The two most beautiful equations in mathematics

This goes out to all of you mathematicians - academic, professional, amateurs or aspiring wannabes. Can you name the most beautiful equation(s) you have ever encountered?

Here are my two (2) votes:

Euler's equation:


Lagrange's equation:


Next question: Why did I choose them?

Euler's equation provides the link between three areas of mathematics: complex numbers, trigonometric functions and the exponential function. Amongst many other applications, this has provided us with the key to solving 2nd order differential equations. And in the specific case of



we obtain the equation that links the five major constants in mathematics:



Lagrange's equation successfully reduced Newtonian mechanics from graphical to purely analytical. While credit for pioneering analytical mechanics is due to Euler, the true champion is Lagrange. Lagrange's equation eliminated the need for an inertial axes of reference.