MEC. Addendum on Complex Variables I

Mathematics Education Centre
Culture, Pedagogy and Identity Interest Group
Making sense of solving equations visually using functions and mapping diagrams¤

March 29, 2017

Addendum on GeoGebra Examples and Complex Variables

Part I
Mapping Diagrams for Real Functions
Complex Arithmetic

Part II
Complex Functions

Martin Flashman
Professor of Mathematics
Humboldt State University

http://flashman.neocities.org/Presentations/MEC2017/MEC.MD.CV.I.html

Background and References to other work on Mapping Diagrams for real variables ¤

1. Mapping Diagrams. ¤

Beginning example:
Understanding Functions: Tables, Mapping Diagrams, and Graphs

2. Linear Functions. ¤
2.1 Linear functions are the key to much of mathematics.
Linear functions are traditionally expressed by an equation like : $f(x)= mx + b$.
Mapping diagrams for linear functions have a simple unifying feature- the focus point, determined by the numbers $m$ and $b$, denoted here by $[m,b]$.
Mapping Diagrams and Graphs of Linear Functions

Visualizing linear functions using mapping diagrams and graphs.

Notice how points on the graph pair with arrows and points on the mapping diagram.

2.2 Solving linear equations. Visualizing the solution of equations.
Every linear function can be considered a composition:
If $f_{+b}(u) = u+b$ and $f_{m*}(x)= mx$ then $f(x)= mx+b = f_{+b}(f_{m*}(x))$ .
The composition allows the visualization of the traditional steps in solving a linear equation. ¤

3.1 Complex Numbers: $\mathbb{C}$ ¤
Definition: The "number" $i$:

To solve the equation $x^2 +1 =0$ we create a symbol $i$ and declare that $i^2 = -1$ so $i$ solves the equation and we write $$ i = \sqrt{ -1 }.$$

Definitions: A complex number $z$ is a number that can be expressed in the form $z =a +bi$ where $a$ and $b$ are real numbers.

If $z =a +bi$ then $a$ is called the real part of $z$ and $bi$ (and sometimes $b$) is called the imaginary part of $z$.
Examples: $3 + 2i$ and $-2 - i$.

3.2 Complex Geometry- The Complex Plane
Complex numbers can be identified with points in a cartesian plane by having $a +bi$ identified with the point with coordinates $(a,b)$ or with position vectors by identifying $a+bi$ with the vector $<a,b>$. With this identification $i$ is identified with the point $(0,1)$ and $-i$ is identified with $(0,-1)$.

3.2.1 Complex Number Norm (Magnitude):
The norm of $z$ is defined by $|z| = |a+bi| = \sqrt{ a^2 +b^2}$

3.2.2 Polar Representation of $z$: ¤
Using trigonometry we have the identification:
$$z = |z| \cos( \theta) + |z| \sin(\theta) i = |z| [\cos( \theta) + \sin(\theta) i] = |z| cis( \theta) $$ where $a = \cos( \theta ), b = \sin(\theta)$.
$z$ is sometimes represented as an ordered pair $(|z|, \theta)$.

The angle $\theta$ determined by $z$ can be measured in degrees or radians and restricted to be in a specific interval.

For example $\theta \in [0,2 \pi)$ or $\theta \in (-\pi, \pi]$.
Thus the angle can be considered a function of $z$, called the argument of $z$: $Arg(z) =\theta$.

3.2.3 Exponential Representation of $z$: ¤
One can show with the series using $\theta$ in radian measure: $$e^{i\theta}= \cos(\theta) + \sin(\theta)i = cis (\theta)$$
and thus $$z =|z|e^{i \theta}$$.
Note: When $\theta =\pi$ this equation demonstrates that $e^{\pi i} = \cos(\pi) + sin(\pi)i = -1$.

3.3 Complex Arithmetic

3.3.1 Complex Addition: $z_1+z_2$ ¤
If $z_1 = a_1 + b_1i$ and $z_2 = a_2 + b_2i$ then

$$z_1+z_2 =a_1 + a_2 + (b_1+ b_2) i $$

Example: If $z_1 = 2+3i; z_2= 1 - i$ then $z_1+z_2= (2+1) + (3-1) i = 3 + 2i $.

The addition can be thought of as vector addition
- with separate addition of the real and imaginary parts.
Addition can be visualized in the complex plane by using parallelograms.

3.3.2 Complex Multiplication: $z_1 \cdot z_2$ ¤

Algebraically:If $z_1 = 2+3i; z_2= 1 - i$ then $z_1 \cdot z_2= (2+3i) \cdot (1-i) = 2 \cdot 1 - 3\cdot i^2 + 3i \cdot 1 +2 \cdot (-i) = (2+3) +(3- 2)i = 5+i$.

It is possible understand complex multiplication geometrically using the polar or the exponential representation (in radian measure):
$z_1=|z_1| [\cos( \theta_1) + \sin(\theta_1) i] = |z_1|e^{i\theta_1} ;\ z_2=\cdot |z_1| [\cos( \theta_2) + \sin(\theta_2) i] = |z_2|e^{i\theta_2}$
$z_1 \cdot z_2= |z_1| [\cos( \theta_1) + \sin(\theta_1) i] \cdot |z_1| [\cos( \theta_2) + \sin(\theta_2) i] $
   $ = |z_1|\cdot |z_1| [\cos( \theta_1) + \sin(\theta_1) i] \cdot [\cos( \theta_2) + \sin(\theta_2) i]$
       $ = |z_1|\cdot |z_1| [\cos( \theta_1)\cos( \theta_2) - \sin(\theta_1)\sin(\theta_2) + (\sin(\theta_1)\cos( \theta_2) + \sin(\theta_2) \cos(\theta_1) )i]$
       $ =|z_1|\cdot |z_1| [\cos( \theta_1 + \theta_2) + \sin(\theta_1 +\theta_2) i]$
       $ =|z_1|\cdot |z_1| cis (\theta_1 +\theta_2) $
or more simply using $\theta$ in radian measure:
$z_1 \cdot z_2= |z_1|e^{i\theta_1}\cdot |z_2|e^{i\theta_2}$
        $ =|z_1|\cdot |z_1| e^{(\theta_1 +\theta_2)i}$

Example: If $z_1 = \sqrt 2\ cis(45°); z_2= 2\ cis(30°)$ then $z_1\cdot z_2= 2\sqrt 2\ cis (75 °) $.¤

3.3.3 Complex Inverses: $ \frac1z, z \ne 0$ ¤

Example: Express $\frac 1{2+3i}$ in the standard form of $a + bi$.

Solution: $\frac 1{2+i} =\frac 1{2+i} \cdot \frac {2-i}{2-i} = \frac{2-i}{4+1} = \frac 2{5} - \frac 3{5}i$.

Fact: If $a^2+b^2 \ne 0$ then $\frac1{a+bi} = \frac a{a^2 + b^2} - \frac b{a^2 + b^2}i $

3.3.4 Geometry for $ \frac1z, z \ne 0$
Geometric understanding of $ \frac1z, z \ne 0$ comes best by using the polar or exponential representation and the multiplicative inverse property:$ z \cdot\frac1z = 1$.

If we let $z=|z [\cos( \theta) + \sin(\theta) i] ;\ z_1=\frac1z= |z_1| [\cos( \theta_1) + \sin(\theta_1) i] $ then
$$z \cdot z_1= |z|\cdot |z_1| [\cos( \theta + \theta_1) + \sin(\theta +\theta_1) i]$$ $$ =|z|\cdot |z_1| cis (\theta +\theta_1) = 1=1\ cis(0^{\circ} ) $$ then $|z_1| = \frac 1{|z|}$ and $\theta_1 =-\theta$.

That is, if $z=|z|\ cis(\theta) = |z|e^{i\theta}$ then $\frac1z = \frac1{|z|} \ cis(-\theta) = \frac1{|z|} e^{-i\theta}$

On to Part II Complex Functions

References:¤

M. Flashman GeoGebra Book [in development]: Mapping Diagrams to Visualize Complex Analysis http://ggbtu.be/bNi69jyKs

AMATYC Webinar Martin Flashman - Using Mapping Diagrams to Understand Functions (YouTube)

Mapping Diagrams from A(lgebra) B(asics) to C(alculus) and D(ifferential) E(quation)s.
A Reference and Resource Book on Function Visualizations Using Mapping Diagrams

The Sensible Calculus Program