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 **C**^{2} (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 **C**^{2} 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 **R**^{3}, 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.

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.

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 **R**^{2} 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.