Practical uses for the number i

As noted above, it’s all over high school physics, in the form of rotation (aka, trigonometry).

This also pops up in chemistry, incidentally; I remember being pleased as a high schooler to realize that the “109.5 degrees” which we were made to memorize (for some bizarre reason…) as the bond angle of methane wasn’t some arbitrary quantity which could only be discovered empirically, but in fact could be calculated quite straightforwardly.

[Complex numbers also arise, though perhaps less obviously, in any phenomenon whose acceleration is proportional to its negation (e.g., Hooke’s law, that staple of high school physics). I just keep harping on rotation because it’s so intuitive to us; it’s something our brains are already hardwired for fluency with.]