A few years ago, mathematician Ian Stewart published a book titled 17 Equations That Changed The World. It's an interesting list:
I would go beyond these 17 and add:
- 1+1=2, which some unknown someone realized long ago was profound and incredibly useful. The logical proof of this assertion, though, didn't come until Bertrand Russell and Alfred North Whitehead three-volume Principia Mathematica. It's a crazy book -- the text looks almost alien or elvish, and it took them several hundred pages of work before they could prove that 1+1=2. Russel said his work on this book "had actually damaged his brain." Here's an interesting discussion of the Principia Mathematica from NPR in 2010.
- Newton's second law of motion, F=ma (really, F=dp/dt, the time rate of change of momentum).
- Planck's Law of radiation (probably).
- The Lagrangian of quantum electrodynamics, because it changed how quantum physics was done ever afterward.
- Einstein's equations of general relativity, because they changed our view and understanding of the universe, and for their sheer elegance and sophistication.