> The concept you are looking for here is algebraic closure. Yes, I know that the complex numbers are an ACF. Why apply the Cayley-Dickson construction [1] on

> The concept you are looking for here is algebraic closure. Yes, I know that the complex numbers are an ACF. Why apply the Cayley-Dickson construction [1] only once? The complex numbers may be an algebraically closed field, but they aren't a Central Simple Algebra over the Reals [2]. Apply the Cayley-Dickson construction again and you get the quaternions, which are real central simple. Apply it a third time and you get the octonions. You can keep doing this over and over and over getting more and more closure properties... > Without complex numbers, you wind up only being able to describe real exponential responses Why can't you describe them using pairs of real numbers in polar coordinates? [1] https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_constru... [2] https://en.wikipedia.org/wiki/Central_simple_algebra https://en.wikipedia.org/wiki/Central_simple_algebra