The greatest hits of Carl Friedrich Gauss

Question

Without any doubt, Gauss is considered one of the greatest mathematicians in history. However, digging into my mind, I don't remember any iconic theorem that was proved by him.

On the contrary, for example, Euler has a few famous results that everyone remembers: Formulas like $e^{i\pi}+1=0$ or $V-E+F=2$ or $\zeta(2)=\frac{\pi^2}{6}$, etc.

My question is that what are the greatest and most beautiful works of Gauss? Consider that you want to explain the majesty of Gauss to some fresh math student. What will you show him?

Answer 1

I recommend to you Chapter 10 from Barry Mazur and William Stein, Prime Numbers and the Riemann Hypothesis, Cambridge University Press (2016). Also you can find more about Gauss work on prime numbers from different references as companion of this nice book, I say Gauss law for the distribution of prime numbers.

Answer 2

Gauss develop (among many other things) a method called row reduction that you can find here. This method allows you do get several information of a matrix.

This is at the base of linear algebra, so it is very important. It allows you to calculate the rank, to invert the matrix, and to do, again, many other things.

Plus it is use daily in every electronic device around the world to solve linear systems of equations.

Answer 3

If you want single formulas, you might try

$$ \iiint_V \nabla \cdot {\bf F} \; dV = \iint_S {\bf F} \cdot {\bf n}\; dS $$ (Gauss's theorem, aka the divergence theorem)

or

$$f(x) = \dfrac{1}{\sqrt{2\pi \sigma^2}} e^{-\dfrac{(x-\mu)^2}{2\sigma^2}}$$

(the normal or Gaussian probability density)

Answer 4

Many will call the quadratic reciprocity law Gauss' greatest hit. For students that know some calculus, but no number theory, you might think of the Arithmetic-Geometric mean, at first sight a "recreational" problem, that Gauss solved when under twenty.

Answer 5

Gauss discovered which regular polygons can be constructed using compass and straightedge. The answer is surprisingly deep and algebraic, and superficially does not even seem to use geometry at all. Today Gauss's investigations of regular polygons form part of the basis of algebraic number theory. I think this is an extremely beautiful and surprising result-- easy to state, but with a proof that comes out of nowhere. Gauss discovered all this when he was in his 20s. In retrospect the proof is not so hard to follow with the right background, but Gauss had to lay the theoretical groundwork himself to even begin formulating it. Exactly why the problem lead him so far afield from standard geometry, I have no idea, but I wish I could look at problems from such a high vantage.