4 ms·

The concept you are looking for here is algebraic closure. If you want "ax^2 + bx + c" to have numeric roots and be factorable for all coefficients, you have to

The concept you are looking for here is algebraic closure. If you want "ax^2 + bx + c" to have numeric roots and be factorable for all coefficients, you have to expand your definition of number from real to complex. Probably the most blatant physical example of this is modeling mass-spring-damper systems. The differential equations will map the physical coefficients to such a polynomial, and the roots of that polynomial determine the system's behavior. Without complex numbers, you wind up only being able to describe real exponential responses and don't have the vocabulary to describe the underdamped (and oscillating) system responses with complex exponentials in the results.

KirillPanov 5y ago
> 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