Defining the Complex Plane

[Destroxia]

So I know that a complex number can be represented by $z=x+iy$, where $z = x + iy \in \mathbb{C}$.

Would it be okay to then state that $z = x + iy \in \mathbb{C} := (x,y) \in \mathbb{R}^2$?

If we can just look at complex numbers as coordinates in $\mathbb{R}^2$ what is the point of even defining a complex plane? (just started learning these math logic notations, so pardon me if my intuition is incorrect)

 

[wrobel]

If we can just look at complex numbers as coordinates in R2\mathbb{R}^2 what is the point of even defining a complex plane?

multiplication

 

[SteamKing]

And mapping a function f(z) = w onto a companion complex plane which consists of the points w = u + i v

 

[Destroxia]

multiplication

Okay, but couldn't you just use properties of vectors in $\mathbb{R}^2$, and treat each point as a position vector?

 

[SteamKing]

Okay, but couldn't you just use properties of vectors in $\mathbb{R}^2$, and treat each point as a position vector?

It's a lot less abstract with a simple graphical representation.

Also, it makes finding all the roots of zⁿ + k = 0 a snap.

 

[Destroxia]

And mapping a function f(z) = w onto a companion complex plane which consists of the points w = u + i v

So there would be no way to map a $\mathbb{R}^2$ vector function onto another companion plane of $\mathbb{R}^2$ ? I feel like I'm not understanding something...

 

[SteamKing]

So there would be no way to map a $\mathbb{R}^2$ vector function onto another companion plane of $\mathbb{R}^2$ ? I feel like I'm not understanding something...

Look, these are ways of simplifying things. Why do you want to make stuff harder than it should be?

We could replace plane geometry with analytic geometry and make the former 10 times harder than it already is. What would be the point?

 

[FactChecker]

The most important difference between C and R² is the way multiplication is defined. For any complex number, z, multiplying other numbers by z will rotate them by the argument of z. Vectors in R² don't have anything simple like that. Making a connection between the fundamental geometric property of rotations and the fundamental algebraic property of multiplication has profound consequences. For instance:
1) Every nonzero complex number z has a multiplicative inverse. (Just like every rotation has a rotation in the opposite direction. Likewise for the scaling.)
2) Every nonzero complex number z has an n'th root. (Just like every rotation can be done in n smaller rotations. Likewise for the scaling.)
3) The complex derivative of a complex values function, f, can be defined mimicking derivatives of real functions. The existence of a complex derivative in a disk has surprising consequences:
a) All higher order derivatives exist in the disk.
b) The Taylor series converges in the disk and represents the function.
c) The value of line integrals within the disk have simple values. Loop integrals are 0. Other line integrals from a to b are independent of the line path between a and b.
d) Maximums and minimums of the real and imaginary parts of f occur on the circumference of the disk.
e) The values of the function inside the disk is completely determined by its values on the circumference of the disk.

 

[Destroxia]

Look, these are ways of simplifying things. Why do you want to make stuff harder than it should be?

We could replace plane geometry with analytic geometry and make the former 10 times harder than it already is. What would be the point?

I just didn't really see the difference, because I've just started complex analysis, they seemed like the same process to me, not one was more simple than the other. I think I understand a bit now.

The most important difference between C and R² is the way multiplication is defined. For any complex number, z, multiplying other numbers by z will rotate them by the argument of z. Vectors in R² don't have anything simple like that. Making a connection between the fundamental geometric property of rotations and the fundamental algebraic property of multiplication has profound consequences. For instance:
1) Every nonzero complex number z has a multiplicative inverse. (Just like every rotation has a rotation in the opposite direction. Likewise for the scaling.)
2) Every nonzero complex number z has an n'th root. (Just like every rotation can be done in n smaller rotations. Likewise for the scaling.)
3) The complex derivative of a complex values function, f, can be defined mimicking derivatives of real functions. The existence of a complex derivative in a disk has surprising consequences:
a) All higher order derivatives exist in the disk.
b) The Taylor series converges in the disk and represents the function.
c) The value of line integrals within the disk have simple values. Loop integrals are 0. Other line integrals from a to b are independent of the line path between a and b.
d) Maximums and minimums of the real and imaginary parts of f occur on the circumference of the disk.
e) The values of the function inside the disk is completely determined by its values on the circumference of the disk.

Thank you for this list, the concept of the rotation of values makes the difference a lot more clear to me, also the part on line integrals.