johncarlosbaez, 2 months ago (edited 2 months ago)

So you wake up one day wanting to invent a 2-dimensional number system. This requires a new number 𝑖 that's at right angles to 1. So you figure multiplying by 𝑖 must rotate numbers by 90°. So multiplying by 𝑖² rotates by 180°, so

𝑖² = -1

Cool!

Then you notice something else. The derivative of a function in the 𝑦 direction must be 𝑖 times its derivative in the 𝑥 direction, because the derivative is linear and you get the 𝑦 direction by rotating the 𝑥 direction by 90°: that is, multiplying it by 𝑖. So you get this equation:

[ \frac{\partial f}{\partial y} = i \frac{\partial f}{\partial x} ]

Cool!

Then you notice something else. If you use this equation twice you get

[ \frac{\partial^2 f}{\partial y^2} = i \frac{\partial f}{\partial x\partial y} = i^2 \frac{\partial^2 f}{\partial x^2} = - \frac{\partial^2 f}{\partial x^2} ]

so

[ \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 0 ]

Wow! Every function with a second derivative obeys the Laplace equation!

You decide this one is a keeper.

https://en.wikipedia.org/wiki/Cauchy%E2%80%93Riemann_equations