Math 629: History of Mathematics
Frank Sottile

• Opening Remarks:
      This week's readings are on various aspects of the complex numbers and span several centuries. We already saw in the solution for the cubic how the complex numbers play a role even in finding real (as in real-number, but back then `actual') solutions. This is reprised in Chapter 14, which describes the rise of complex numbers in algebra, culminating in the Fundamental Theorem of Algebra, that every polynomial of degree n in one variable has n complex roots, counted with multiplicity. I need to point out a small problem with Stillwell: his `reason' for this–that the complex numbers have two real dimensions, so that there is more room for zeroes to appear. (This cannot be relevant. The fundamental theorem of algebra holds in many fields which do not have any topoligical dimension, or are in fact infinte-dimensional over the base fields; e.g. the algebraic closure of the rational numbers. Stilwell is simply mistaken anbout this point.) The middle parts of the chapter discuss how we came to view the complex numbers geometrically, as points in the plane. The mathematical meat in the chapter is his description of the proof of the Fundamental Theorem of Algebra by d'Alembert–at least how d'Alembert's proof was made rigorous, using only Taylor series, a tiny bit of geometry, and Weierstraß's Theorem that a continuous function on a bounded set takes a maximum and a minimim (This is referred to as the Bolzano-Weierstraß's Theorem in analysis classes). It is worth re-reading that part to understand it better, as the Fundamental Theorem of Algebra arises in the curriculum in secondary schools. As someone who works in real algebraic geometry, I can confirm that the work needed to make Gauß's proof rigorous is substantial. By now there are many, many proofs of this basic result. One of my favourites uses topology.
  
      Chapter 15 describes how the study of algebraic curves moved from the real plane to the complex projective plane, culminating in a topological understanding of algebraic curves by Riemann and others. We are skipping that, except for the Historical Notes at the end of the chapter.
  
      Chapter 16 continues this complex-number theme, but into the realm of complex functions. This is a great story; I am still amazed by the result that if a complex function has one derivative, then it has infinitely many derivatives and thus a power series expansion and thus.... It begins with a discussion of the complex exponential, then mentions this above result (due to Cauchy) and the Cauchy-Riemann equations. The relation to Green's Theorem and fluid flow are also mentioned. These Cauchy-Riemann equations also arise in electro-magnetism and are one of the reasons (the other is the complex exponential) that complex functions continue to be used among electrical engineers. The last topic, on elliptic functions and lattices Λ in C, along with moduar functions is a link back to number theory, which studies these objects. This may be revisited when we discuss more modern number theory in the waning weeks of this course. The last section of Chapter 16 touches on linear fractional transformations, also called Möbius transformations. There is an interesting and award-winning video about them that you may wish to watch on you tube.
  
  
• Reading:
  • Chapters 14, 16 of Stillwell, and the Historical Notes at the end of Chapter 15.
• Assignment: Due Monday, 25 March 2024. (HW 10)
  Here is a .pdf and a LaTeX source of the assignment.
  To hand in: We are using Gradescope for homework submission.
  1. The exercises in Chapter 14 on the Fundamental theorem of Algebra do not touch on essential parts of that result. How the desire to show that all rational functions may be integrated in elementary terms was a motivation is interesting, and puts the technical topic of partial fractions from Calculus in an interesting light.
       To that end, do exercises 14.5.2, 14.5.3, and 14.5.4 on Bernoulli's undertstanding of the multiple angle formulas for the tangent function.
  2. With this point in mind again, as well as hewing closer to Calculus and topics suitible for good high school students, I will assign the first two exercises in Section 16.1 involving the complex exponential and Euler's equation e^iπ + 1 = 0. The first two, 16.1.1 and 16.1.2 show how to deduce the form of the complex exponential from the power series for the exponential function. An easy justification for the validity of the series is that it converges absolutely. The last two, 16.1.3 and 16.1.4, which involve figuring out the values of iⁱ, are a little mind-bending, and you might feel some kinship with Cardano, whose use of complex numbers caused him "mental tortures".
  3. DO the three exercises at the end of Section 16.6: 16.6.1–16.6.3. They foreshadow the non-commutativity in algebra discovered by Hamilton in the mid 19th century.

Last modified: Mon Mar 18 14:05:08 PDT 2024 by sottile