Unit 1: Complex Numbers *The set of real numbers is a field that is not algebraically closed. In other words, some polynomials with real coefficients do not have a root in the real numbers (e.g., x^{2}+1=0 has no solution in the real numbers). In order to extend the set of real numbers to a larger field that is algebraically closed, we will consider the plane equipped with an addition rule and a multiplication rule that extends the corresponding operations in the real numbers. This new field is known as “the complex numbers.” In the complex numbers, there is a number whose square is equal to negative one: the “imaginary number” i.
In this unit, we will study the algebraic and geometric properties of the complex numbers. Complex numbers are akin to 2D vectors, and therefore addition and multiplication of complex numbers have a very nice geometric interpretation. We will also consider some basic topology of the complex plane: open sets, paths, etc. In order to discuss functions that become infinite as the variable approaches a given point, we will introduce the extended complex plane and its representation as the unit sphere in the three dimensional real space by means of the stereographic projection.*
Unit 1 Time Advisory
This unit will take you 10 hours to complete
☐ Subunit 1.1: 1.5 hours
☐ Subunit 1.2: 5.5 hours
☐ Subunit 1.2.1: 2 hours☐ Subunit 1.2.2: 0.5 hours
☐ Subunit 1.2.3: 2 hours
☐ Subunit 1.2.4: 1 hour
☐ Subunit 1.3: 3 hours
Unit1 Learning Outcomes
Upon successful completion of this unit, the student will be able to:
- State the field axioms
- Perform arithmetic calculations with complex numbers
- Interpret complex arithmetic geometric
- Represent complex numbers in Cartesian and polar form
- State De Moivre’s Theorem
- Define elementary topological objects and concepts (e.g. open sets)
in the complex plane
- Define the stereographic projection and explain its relationship to
the point at infinity.
1.1 The Field of Complex Numbers
- Reading: San Francisco State University: Matthias Beck, Gerald
Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in
Complex Analysis: “1.0-1.1: Introduction and Definitions and
Algebraic Properties”
Link: San Francisco State University: Matthias Beck, Gerald
Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in
Complex Analysis: “1.0-1.1: Introduction and Definitions and
Algebraic
Properties”
(PDF)
Instructions: Scroll down to page 5 (marked page 1) of the document
and read the indicated sections. Please note that you will be
returning to this resource throughout the course, so you may prefer
to save the PDF to your desktop for quick reference.
Terms of Use: The material above has been reposted with permission
by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka.
It can be viewed in its original form
here (PDF). It may not be
altered in any way.
- Lecture: Suffolk University: Professor Adam Glesser’s F09 Suffolk
Math 481, Lecture 1: Introduction
Link: Suffolk University: Professor Adam Glesser’s F09 Suffolk Math
481, Lecture 1: Brief
Outline (YouTube)
Instructions: In this 30 minute lecture, Professor Glesser will briefly define the complex numbers and then lay out the key concerns of Complex Analysis. Pay special attention to his explanation of the difficulties of establishing the existence of limits of functions on the complex plane.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
1.2 The Complex Plane
1.2.1 Complex Arithmetic and Polar Representation
- Reading: San Francisco State University: Matthias Beck, Gerald
Marchesi, Dennis Pixton, and Lucas Sabalka’s: A First Course in
Complex Analysis: “1.2: From Algebra to Geometry and Back”
Link: San Francisco State University: Matthias Beck, Gerald
Marchesi, Dennis Pixton, and Lucas Sabalka’s: A First Course in
Complex Analysis: “1.2: From Algebra to Geometry and
Back”
(PDF)
Instructions: Scroll down to page 7 (marked page 3) of the document
and read the indicated section.
Terms of Use: The material above has been reposted with permission
by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka.
It can be viewed in its original form
here (PDF). It may not be
altered in any way.
- Lecture: Suffolk University: Professor Adam Glesser’s F09 Suffolk
Math 481, Lecture 2: The Field of Complex Numbers
Link: Suffolk University: Professor Adam Glesser’s F09 Suffolk Math
481, Lecture 2: The Field of Complex
Numbers (YouTube)
Instructions: Click on the link above to play the video. (Time: 1 hour, 12 minutes)
About the Media: In this lecture, Professor Glesser will establish the fact that the complex numbers are a field.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
1.2.2 DeMoivre’s Theorem
- Reading: Imperial College, London: W.W.L.Chen’s Introduction to
Complex Analysis: “1.3: Complex Numbers: Rational Powers”
Link: Imperial College, London: W.W.L.Chen’s Introduction to Complex
Analysis: “1.3: Complex Numbers: Rational
Powers”
(PDF)
Instructions: Click on the link above, then click on “Chapter 1:
Complex Numbers.” The reference will open in PDF. Scroll down to
page 2 of the document and read the indicated section.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.
1.2.3 Geometric Properties
- Reading: San Francisco State University: Matthias Beck, Gerald
Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in
Complex Analysis: “1.3: Geometric Properties”
Link: San Francisco State University: Matthias Beck, Gerald
Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in
Complex Analysis: “1.3: Geometric
Properties”
(PDF)
Instructions: Scroll down to page 10 (marked page 6) of the
document and read the indicated section.
Terms of Use: The material above has been reposted with permission
by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka.
It can be viewed in its original form
[here](http://math.sfsu.edu/beck/complex.html) (PDF). It may not be
altered in any way.
- Lecture: Suffolk University: Professor Adam Glesser’s F09 Suffolk
Math 481, Lecture 3: Polar Form and Lecture 4: The Geometry of
Complex Numbers
Link: Suffolk University: Professor Adam Glesser’s F09 Suffolk Math
481, Lecture 3: Polar
Form and Lecture 4:
The Geometry of Complex
Numbers (YouTube)
Instructions: Click on the links above to play the videos. (Time: 1 hour, 9 minutes; 1 hour, 50 minutes)
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
1.2.4 Elementary Topology
- Reading: Imperial College, London: W.W.L.Chen’s Introduction to
Complex Analysis: “2.1-2.2: Foundations of Complex Analysis: Three
Approaches and Point Sets in the Complex Plane”
Link: Imperial College, London: W.W.L.Chen’s Introduction to Complex
Analysis: “2.1-2.2: Foundations of Complex Analysis: Three
Approaches and Point Sets in the Complex
Plane”
(PDF)
Instructions: Click on the link above, then click on “Chapter 2:
Foundations of Complex Analysis.” The reference will open in pdf.
Scroll down to page 1 of the document and read the indicated
sections.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.
1.3 The Extended Plane and the Stereographic Projection
- Reading: Imperial College, London: W.W.L.Chen’s Introduction to
Complex Analysis: “2.4: Foundations of Complex Analysis: Extended
Complex Plane”
Link: Imperial College, London: W.W.L.Chen’s Introduction to Complex
Analysis: “2.4: Foundations of Complex Analysis: Extended Complex
Plane”
(PDF)
Instructions: Click on the link above, then click on “Chapter 2:
Foundations of Complex Analysis.” The reference will open in pdf.
Scroll down to page 6 of the document and read the indicated
sections.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.
Assessment: Washington University in St. Louis: Professor M. Victor Wickerhauser’s Complex Variables: “HW # 1” Link: Washington University in St. Louis: Professor M. Victor Wickerhauser’s Complex Variables: “HW # 1” (PDF)
Instructions: Click on the first link and scroll down to the link to HW#1, which will open in PDF. Work through all problems. When finished, return to the first page and click on the “solutions” link.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.Assessment: University of Illinois, Urbana-Champaign: Professor Robert Ash’s and W.P. Novinger’s Complex Variables: “Chapter 1, Problems 1, 2, 3, 10, 12, 13” Link: University of Illinois, Urbana-Champaign: Professor Robert Ash’s and W.P. Novinger’s Complex Variables: “Chapter 1, Problems 1, 2, 3, 10, 12, 13” (PDF)
Instructions: Click on the link and select “Chapter 1” which will open in PDF. Then scroll down to page 9 and work through the indicated problems. When finished, return to the main page and click on the “Solutions” link.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.