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By "fundamental", I mean that it appears frequently in sciences and engineering. It seems that without complex numbers, we wouldn't be able to describe a lot of stuffs such as electromegnetic waves, quantum states.

Of course, we can always use some matrix group isomorphic to complex numbers, but that doesn't change the question radically, because then we still have to ask: Why these matrices? Especially, why these $2\times2$ matrices?

By "complete", I don't mean it in a mathematical sense, but rather an intuitively sense. I call it complete because all of the functions that we know of, gives a complex result, no matter what function we use and how we combine them: exponentiate it, log it, finding the root of some crazy equation, or maybe do some crazy integrals.

Of course, there are more complicated "number systems" such as, but not limited to, quaternions, hyperbolic complex numbers, etc. But these are far away from being as natural as complex numbers. They don't rise from any form of calculation. Instead, you have to "set them up" first, or "construct them" first, then will you be able to see there existence and relations.

With all the discussions and doubts above, my questions are follows. Why are complex numbers so fundamental and complete? Why, specifically, 2D transformations (geometric interpretations of $\mathbb{C}$) make up the foundation of our universe? What about 3D, 4D transformations?

Neo
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    The complex numbers form an algebraically closed field. Such a field has welcome properties like that every polynomial $\mathbb C[X]$ splits into linear factors over $\mathbb C$. Prize : there is no order in $\mathbb C$. If we extend to quaternions , we lose moreover the commutativity of multiplication. In physics, we only need real numbers. Complex numbers are sometimes used only to describe formulas more compact. – Peter Aug 07 '21 at 09:22
  • Not every equation has a solution, even in $\mathbb C$, for example $e^z=0$ has no solution. – Peter Aug 07 '21 at 09:25
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    Just a little idea, which surely doesn't explain everything, but is interesting nonetheless: you can think of the complex numbers as the rotations and dilations of a 2d plane centered at a specific point. In a plane, all rotations commute. But if we go to any higher dimension, they no longer do. So if we generalize the geometric view of complex numbers to higher dimensions, we must lose the immensely useful property of commutativity. – Vercassivelaunos Aug 07 '21 at 09:52
  • @Peter What do you mean by "there is no order in $\mathbb{C}$? – Neo Aug 08 '21 at 02:22
  • @Peter And if you mean there is no total ordering in $\mathbb{C}$, then my question is: How is this a prize? – Neo Aug 08 '21 at 02:30

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