Favorite Equation

What is your favorite equation? What makes it special to you? What can you share with us about your favorite equation?

 

I'll pick on Euler's identity as an example.

e^(iπ)+1=0

It's elegant and sweet. But I suspect many people who say they love it are really just reacting to its typography - "Wow, it contains five of the most fundamental constants in math!" - without necessarily grokking what it says, why it's true, and how deep it is. The truth is, it's not actually very deep, and it doesn't reveal much that is surprising or interesting.
It follows almost directly from the definitions; I say "almost" because just how directly depends on the path taken to define the various participants. In some paths, it's totally trivial (consider that π is defined as the number which makes this correct in Rudin's Real and Complex Analysis,); in others, it's an effort which requires a few lines and a diagram (see for instance the purely elementary proof in Numbers by Conway and Guy).

 

I have not seen this equation in any precalculus textbook or even calculus textbooks I have downloaded in the last couple of months.

 

In mathematics, Euler's identity (also known as Euler's equation) is the equality. where e is Euler's number, the base of natural logarithms, i is the imaginary unit, which by definition satisfies i^2 = −1, and. π is pi, the ratio of the circumference of a circle to its diameter.

Euler’s formula allows us to express complex numbers as exponentials, and explore the different ways it can be established with relative ease. You can find more about in CALCULUS, COMPLEX NUMBER
It's used in the most practical sense for working with radioactive decay, including in the commonly used formula Ce^(kt). In mathematics, it is a crucially important tool that can allow mathematicians to convert complex numbers to a more usable form.

 

I may decide to explore Euler's Equation in the future. Right now, precalculus is sufficient. Thank you for reply.