A nice companion to this, which I just found recently, is Blue1Brown3’s explanation of Euler’s formula and what it means to take something to the power of i: https://www.youtube.com/watch?v=mvmuCPvRoWQ

I’m not sure if complex numbers should be introduced to people without calculus background. Without motivating example, it seems like a puzzle game, rather than something that can help us better understand existing mathematical objects, or model some objects and processes of the real world. Electrical applications can be helpful but can also be introduced only after teaching enough electromagnetism. Even the Euler’s formula requires some understanding of the Taylor series.

I remember how fascinated I’ve been by the complex analysis and its applications, such as calculating improper integrals with residues, or analyzing series that are impossible to analyze within real analysis. Uniqueness of the analytic continuation when it exists was a truly exciting result. It was no longer a puzzle game, but a key to the world beyond the real numbers and a useful tool. But if I attempted to enter that world without preparation, it would make no sense.

A nice companion to this, which I just found recently, is Blue1Brown3’s explanation of Euler’s formula and what it means to take something to the power of i: https://www.youtube.com/watch?v=mvmuCPvRoWQ

I’m not sure if complex numbers should be introduced to people without calculus background. Without motivating example, it seems like a puzzle game, rather than something that can help us better understand existing mathematical objects, or model some objects and processes of the real world. Electrical applications can be helpful but can also be introduced only after teaching enough electromagnetism. Even the Euler’s formula requires some understanding of the Taylor series.

I remember how fascinated I’ve been by the complex analysis and its applications, such as calculating improper integrals with residues, or analyzing series that are impossible to analyze within real analysis. Uniqueness of the analytic continuation when it exists was a truly exciting result. It was no longer a puzzle game, but a key to the world beyond the real numbers and a useful tool. But if I attempted to enter that world without preparation, it would make no sense.