.....A complex number in complex plane is certainly a vector, both a and b define the complex number, it is not that by transforming the origin you can change the magnitiude or spatial position of a complex number, on the other hand a point (a,b) in real plane can have any arbitrary values of a and b depending on the selection of origin.
still to be safe I will choose 5)...
I thought a complex number in a complex plane is a point, not a vector. I must be wrong, since you are never wrong.
I have chosen the safer side too...
But you are missing a point, a complex number (a,b) is a number, but (a, b) on a real plane is a point......so how do you write a number with two disjoint parts ? like (component a+i component b), but a point (a,b) has no meaning as a+b...
No, the point was you said a complex number in a complex plane is a vector and I say it is a point. Do you think your statement was correct?
A number cannot be a point.....Complex number is a number (having magnitude) not a point. Just because it is represented on Complex Plane as a point does not make it a point...It is still OA only.
Sigh.
The Euclidean space $$E^1$$ is a line and $$E^2$$ is a plane. The points of $$E^1$$ and $$E^2$$ are the points of geometry. There is also a notion of distance between points and in $$E^2$$ a notion of lines intersecting in points. The Euclidean spaces support the continuous transforms of translation and (for $$E^2$$ ) rotation as isometries — they leave distances unchanged — so no point and no direction is preferred. The Euclidean spaces also support the isometries of reflection in a point or line.
The real numbers $$\mathbb{R}$$ are a model for the Euclidean line. Corresponding to points in $$E^1$$ we have real numbers and corresponding to distance we can add the Euclidean metric for distance $$\left| x_2 - x_1 \right| = \sqrt{ \left( x_2 - x_1 \right)^2 }$$. So our model* of $$E^1$$ is not only complete, it is more than complete in that $$\mathbb{R}$$ has a distinguished point $$0$$ and a notion of addition which lets it satisfy the axioms of a vector space. So a real number is a number, a point and a vector. Likewise in $$\mathbb{R}^2$$ we have ordered pairs of numbers, points and vectors (with the notion of pair-wise addition).
Because numbers support addition where $$1 + 1 = 2$$, and multiplication, where $$-1 \times -1 = 1$$, a transformation of translation or rotation or reflection alters (at least some) numbers or pairs of numbers so they no longer have their original relations as numbers, nevertheless such translations and rotations are still isometries as they preserve the Euclidean metric for distance between pairs of points.
We also have the mechanisms of linear algebra available, and we can build arbitrary projection operators. Pick a number, $$\theta \in \left( - \frac{\pi}{2}, \frac{\pi}{2} \right] $$, then $$(x(\lambda) ,y (\lambda)) = \lambda \left( \cos \theta, \sin \theta \right) = \lambda \hat{n} $$ is the equation of a line through the origin. We can decompose any vector into components parallel to that line and orthogonal to that line.
$$\vec{v}_{\parallel} = \left( \vec{v} \cdot \hat{n} \right) \hat{n} = ( v_x \cos \theta + v_y \sin \theta ) ( \cos \theta, \sin \theta ) = \begin{pmatrix} \cos^2 \theta & \cos \theta \sin \theta \\ \cos \theta \sin \theta & \sin^2 \theta \end{pmatrix} \begin{pmatrix} v_x \\ v_y \end{pmatrix}, \\ \vec{v}_{\perp} = \vec{v} - \vec{v}_{\parallel} = \begin{pmatrix} \sin^2 \theta & -\cos \theta \sin \theta \\ -\cos \theta \sin \theta & \cos^2 \theta \end{pmatrix} \begin{pmatrix} v_x \\ v_y \end{pmatrix}$$
So I don't see what is so special about complex numbers decomposing as two real numbers when ordered pairs of real numbers can be decomposed in a continuum of ways.
The complex numbers $$\mathbb{C}$$ also has a notion of distance, $$\left| z_2 - z_1 \right| = \sqrt{ z \, \bar{z} }$$, and so can also be used as a model for the Euclidean plane and also have a notion of addition and distinguished point $$0$$. So a complex number is likewise a number, a point and a vector. Indeed it can be vector of a 2-d real vector space as the notion of multiplication by real scalars is supported, or a vector of a 1-D complex vector space as multiplication by complex scalars is likewise supported. The complex numbers support an important isometry, the reflection in the real line called complex conjugation $$z \mapsto \bar{z}, \; (a + i b) \mapsto (a - i b)$$. This leaves addition and multiplication relations unchanged. $$ \overline{z_1 + z_2} = \bar{z_1} + \bar{z_2}, \; \overline{z_1 \times z_2} = \bar{z_1} \times \bar{z_2}$$. This allows to build the special decomposition operators: $$\Re z = \frac{1}{2} \left( z + \bar{z} \right), \Im z = - \frac{i}{2} \left( z - \bar{z} \right), z = \Re z + i \Im z$$ And the arbitrary projection operator: $$z_{\parallel} = \frac{1}{2} \left( z + e^{2 i \theta} \bar{z} \right), \; z_{\perp} = \frac{1}{2} \left( z - e^{2 i \theta} \bar{z} \right)$$
*Since a mathematical model satisfies the defining axioms of another mathematical object, for all practical purposes the map can be the territory in math.
