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I'm taking an introductory course in Complex Analysis. Close to the beginning of the term, we had a review of complex numbers, but we weren't expected to know any of this crazy number theory stuff, and as far as the reasons for this addition to the real numbers, my prof offered only this interesting tidbit (recalling from memory): [itex] \mathbb{C} [/itex] is closed under the operators [itex] \cdot \ \ and \ \ + [/itex] and is therefore a number "field" (whatever the heck that means). He sometimes spoke of [itex] \mathbb{R}^2 [/itex] instead of [itex] \mathbb{C} [/itex], during that discussion and explained that there was no reason for alarm, as they were essentially interchangeable.
Okay, the point of my post is that I'm confused as to what that means, and why it should be the case. Isn't [itex] \mathbb{R}^2 [/itex] a vector space, not a "field"? Furthermore, my confusion was compounded when we went on to such things as complex valued functions, and calculus in the complex plane, which I admit seems to be developed in the same way, and have similar qualities as vector calculus. But if they're so interchangeable, why do the two forms of calculus exist independently. Why formulate things in terms of complex numbers if you can just use vectors? All my math classes are so frustrating because they're just a hodgepodge of branches of math that have interrelationships and coincidences of terminology that are interesting to ponder, but that I feel I will NEVER understand them because nobody outright states them. (eg. eigenvalues in DE's vs. Lin algebra, the term "singular" appearing all over the place etc). But let's not get sidetracked. I'm rambling because I'm confused. Let me offer a specific example of this confusion.
A complex valued function f is a function of an independent complex variable z with real and imaginary parts such that:
z = x + iy
Therefore f: [itex] \mathbb{C} [/itex] ----> [itex] \mathbb{C} [/itex] (in general anyway. So the first [itex] \mathbb{C} [/itex] or some subset of it is the domain on which the values of z for which f is defined reside. And the second [itex] \mathbb{C} [/itex] is the range of f, ie in general it takes on complex values. But then we wrote f as:
f(z) = u(x,y) + iv(x,y), where u and v are real valued functions of x and y that represent the real and imaginary parts of f respectively. So the domain of u is [itex] \mathbb{R}^2 [/itex], same with v, yet we seem to be claiming that that plane is the same as the plane [itex] \mathbb{C} [/itex] in which all the z's given by z = x + iy reside! After all, it's the same damn x and y that you're plugging into u and v to evaluate them as the x and y you would plug into z to evaluate f(z) directly! So now I'm totally confused, because they are in fact NOT the same plane. We don't say u = u(z), because u is not a function of the complex variable x + iy, but of the two variables x and y SEPARATELY. You might argue that these planes are all just abstractions, so what difference does it make whether we label the y-axis with real or imaginary numbers? But I just don't GET it. It makes a big difference...or does it?
Okay, the point of my post is that I'm confused as to what that means, and why it should be the case. Isn't [itex] \mathbb{R}^2 [/itex] a vector space, not a "field"? Furthermore, my confusion was compounded when we went on to such things as complex valued functions, and calculus in the complex plane, which I admit seems to be developed in the same way, and have similar qualities as vector calculus. But if they're so interchangeable, why do the two forms of calculus exist independently. Why formulate things in terms of complex numbers if you can just use vectors? All my math classes are so frustrating because they're just a hodgepodge of branches of math that have interrelationships and coincidences of terminology that are interesting to ponder, but that I feel I will NEVER understand them because nobody outright states them. (eg. eigenvalues in DE's vs. Lin algebra, the term "singular" appearing all over the place etc). But let's not get sidetracked. I'm rambling because I'm confused. Let me offer a specific example of this confusion.
A complex valued function f is a function of an independent complex variable z with real and imaginary parts such that:
z = x + iy
Therefore f: [itex] \mathbb{C} [/itex] ----> [itex] \mathbb{C} [/itex] (in general anyway. So the first [itex] \mathbb{C} [/itex] or some subset of it is the domain on which the values of z for which f is defined reside. And the second [itex] \mathbb{C} [/itex] is the range of f, ie in general it takes on complex values. But then we wrote f as:
f(z) = u(x,y) + iv(x,y), where u and v are real valued functions of x and y that represent the real and imaginary parts of f respectively. So the domain of u is [itex] \mathbb{R}^2 [/itex], same with v, yet we seem to be claiming that that plane is the same as the plane [itex] \mathbb{C} [/itex] in which all the z's given by z = x + iy reside! After all, it's the same damn x and y that you're plugging into u and v to evaluate them as the x and y you would plug into z to evaluate f(z) directly! So now I'm totally confused, because they are in fact NOT the same plane. We don't say u = u(z), because u is not a function of the complex variable x + iy, but of the two variables x and y SEPARATELY. You might argue that these planes are all just abstractions, so what difference does it make whether we label the y-axis with real or imaginary numbers? But I just don't GET it. It makes a big difference...or does it?