## Classes with Punch Lines

Let me tell you, Physics 11 and Math 36 can pack a punch. As my first semester draws to a close, I am amazed by how much I have learned in such a short…

Real Analysis I, Real Analysis II, and Complex Variables have several things in common:

- They are upper-level math classes.
- Much of their material consists of proofs written years ago by now-dead mathematicians, usually from the nineteenth century.
- My friend Alexandra and I took all of these classes with Professor Gonzalez, who loves to remind us of #2.

Here are some of our favorite things about the dead mathematicians you learn about in these math classes. Enjoy!

**Archimedes**

This guy has been dead since roughly 212 B.C.E. But he did a lot of important things in mathematics, including coming up with the Archimedean sequence of partitions (whose existence for a particular bounded function proves that the function is integrable). This is an important concept and once led Prof. Gonzalez to ask us, “If a sequence of partitions hits you in the face, how do you know it’s Archimedean?”

**Marc-Antoine Parseval**

Parseval’s Theorem (which states the necessary and sufficient conditions for an orthonormal system to be complete) was the origin of the dead nineteenth century mathematician joke. The two most commonly used orthonormal systems are the exponential system and the trigonometric system. According to Prof. Gonzalez, “[the trigonometric system] is used by engineers and dead nineteenth century mathematicians.” Who knew?

**Carl Friedrich Gauss**

Gauss proved the Fundamental Theorem of Algebra (which states that a complex polynomial function a degree of at least 1 has at least one root) at age 19, causing Prof. Gonzalez to feel that his life was a failure when he had not made such an accomplishment at 19. I do not share that sentiment; I was just trying to get through Comp 40 at age 19!

**Augustin-Louis Cauchy**

Seriously everything we talk about in complex analysis came from this dead nineteenth-century mathematician, who died in 1857. The Cauchy Integral Formula is a “beautiful and deep result,” according to my professor.

**Bernhard Riemann**

Riemann was the other dead nineteenth-century mathematician who, with Cauchy, came up with the Cauchy-Riemann equations (and a really original name for them). These equations are used to determine if a complex function is analytic—similar to being differentiable, except that being analytic at a point means that the function is differentiable at that point *and *analytic everywhere in a small region around that point.

**Georg Cantor** (of the Cantor Diagonalization)

I found a note in my notes to Google this guy because he was “mentally unstable.” Apparently, he suffered from depression, was admitted to hospitals for the mentally ill several times, and may have had bipolar disorder. Despite that, he proved that the open interval of real numbers between 0 and 1 is uncountable (meaning that it does not have a one-to-one correspondence with the set of natural numbers). He died in 1918, but the height of his mathematical career was in the late 1800s, making him a true dead nineteenth century mathematician.

Do you have a favorite mathematician (dead or otherwise)? Let us know in the comments!

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