$ in the complex plane forms either one single point, a straight line, a circle or fills the entire complex plane.