What makes complex numbers so special?

[Lapidus]

Is there in a nutshell an explanation or even a single reason why complex numbers have so many fascinating consequences and give rise to so much deep stuff like analytic functions (with all its stunning properties), Riemann surfaces, analytic continuations, modular forms, zeta function, its central role in quantum physics and many more?

thanks

 

[chiro]

Hey Lapidus.

I think understanding how complex numbers relate to periodicity and geometry definitely helps putting them into context.

Complex numbers give periodicity through the i^2 = -1 and the multiplication of a number by i which rotates the quantity. All wave mechanics are based on periodicity and this is why all the quantum stuff and general wave equations and formulae have imaginary numbers in them.

Geometrically complex numbers provide a way of understanding inner and outer products and you also have two dimensions being the minimum for geometry since geometry requires not only distance, but also angle as well.

The idea of periodicity is seen in number theory since division by a number has a period that corresponds to having a proper factor. For example dividing by two always gives a remainder of 0,1,0,1,0,1 and so on. The nature of waves has a close correspondence with divisibility and factorization and this all comes down to periodicity in one form or another.

The other thing is that they are algebraically complete so understanding complex numbers helps understanding general forms of algebra.

If you look at vector algebra's with real and complex numbers then you will notice that the complex form allows an understanding with a complex variable instead of a pair of real variables and this can give additional insights - particularly when you have tensor products of complex variables as opposed to tensor products involving only real variables. The understanding comes from the above by understanding what complex numbers do geometrically and also with periodicity. A lot of the higher dimensional algebra's like quaternions and the clifford algebras can make sense in complex numbers by understanding the above.

You also have to keep in mind that the prime number decomposition applies to group theory and general algebra in which complex numbers can play a big role.

 

[Lapidus]

Awesome answer, Chiro! Thanks!

 

[HallsofIvy]

The reason why "give rise to so much deep stuff" is that the complex plane is two dimensional. To show that a function is continuous at a given point, x_0 or to determine the derivative, we have to take a limit as x approaches x_0. On the real number line, which is one dimensional, we can only take the limit "from above" or "from below" and those have to be the same. In the complex plane, the limit , as we approach the point along any curve, must be the same. We have immediately jumped from "2 cases" to "an infinite number of cases". For example, we can derive the Cauchy-Riemann equations for a differentiable function, which then shows that any differentiable function must be infinitely differentiable and even analytic, by taking the limit first parallel to the real-axis and then parallel to the imaginary axis.

 

[FactChecker]

I would like to combine the important points that @chiro and @hallsodivy made:

i^2 = -1 and the multiplication of a number by i which rotates the quantity.

the complex plane is two dimensional ... In the complex plane, the limit , as we approach the point along any curve, must be the same.

We have two dimensional space with all directions and multiplication/division causes rotations in that space. It's the introduction of those rotations and the consequences when lim_z->a(f(z)-f(a))/(z-a) have to be the same from any direction that cause the powerful results.