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 cannot say that I agree with Stillwell's `reason' for this–that the complex numbers have two real dimensions, so that there is more room for zeroes to appear. (I do not see how this can be relevant for why the fundamental theorem of algebra holds in the algebraic closure of the field with 2 elements.) 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 secondary curriculum. 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.
  
      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. There is a nice page about curves in the real projective plane that can help understand some of this. The topological description of Section 15.2 can be used to show that any sphere S^n-1 in Rⁿ may be identified with R^n-1 with a single point attached at infinity ∞ (the north pole of the sphere S^n-1). Riemann's topological description of algebraic curves coming from branch points, cutting, and glueing, continues to be a powerful technique to study topological spaces (google Surgery in topology, or look up handlebodies or `pair of pants decomposition'). I also wish to point out that Riemann's decomposition enables us to understand the shape, for example, of a smooth cubic curve (as in Figure 15.11), while trying to understand how the donut maps 2 to 1 to the sphere with four branch points is quite a bit harder (I still cannot quite see this).
  
      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, 15, and 16 of Stillwell. Re-read Sections 8.6 and 8.7 when reading 15.1, and try to understand Exercise 8.7.3.
• Assignment: Due Monday, 30 March 2020. (HW 11)
  To hand in: We are using E-campus 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 the newness of branch points, I'll assign the three exercises in Section 15.2, which are mostly computational. That is, 15.2.1, 15.2.2, and 15.2.3. Possible sticking points in the first two are understanding what happens near 0 and near ∞.
  3. 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".