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I am still struggling to convince myself that complex numbers are more than just clever notation for polar coordinates. I know that the real numbers aren't "ju

I am still struggling to convince myself that complex numbers are more than just clever notation for polar coordinates. I know that the real numbers aren't "just notation" because things like $\pi$, e, $\sqrt{2}$ etc can't be expressed using pairs of natural numbers (i.e. as rational numbers). I'm having trouble finding an equally simple example of "can't do this using pairs of reals" justification for complex numbers. Are there any simple and obvious applications that require continuity/calculus or topology on the complex plane that can't be restated in terms of real-pairs (and calculus/topology over those)? Trigonometry arises whenever you are able to take limits over any kind of cyclical/repeating process, so the fact that sin and cos appear in Euler's Identity does not seem terribly profound to me. Also, why stop at complex numbers, since it's "turtles all the way"? Quarternions, octonions are neat too. Why doesn't Feynman think octonions are "our jewel" too? At each step (real -> complex -> quaternion -> octonion) you get a larger class of formulae having roots, but sacrifice more properties of the ring structure. For example, the reals have a total order but the complex numbers do not.

ThrustVectoring 5y ago
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
Tainnor 5y ago
Complex numbers are pairs of real numbers. As a vector space they are isomorphic to R^2 and one construction of the complex numbers is, in effect, just as pairs of real numbers. (There are of course other ways of constructing the complex numbers, too, such as via matrices. These are also all isomorphic to R^2 as a vector space.) Of course, what makes complex numbers special is that you can multiply them, and that gives you a field. Following that train of thought, > Are there any simple and obvious applications that require continuity/calculus or topology on the complex plane that can't be restated in terms of real-pairs (and calculus/topology over those)? You can of course develop all of complex analysis without introducing the traditional a+bi notation and just referring to vectors in R^2. That's notation. However, complex differentiability (being holomorphic) is a stronger definition than regular differentiability in C(R^2, R^2), in particular, the Cauchy-Riemann equations need to be satisfied. From that stronger definition, you get all sorts of other theorems such that all holomorphic functions are analytic and so on.
- KirillPanov 5y ago
  Thank you for bringing up complex differentiability. Complex analysis is the only field I've found that seems to genuinely require complex numbers rather than just polar coordinate notation. However I get the impression that complex analysis, as a research field, has been a dead end for a long time now with few people working on it [1]. If pure research has come to a halt then it's fair to judge the field based on the practical applications it has achieved. Are there many? The class of holomorphic functions is pretty rigidly constrained -- infinitely differentiable, equivalent to a Taylor series at each point. I once heard that even physicists doubt the legitimacy of assuming that all physical functions are holomorphic (but now I'm speaking beyond my expertise). Continuing to speak beyond my expertise, I don't think Feynman had holomorphic functions in mind when he said "our jewel". I have to say that the definition of limits for complex numbers always seemed like an artificial straitjacket. I understand why you can't have complex limits defined any other way, but maybe the take-away here isn't "complex limits should work like this" so much as "complex limits, and therefore complex analysis, is not very useful". [1] "Complex analysis is pretty much dead. There are some remaining problems but most of them are either too specialised or too difficult. This is not a very healthy situation, and there are relatively few people working in complex analysis nowadays.", Dmitry Beliaev , former Assistant Professor (Mathematics) at Princeton University, https://www.quora.com/Are-there-any-mathematics-faculty-at-Princeton-specializing-in-complex-analysis?share=1 https://www.quora.com/Are-there-any-mathematics-faculty-at-P...

ThrustVectoring 5y ago

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

Tainnor 5y ago

Complex numbers are pairs of real numbers. As a vector space they are isomorphic to R^2 and one construction of the complex numbers is, in effect, just as pairs of real numbers. (There are of course other ways of constructing the complex numbers, too, such as via matrices. These are also all isomorphic to R^2 as a vector space.) Of course, what makes complex numbers special is that you can multiply them, and that gives you a field. Following that train of thought, > Are there any simple and obvious applications that require continuity/calculus or topology on the complex plane that can't be restated in terms of real-pairs (and calculus/topology over those)? You can of course develop all of complex analysis without introducing the traditional a+bi notation and just referring to vectors in R^2. That's notation. However, complex differentiability (being holomorphic) is a stronger definition than regular differentiability in C(R^2, R^2), in particular, the Cauchy-Riemann equations need to be satisfied. From that stronger definition, you get all sorts of other theorems such that all holomorphic functions are analytic and so on.

KirillPanov 5y ago
Thank you for bringing up complex differentiability. Complex analysis is the only field I've found that seems to genuinely require complex numbers rather than just polar coordinate notation. However I get the impression that complex analysis, as a research field, has been a dead end for a long time now with few people working on it [1]. If pure research has come to a halt then it's fair to judge the field based on the practical applications it has achieved. Are there many? The class of holomorphic functions is pretty rigidly constrained -- infinitely differentiable, equivalent to a Taylor series at each point. I once heard that even physicists doubt the legitimacy of assuming that all physical functions are holomorphic (but now I'm speaking beyond my expertise). Continuing to speak beyond my expertise, I don't think Feynman had holomorphic functions in mind when he said "our jewel". I have to say that the definition of limits for complex numbers always seemed like an artificial straitjacket. I understand why you can't have complex limits defined any other way, but maybe the take-away here isn't "complex limits should work like this" so much as "complex limits, and therefore complex analysis, is not very useful". [1] "Complex analysis is pretty much dead. There are some remaining problems but most of them are either too specialised or too difficult. This is not a very healthy situation, and there are relatively few people working in complex analysis nowadays.", Dmitry Beliaev , former Assistant Professor (Mathematics) at Princeton University, https://www.quora.com/Are-there-any-mathematics-faculty-at-Princeton-specializing-in-complex-analysis?share=1 https://www.quora.com/Are-there-any-mathematics-faculty-at-P...