We all know, that all real numbers are complex numbers also. But in real numbers, there are multi-dimensional coordinate planes. Is there such thing in complex numbers also? If yes, what is there meaning and how do we represent numbers over there?
Edit: Things are getting clearer now. But my question slightly changes,
1. Are multidimensional complex planes used in real? What is there use? If no, why not?
2. Please give an example of a "line" in a 3D complex plane. And what is the meaning of coordinates in a multidimensional complex plane?
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Aditya Agarwal
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Perhaps use co-ordinates like $(z_1,z_2,z_3$) where each $z_i$ can be a complex number – Henry Sep 16 '15 at 07:20
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But what would be there meaning? Complex numbers cant be compared. – Aditya Agarwal Sep 16 '15 at 07:21
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You can answer questions like "How far apart are two points?" and "Do three points lie on a straight line?" – Henry Sep 16 '15 at 07:26
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Is it used in real? – Aditya Agarwal Sep 16 '15 at 07:27
1 Answers
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The complex plane is a two dimensional real vector space (using the natural identification $(x,y)=x+iy$). Of course one can form the (complex) vector spaces $\mathbb C^n$ for each positive integer $n$, that is, a complex space of dimension $n$; the set of all $(z_1,\dots,z_n)$ for $z_j\in\mathbb C$. However$,$ since $\mathbb C$ is identified with $\mathbb R^2$, $\mathbb C^{n}$ is identified with $\mathbb R^{2n}$ in the same natural way.
When discussing dimensions, you need to specify what field you are considering (the real numbers, complex numbers, rational numbers, etc).
SamM
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1Using the map $(x,y)\mapsto x+iy$. And in this context, identified means that there is a bijective linear map from $\mathbb R^2\to\mathbb C$ (in terms of vector spaces, this is equivalence). One can also show that the map, as above, has many other "structure preserving" properties. – SamM Sep 16 '15 at 07:27
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Can you please elaborate more? I don't seem to understand what maps are? Linear maps? – Aditya Agarwal Sep 16 '15 at 07:28
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A map (function) is a relation which relates a point with another point, see https://en.wikipedia.org/wiki/Function_(mathematics). A linear map $f$ is a function which preserves the structure of a vector space, like the spaces discussed in my answer. – SamM Sep 16 '15 at 07:31