There have been a number of Scientific discoveries that seemed to be purely scientific curiosities that later turned out to be incredibly useful. Hertz famously commented about the discovery of radio waves: “I do not think that the wireless waves I have discovered will have any practical application.”
Are there examples like this in math as well? What is the most interesting “pure math” discovery that proved to be useful in solving a real-world problem?
And you can also just write it out using real numbers if you wish, it’s just more mathematically concise to use complex numbers. It’s a purely subjective, personal choice to choose to use complex-valued notation. You are trying to argue that making a personal, subjective, arbitrary choice somehow imposes something upon physical reality. It doesn’t. There isn’t anything wrong with the standard formulation, but it is a choice of convention, and conventions aren’t physical. If I describe my losses in a positive number, and then later change convention and describe my winnings with a negative number, the underlying physical reality has not changed, it’s not going to suddenly transmute into something else because of a change in convention in how I describe it.
The complex numbers in quantum theory are not magic. They are also popular in classical mechanics as well, and are just quite common in wave mechanics in general (classical or quantum). In classical wave mechanics, in classical computer science, we use the Fourier transform a lot which is typically expressed as a complex number. It’s because waves have two degrees of freedom, and so you could describe them using a vector of two real numbers, or you could describe them using complex numbers. People like the complex-valued notation because it’s more concise to write down and express formulas in, but at the end of the day it’s just a convention, a notation created by human beings which many other mathematically equivalent notations can describe the same exact thing.
Yeah, I agree it can all be mapped onto positive reals. How about zero? Would that be needed to be having a physical reality ?
We all have an intuitive grasp of 0, it’s just when you define what it looks like for something to change in a particular situation, i.e. you define what x>0 or x<0 looks like, and then x=0 is just when it hasn’t changed at all.
I feel like this discussion is getting too philosophical. My point isn’t that deep, it is just to keep complex numbers intuitive and tied to physical reality. We shouldn’t treat imaginary numbers like some out there, almost mystical thing we should just accept at face value. When we realize that they are mathematically equivalent to a set of operations on a vector of two real numbers, we can then get an intuitive understanding of what they actually represent in the real world. You can visualize a complex number as a vector representing on a plane (called the complex plane), and then visualize operations on the complex numbers as manipulations of that vector.
The Fourier transform has complex numbers in it. This isn’t mysterious, it’s just that the Fourier transform deals with waves, and waves are two-dimensional, so they need to be described by a vector of two numbers. The Fourier form effectively wraps the wave around a circle, and if the rate of wrapping is different from the wavelength of the wave, then every time you complete a revolution of the circle, you will either have overshot or undershot a complete cycle of the wave, causing your second wrapping to be off-center, and if you repeat this indefinitely, then all the off-center wrappings will cancel each other out, giving you 0 in the limit. But if the rate of wrapping is equivalent to the wavelength, then a revolution around the circle would exactly correspond to a cycle of the wave, so it would you would not get this cancelling and it would blow up to infinity in the limit.
You can get very intuitive mental images of what complex numbers are actually doing when you recognize this. There shouldn’t be a layer of mystery put on top of them. People often act like they are something so mysterious we just have to accept at face-value, and others will even justify this by pointing out that they’re used in quantum mechanics, and quantum theory is “weird,” therefore we should just accept this weird thing at face-value and not question it.
All I am trying to point out is that complex numbers are not “weird,” they have clear meaning you can visualize them and get an intuition for them, and the reason they show up in certain equations always has very good and intuitive explanations for it. I am not making a deep philosophical point here. I am only arguing against the notion of obfuscating the meaning of imaginary numbers. The term “imaginary” is honestly not a good name. Complex numbers probably should just be called 2D numbers, with the real and imaginary components called the X and Y component or something like that. They are just a way of concisely representing something that is two-dimensional. There are also quaternions which are 4D numbers.