"Lecture" for Week 9

Complex Numbers

Projective spaces

I'm trying to suppress the recondite aspects of projective geometry and elliptic functions by leaving sections out of the reading. Unfortunately, Stillwell assumes in Chap. 15 that you've read Chap. 8, and the problem is compounded because the reference to Sec. 8.6 on p. 297 should be to Sec. 8.7 and the two references to Sec. 8.5 on pp. 298-9 should be to Sec. 8.6. (He added a section in the new edition and forgot to renumber the cross-references.) So, I need to give a quick survey of projective lines and planes.

There are (at least) 3 ways to think about the real projective line: (Draw pictures.) (1) To the usual real line, add a point at infinity. Don't distinguish between plus and minus infinity, so the result is like a circle, not a closed interval. (2) In the real PLANE, identify two points (vectors) if they are nonzero scalar multiples of each other. (The multiplying scalar may be negative.) In other words, the points of the projective line are equivalence classes of the plane under the equivalence relation (see p. 20) "same direction". In still other words, the points of the projective line are the lines through the origin in the plane. (3) Identify each of these lines with the place where it intersects the unit circle in the plane, but remember that antipodal (opposite) points are on the same line. Thus the projective line can be identified with half of the unit circle (including ONE of the endpoints). Topologically there is no break in the space: the missing endpoint is the "same point" as the endpoint that is there on the other end, so we're back to the circle model.

With that under your belt, you're ready to contemplate the cases of genuine interest, the complex projective line and the real projective plane. First, an unavoidable terminological ambiguity: The set of ordinary complex numbers is usually called "the complex plane" because it consists of PAIRS of real numbers, z = x+ iy, but as a complex object it is a LINE, because one-dimensional. To make the complex projective line we add just ONE point at infinity, getting a sphere (see p. 300). Every direction to infinity goes to the same point. We could also mimic the second definition above: Two points in C2 (the space of pairs of complex numbers, also confusingly called "the complex plane") are equivalent if one is a nonzero COMPLEX multiple of the other. The resulting "lines" through the origin in C2 have real dimension 2, and the space of all of them also has real dimension 2 and can be identified with the complex sphere.

Finally, the real projective plane also has real dimension 2, as you'd expect, but it is not the same thing as the complex projective line. The three definitions of the real projective line have analogs: (1) On any line in the plane, add a point at infinity. Parallel lines share the same point at infinity, but intersecting lines have different infinite points. Thus every line becomes a circle, there is a new line at infinity that also has the structure of a circle, and any two distinct lines intersect at exactly one point, which may be at infinity. (2) In real 3-space R3, identify two vectors if they are nonzero scalar multiples. Thus the points in the projective plane are the lines through the origin in 3-space. (3) Identify those lines with their intersections with the unit sphere, but remember that antipodal points must be identified, since there are always two such intersections. So the real projective plane can be parametrized as a hemisphere, including exactly half of its boundary (think the geographical northern hemisphere plus the half of the equator stretching from the Greenwich meridian to the international date line (including the Greenwich-longitude endpoint but not the other one)). Each boundary point that is present is identified with the missing boundary point directly opposite it. (From the point of view of its internal geometry, this picture of the space is misleading, since in fact all its points and regions are alike.) Interestingly, the real projective plane is nonorientable, like a Moebius strip (see pp. 142-144). Note that it contains an entire circle "at infinity", not just one point as the complex sphere does.

We could go on to discuss a complex projective plane and real and complex projective spaces of higher dimensions, but let's not.

Bezout's theorem

The reason that complex projective spaces are nice is that it becomes correct to say that the locus of a polynomial equation of degree n and that of a polynomial equation of degree m intersect in exactly mn points, provided that when the equations are written in the form p(x,y) = 0 they have no factor in common. (And similarly for different numbers of equations and variables. See Stillwell pp. 118-120 and 148-149 and the Wikipedia page.) For this to be true in all cases we need to allow (1) complex roots, (2) intersections at infinity, and (3) multiple roots counted with their multiplicity. Caveats (1) and (3) are already familiar from high-school algebra for one equation of degree 2: ax2 + bx + c = 0. If the discriminant, b2 - 4ac, is negative, the roots are complex; if the discriminant is 0, there is only one solution, but it is a double root (the graph is tangent to the horizontal axis). Point (2) occurs for two equations (in 2 unknowns) of degree 1: The theorem says that the two lines should have one intersection. This fails for parallel lines until the line at infinity is added; that is why the real projective plane was invented. (It also fails resoundingly if the two lines are the same, but that case is excluded by the rule against common factors.)

Unsung heroes

Stillwell's format provides us with biographies of major mathematicians but tends to give short shrift to minor players. Here are several who had interestingly unusual career trajectories.
Étienne Bézout (1730 - 1783)
Both the Wikipedia and the St. Andrew's MacTutor articles on Bezout appear to be based largely upon the article by Judith Grabiner in the Dictionary of Scientific Biography.

Although the family of successively more general theorems about the number of solutions of a system of polynomial equations is universally known as "Bezout's theorem", Bezout the person is seldom given much attention. In part this is true because (1) Newton had stated the essence of the theorem long before Bezout (Stillwell, p. 119) and (2) Bezout's treatment was not completely rigorous. Thus Bezout appears in perspective as just one in a long sequence of people who built up the modern understanding of the topic.

In his lifetime Bezout was known primarily as a teacher, not a researcher. He was employed by the Marine and Artillery military academies of (pre-revolutionary) France and published his lectures in six volumes, Cours complet de mathématiques à l'usage de la marine et de l'artillerie, perhaps the first "engineering calculus" textbook. (An English translation was adopted at Harvard and influenced the teaching of mathematics in America in the 19th century.) According to Grabiner, "The orientation of these books is practical,... He eschewed the frightening terms “axiom,” “theorem,” “scholium,” and tried to avoid arguments that were too close and detailed" and therefore was occasionally criticized for lack of rigor. Apart from "scholium", this sounds very familiar to freshman calculus teachers today.

Bezout's big theorem appeared in his book, Théorie des équations algébriques, published in 1779 after almost 20 years of work on polynomial equations. His overall approach to equations is called "elimination theory" -- solving for one unknown in terms of the others to get higher-degree equations in fewer variables. Along the way he encountered determinants, the systematic theory of which was not developed until the mid-19th century. The writing and reading of his work were made more difficult by the fact that numerical subscripts were not yet in use.

Caspar Wessel (1745 - 1818) and Jean-Robert Argand (1768 - 1822)
(Sources: MacTutor; M. J. Crowe, A History of Vector Analysis.)

Wessel was born in Norway, which was politically a part of Denmark until shortly before his death, and spent most of his career in, or associated with, Denmark. He became a surveyor to pay his way through law school, but he decided to stay in that profession. He developed some sophisticated mathematical methods for surveying, and in this way he was led to the geometrical interpretation of complex numbers, the subject of his only published mathematical research paper (1799). The paper was in Danish and was essentially forgotten until 1895. By that time the "complex plane" had become known as the "Argand diagram".

Argand was born in Switzerland but spent his adult life in France. As a mathematician, he was only slightly less obscure than Wessel. His version of the complex number plane was published in 1806 in a privately printed book that did not even carry his name as author. In 1813 Jacques Francais called attention to Argand's idea in a book of his own and ultimately tracked down Argand as the author. Later Argand made contributions to the fundamental theorem of algebra (being the first to state it for the case where the coefficients in the original polynomial are complex) and combinatorics.

Argand's approach to complex numbers was in the tradition of de Moivre and Euler, with emphasis on the Euclidean geometry and trigonometry of the plane rather than its vector-space structure. He is credited with introducing the concept of the modulus of a complex number (which is simply the distance of the corresponding point from the origin). In retrospect, Wessel's earlier work was more modern and far-reaching. The concepts in it (at least as rediscovered by later mathematicians) are fundamental not only for complex analysis but also for linear algebra: Wessel seems to have been the first to think of the elements of R2 as VECTORS (things that can be ADDED) rather than merely as points. Indeed, the modern concept of a vector space, so central to college-level mathematics since the mid-20th century, developed surprisingly late; this will be a subject of a later week.