Final Exam

On this part, use chapter one (Incidence Geometry). The rational numbers are \(\mathbb{Q} = \{p/q : p,q \in \mathbb{Z}, q \neq 0 \}\), and two rational numbers \(n/m\) and \(p/q\) are equal if \(pm=nq\).

1. (10 points) Interpret point to be any ordered pair \((x,y)\) of rational numbers. Interpret line as the collection of points which satisfy the equation \(ax+by+c = 0\), where \(a,b\) and \(c\) are rational numbers and \(a\) and \(b\) are not both 0. Interpret incident as \((x_0,y_0)\) lies on a line if that point satisfies the line's equation. Prove or disprove that this is a model for incidence geometry.

On this part, use chapters two (Neutral Geometry) and three (Neutral Geometry Theorems) from the notes.

1. (5 points) Prove that any model of the neutral geometry has infinitely many points.
2. (5 points) Prove that any model of the neutral geometry has infinitely many lines.
3. (5 points) Is it true that any segment in any model of the neutral geometry has infinitely many points? What if the segment has at least two points?

On this part, use chapter four (Complex Numbers) from the notes.

1. (4 points) Let \(\theta\) be a real number. Show that \(\displaystyle \frac{1}{\cos \theta + i \sin \theta} = \cos \theta - i \sin \theta\).
2. (5 points) Write \(i^{2020}\) in \(a+bi\) form.
3. (7 points) Find a Mobius transformation which scales the complex plane by a factor of 3 about the point \(1+i\).
4. (7 points) Find a Mobius transformation which rotations the complex plane by \(\pi/3\) about the point \(1+i\).
5. Let \(\mathbb{Z}[i] = \{n+mi : n, m \in \mathbb{Z}\}\) and \(\mathbb{Q}[i] = \{p+qi : p, q \in \mathbb{Q}\}\). Determine which of the following form a transformation group (on \(\mathbb{C}\)). If one of the following is a transformation group, prove it. If one of the following is not, explain why.
   1. (2 points) \(\{z \mapsto z + b : b \in \mathbb{Z}[i]\}\)
   2. (2 points) \(\{z \mapsto az + b : a,b \in \mathbb{Z}[i]\}\)
   3. (2 points) \(\{z \mapsto az + b : a,b \in \mathbb{Q}[i]\}\)
   4. (2 points) \(\{z \mapsto z + 2b : b \in \mathbb{Z}[i]\}\)
   5. (2 points) \(\{z \mapsto z + 2(b+1) : b \in \mathbb{Z}[i]\}\)
6. (5 points) Let \(\triangle_1\) be the triangle with vertices \(-1+0i, 0+i, 1+0i\) and \(\triangle_2\) be the triangle with vertices \(-3+0i, 0+3i,3+0i\). For which of the following is \(\triangle_1 \cong \triangle_2\)? Give an explanation in each case.
   1. \((\mathbb{C},\mathcal{R})\)
   2. \((\mathbb{C},\mathcal{T})\)
   3. \((\mathbb{C},\mathcal{GL})\)
   4. \((\mathbb{C},E_s)\)
   5. \((\mathbb{C},E)\)
7. The distance between two points \(z, w \in \mathbb{C}\) is given by \(|z-w|\).
   1. (2 points) Verify that complex distance corresponds to the usual Euclidean distance if the points are viewed as being in \(\mathbb{R}^2\) rather than points in \(\mathbb{C}\).
   2. (5 points) Show that special Euclidean transformations preserve Euclidean distances. Recall that \(E_s = \{z \mapsto e^{i\theta}z + b : \theta \in \mathbb{R}, b \in \mathbb{C}\}\).
8. (8 points) Give a Mobius transformation which sends \(1 \mapsto 3, i \mapsto 0\), and \(2 \mapsto -1\).
9. For any complex number \(z\), we define \[\sin z = \frac{e^{iz}-e^{-iz}}{2i}, \cos z = \frac{e^{iz} + e^{-iz}}{2}.\] Prove the following analogs of regular trigonometry:
   1. (3 points) \(\cos(z + 2\pi) = \cos z\)
   2. (4 points) \(\cos^2 z + \sin^2 z =1\)
   3. (5 points) \(\cos(z_1 \pm z_2) = \cos z_1 \cos z_2 \mp \sin z_1 \sin z_2\)
10. (10 points) Invent a non-trivial geometry (in the sense of Definition4.48) of your own. Choose a set and transformations on your set. Prove that your transformations form a transformation group. Give an example of a figure in your geometry.