Geometry :: Notes :: Complex Numbers

1 Complex Numbers

1.1 Basic Definitions

The imaginary unit is \(i\) and satisfies \(i^2 = -1\).

A complex number is a number \(z=(x,y)\) in the cartesian plane. The horizonal axis measures the real part of \(z\) (\(\Re(z) = x\)), and the vertical axis measures the imaginary part of \(z\) (\(\Im(z) = y\)).

The cartesian form of a complex number \(z = (x,y)\) is \(z=x+yi\).

Addition and subtraction of complex numbers \(z=a+bi\) and \(w=c+di\) is defined by

\begin{equation*} z \pm w = (a + c) \pm (b+d)i. \end{equation*}

Multiplication of complex numbers \(z=a+bi\) and \(w=c+di\) is defined by

\begin{equation*} zw = (ac-bd) + (ad+bc)i. \end{equation*}

The absolute value or modulus of a complex number \(z=a+bi\) is

\begin{equation*} |z| = \sqrt{a^2 + b^2}. \end{equation*}

The distance between \(z=a+bi\) and \(w=c+di\) is given by

\begin{equation*} |z-w| = \sqrt{(a-c)^2 + (b-d)^2}. \end{equation*}

The conjugate of \(z=a+bi\) id defined to be \(\overline{z} = a-bi\).

The quotient of \(z=a+bi\) and \(0 \neq w=c+di\) is

\begin{equation*} \frac{z}{w} = \frac{a+bi}{c+di} = \frac{ac+bd}{c^2+d^2} + \frac{bc-ad}{c^2+d^2}i. \end{equation*}

Euler's Formula is

\begin{equation*} e^{i\theta} = \cos \theta + i \sin\theta. \end{equation*}

The polar form of a complex number \(z=x+yi\) is

\begin{equation*} z = r\cos \theta + ir\sin \theta = re^{i\theta}, \end{equation*}

where \(r = |z|\) and \(\theta\) is the angle with initial side along the positive horizontal axis, and terminal side through the point \((x,y)\).

The argument of the complex number \(z=re^{i\theta}\) is \(\arg(z) = \theta\).

\((\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta)\)

1.1.1 Basic Complex Numbers Problems

Write \(4(1+2i)-2(5-i)\) in cartesian form.

Write \(\displaystyle \frac{2}{3+i}\) in cartesian form.

Write \((1+i)(1-i)\) in cartesian form.

Write \((1+i)^3\) in cartesian form.

Write \(e^{1+i\pi}, e^{i\pi/2}\) and \(e^{e^{ln(\pi + i\pi/2)}}\) in polar and cartesian form.

Verify the identity \(e^{i \pi} + 1 =0\).

Write \(-1+i\sqrt{3}, 4i\), and \(5-5i\sqrt{3}\) in polar form

1.2 Transformations

A transformation is a one-to-one function whose domain and codomain are equal.

A geometric transformation is a translation if it is of the form \(z \mapsto z+b\) for some constant \(b \in \mathbb{C}\).

A geometric transformation is a rotation if it is of the form \(z \mapsto e^{i\theta} z\) for some \(\theta \in \mathbb{R}\).

A geometric transformation is homothetic if it is of the form \(z \mapsto kz\) for some \(0 < k \in \mathbb{R}\). If \(k<1\) then the transformation is called a shrinking, and if \(k>\) the transformation is called a stretching.

Classify each of the following geometric transformations as either translation, rotation, or homothetic:

\(z \mapsto z+2\);
\(z \mapsto 5z\);
\(z \mapsto z + i -3\);
\(z \mapsto iz\).

An isometry of a metric space is a function \(f\) which preserves distances.

A fixed point of a transformation \(T\) is a point so that \(T(x) = x\).

A a geometric transformation is a reflection if it is an isometry whose set of fixed points is a line. This is also called a mirror reflection.

Reflection about the horizontal axis is given by the transformation \(z \mapsto \overline{z}\). Reflection about the line through the origin forming an angle \(\alpha\) with the positive horizontal axis is given by the transformation \(z \mapsto e^{2i\alpha} \overline{z}\).

The geometric transformation \(z \mapsto 1/z\) (for \(z \neq 0\)) is called inversion.

A transformation is euclidean if it is a rotation, translation, or reflection. It is special euclidean if it is a rotation or translation.

A transformation is conformal if it preserves angles.

A mobius transformation is a transformation of the form

\begin{equation*} z \mapsto \frac{az+b}{cz+d}. \end{equation*}

1.2.1 Graphical Transformations

We will now show transformations graphically. In what follows, the blue dots are the set

\begin{equation*} \{z : z=x+iy;\ x,y \in \mathbb{Z};\ -5 \leq x,y \leq 5\}, \end{equation*}

and the red dots are the result of applying said the transformation to the blue dots

The transformation \(z \mapsto z + (2-3i)\).

The transformation \(z \mapsto e^{i\pi/4}z\).

The transformation \(z \mapsto e^{-i\pi/6}z\).

The transformation \(z \mapsto 2z\).

The transformation \(z \mapsto z/3\).

The transformation \(z \mapsto 1/z\). Not the change in scale on this example.

1.2.2 Geometric Transformations

Let \(p,q,\) and \(r\) be complex numbers. Explain why

\begin{equation*} \angle pqr = \arg\left(\frac{r-q}{p-q}\right). \end{equation*}

Here the angle measure is the normal angle measure in radians (as opposed to the one given in the Neutral Protractor Axiom requiring the measure to be between 0 and \(180^\circ\)). Hint: Consider first q=0 and use polar form.

Let \(f(z) = iz+2\) and \(g(z) = -iz+5\). Sketch the composition transformation \(w = f(g(z))\). What type of transformation is this?

Prove that the composition of two [translations, rotations, homothetics] is again a [translation, rotation, homothetic].

Let \(a,b \in \mathbb{C}\). Show that \(w = az+b\) is a composition of rotations, homothetics, and translations.

Prove that the composition of two conformal transformations is conformal.

Find a transformation of the complex plane that takes the unit disk \(\{z : |z| < 1\}\) to the disk \(\{z: |z-5| < 3\}\).

Find transformations for these rotations:

by \(\pi/4\) about the point \(i\);
by \(\pi/2\) about the point \(2+i\);
by \(-\pi/3\) about 3.

1.3 Transformation Groups

Let \(S\) be a nonempty set. A transformation group is a collection \(G\) of transformations on \(S\) so that:

the identity transformation is in \(G\);
the transformations in \(G\) are invertible and their inverses are in \(G\);
\(G\) is closed under composition.

Let \(S\) be a nonempty set and \(G\) a transformation group on \(S\). The pair \((S,G)\) is called a geometry or a model of a geometry, the set \(S\) is the underlying space of the geometry, and the set \(G\) is the transformation group of the geometry.

Let \((S,G)\) be a geometry. If \(A \subset S\), then we say that \(A\) is a figure or that \(A\) is a figure of the geometry.

Two figures \(A\) and \(B\) of a geometry \((S,G)\) are congruent if there is a transformation \(T\in G\) so that \(T(A) = B\). Notice that \(T(A) = \{Tz : z \in A\}\).

The geometry \((\mathbb{C}, E_s)\), where \(E_s\) is the collection of all special euclidean transformations (rotations and translations) and their compositions, is a geometry called the special euclidean geometry. \(E_s\) can also be described as the collection of transformations of the form \(z \mapsto e^{i\theta}z + b\) for \(\theta \in \mathbb{R}\) and \(b \in \mathbb{C}\).

The geometry \((\mathbb{C},E)\), where \(E\) is the collection of all euclidean transformations (rotations, translations, and reflections) and their compositions, is a geometry called the euclidean geometry.

Let \((S,G)\) be a geometry and \(D\) a set of figures in the geometry.

The set \(D\) is invariant if for every figure \(A \in D\) and every transformation \(T \in G\), \(T(A) \in D\).

A function \(f\) defined on \(D\) is invariant if for every figure \(B\in D\) and every transformation \(T \in G\), \(f(B) = f(T(B))\).

Let \(\triangle ABC\) be a triangle. The perimeter of the triangle is \(AB+BC+AB\).

Let \(D\) be the collection of all triangles in \((\mathbb{C},E)\).

Define \(p:D \to \mathbb{R}\) by \(p(\triangle)\) is the perimeter of \(\triangle\). Then \(p\) is an invariant function of \((\mathbb{C},E)\).

Define \(p':D \to \mathbb{R}\) by \(p'(\triangle)\) is the sum of the distance from each vertex of \(triangle\) to the origin. Then \(p'\) is not an invariant function of \((\mathbb{C},E)\).

Two geometries \((S_1,G_1)\) and \((S_2,G_2)\) are isomorphic if there is a continuous invertible function \(f:S_1 \to S_2\) so that

for every transformation \(T_1 \in G_1\), there is a corresponding transformation \(T_2 \in G_2\) so that \(T_1 = f^{-1} \circ T_2 \circ f\), (equivalently \(f \circ T_1 = T_2 \circ f\)), and
for every \(T_2' \in G_2\), there is a corresponding \(T_1' \in G_1\) so that \(T_2' = f \circ T_1' \circ f^{-1} \) (equivalently \(f \circ T'_2 = T'_1 \circ f\)).

The function \(f\) is called an isomorphism. This is equivalent to having the following diagram commute.

The closed unit disk is \(D = \{z \in \mathbb{C} : |z| \leq 1\}\). The open unit disk is \(D^o = \{z \in \mathbb{C} : |z| < 1\}\).

Define \(f : \mathbb{C} \to D^o\) by

\begin{equation*} f(z) = \frac{z}{\sqrt{1+|z|^2}}. \end{equation*}

Then \(f\) is a continuous invertible function which is an isomorphism between models of the euclidean geometry.

The graph of \(\sin(10x)\) is shown in blue below, graphed in the usual model of euclidean geometry. Next, the graph is moved by the function \(f\) given above into the unit disk model of euclidean geometry, and is shown in red.

Some other figures (graphs of functions in this case) are shown both in the standard euclidean plane and in the open disk model.

Let \(\mathcal{T}\) be the set of all translations of \(\mathbb{C}\), so \(\mathcal{T}\) is the set of all transformations of the form \(z \mapsto z+b\) for some \(b \in \mathbb{C}\). The pair \((\mathbb{C}, \mathcal{T})\) is called the translational geometry and is geometry.

1.3.1 Transformation Groups Problems

Let \(\mathcal{GL}\) (\(\mathcal{GL}\) stands for general linear) be all transformations of \(\mathbb{C}\) of the form \(z \mapsto az + b\) for \(a,b \in \mathbb{C}\) and \(a \neq 0\). Show that \(\mathcal{GL}\) is a transformation group (that is show that \(\mathcal{GL}\) contains the identity, show inverses exist and are in \(\mathcal{GL}\), and show compositions are in \(\mathcal{GL}\)).

Convince yourself that \(E_s\) is a transformation group.

Let \(\mathcal{R}\) be all the rotations of \(\mathbb{C}\). Show that \((\mathbb{C},\mathcal{R})\) is a geometry. Hint: show that \(\mathcal{R}\) is a transformation group.

Let \(\mathcal{L}_H\) be the set of all horizontal lines. Is \(\mathcal{L}_H\) an invariant set in:

\((\mathbb{C},\mathcal{R})\)?
\((\mathbb{C},\mathcal{T})\)?
\((\mathbb{C},\mathcal{GL})\)?
\((\mathbb{C},E_s)\)?
\((\mathbb{C},E)\)?

Let \(\mathcal{L}_O\) be the set of all lines through the origin. Is \(\mathcal{L}_O\) an invariant set in:

\((\mathbb{C},\mathcal{R})\)?
\((\mathbb{C},\mathcal{T})\)?
\((\mathbb{C},\mathcal{GL})\)?
\((\mathbb{C},E_s)\)?
\((\mathbb{C},E)\)?

Is \((\mathbb{C}, \{z \mapsto e^{i\theta}z + b : \theta \in \mathbb{R}, b \in \mathbb{C}\})\) a geometry? Have we seen this geometry before?

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 \(\triangle_1 \cong \triangle_2\)?

\((\mathbb{C},\mathcal{R})\)
\((\mathbb{C},\mathcal{T})\)
\((\mathbb{C},\mathcal{GL})\)
\((\mathbb{C},E_s)\)
\((\mathbb{C},E)\)

1.4 Mobius Geometry

\(\mathbb{C}^+ = \mathbb{C} \cup \{\infty\}\) is called the extended complex plane or the Riemann Sphere.

Let \(\mathcal{M}\) be the set of all Mobius transformations, that is \(\mathcal{M}\) is the set of all transformations of the form \[ z \mapsto \frac{az+b}{cz+d}, \] where \(a,b,c,d \in \mathbb{C}\) with \(ad-bc \neq 0\). The pair \((\mathbb{C}^+,\mathcal{M})\) is a model of Mobius Geometry.

If \(T \in \mathcal{M}\) is a a Mobius transformation which is not the identity, then \(T\) has either one or two fixed points.

If a Mobius transformation has three fixed points, it is the identity.

The cross ratio of four complex numbers \(z_0, z_1, z_2,\) and \(z_3\) is \[ (z_0,z_1,z_2,z_3) = \frac{z_0-z_2}{z_0-z_3} \frac{z_1-z_3}{z_1-z_2}. \]

Let \(z_1,z_2,z_3 \in \mathbb{C}^+\) be three distinct points, and \(w_1,w_2,w_3 \in \mathbb{C}^+\) be three distinct points. There is a mobius transformation so that \(z_1 \mapsto w_1, z_2 \mapsto w_2\), and \(z_3 \mapsto w_3\).

A cline is a circle or a line.

A mobius transformation takes clines to clines.

1.4.1 Mobius Geometry Problems

Show that the identity \(z \mapsto z\) is a Mobius transformation.

Show that for \(T,S \in \mathcal{M}\), \(T^{-1} \in \mathcal{M}\) and \(T \circ S \in \mathcal{M}\).

Find a Mobius transformation \(T\) so that \(1 \mapsto 2, 2 \mapsto 3\), and \(3 \mapsto -1\).

Find a Mobius transformation that takes the unit circle \(|z| =1\) to the line \(x+y=1\).

For the following problems, let \(T,S \in \mathcal{M}\) with \[ Tz = \frac{az+b}{cz+d}\\ Sz = \frac{ez+f}{gz+h}, \] \(ad-bc \neq 0\) and \(eh-fg \neq 0\). Also let \[ M = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, \] and \[ N = \begin{bmatrix} e & f \\ g & h \end{bmatrix}. \]

What is \(\det(M)\)? How does this relate to \(T\)?

What is \(M^{-1}\)? How does this relate to \(T\)?

What is \(MN\)? How does this relate to \(T\) and \(S\)?