Posted by: Dave Richeson | March 8, 2010

## Readers’ response: Euler’s greatest hits

My friend Gene Chase is teaching a history of mathematics class at Messiah College this semester. He asked me if I was interested in giving a visiting lecture in his class in a few weeks. The topic: Leonhard Euler.

He said that I could talk about whatever I wanted. Wow, the possibilities!

So I was thinking about giving a biography of Euler followed by “Euler’s greatest hits.” This is where you come in.

You probably know my favorite theorem of Euler’s. What are your favorites? Please leave them in the comments. List as many as you want. The more, the better.

Thanks!

## Responses

1. My vote might be for the Euler-Maclaurin summation formula. But I don’t know that much about which theorems came from Euler.

2. I don’t know if it has a fancy name, but I’m partial to the infinite sum: the reciprocals of the squares of the integers = pi^2/6. What a weird result, and what a genius way he got there.

3. And by integers, I mean integers >= 1, of course. Sorry.

4. I’m still reading Euler’s Gem. The coolest thing I learned in it is that no one ever talked about edges until Euler, even though the Greeks talked a lot about polyhedra. That kind of blew me away.

5. The Euler product.

6. Can’t forget e^(i*pi) + 1 = 0 ! [or more generally e^ (i * theta) = cos(theta)+ i*sin(theta)]

• I vote for this one too! My favorite Euler’s formula (even though Euler didn’t write it in this form). It is also an example of

” Le plus court chemin entre deux vérités dans le domaine réel passe par le domaine complexe. ” (The shortest path between two truths in the real domain passes through the complex domain) – Jacques Hadamard.

It combines i, pi, cos, sin, intertwines complex and real domains in a simple elegant formula.

7. Thanks, everyone. You’ve picked some good ones. Keep them coming!

Here’s one from Twitter: @piggymurph said his favorite is the Euler-Lagrange Equation: http://twitter.com/piggymurph/status/10190591083.

Kate, that’s also one of my favorites. It is known as the “Basel problem.” It was theorem that made everyone take notice of this young mathematician named Euler.

Sue, I hope you enjoy the book. That is a wild fact, isn’t it?

8. One of my personal favorites:
x^phi(n) is congruent to 1 mod n for any x coprime to n with x,n in Z+.

9. I’ve heard it said that Euler was so prodigious that people started naming theorems after the first person *after* Euler to discover them. :)

Be that as it may, I’m an applied guy, so I’ll go with the basic necessary condition on an extreme value of a functional:

$\displaystyle \frac{\partial}{\partial {q}} L - \frac{d}{d{t}} \left( \frac{\partial}{\partial {\dot{q}}} L \right) = 0$