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enter image description hereIs it correct to join the Cartesian Plane with the Complex Plane, the Cartesian plane being composed of the x and y axes (real numbers), and the complex plane being the z axis (imaginary numbers)? I ask this because every real number has its imaginary part.

This representation, in which the z axis is the complex numbers, is this representation correct?

What I want to know is if conceptually I can represent a number as in the graph, since every number has a real and an imaginary part

rafaelcb21
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  • Does this answer your question? What was the motivation for the complex plane? – Lee Mosher Apr 22 '22 at 13:31
  • Actually, I would like to know if I can create a 3-axis plane, z being imaginary numbers? – rafaelcb21 Apr 22 '22 at 13:53
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    Ah, sorry, I misread your question. But still, your question is rather vague. What do you mean by "join"? There certainly is a natural bijection $\mathbb C \times \mathbb R \to \mathbb R^3$ defined by $(x+iy,z) \mapsto (x,y,z)$. Is that all you are asking, or do you want there to be more structure (e.g. vector space structure; field structure; ...)? – Lee Mosher Apr 22 '22 at 14:05
  • I made a graphic representation to facilitate understanding – rafaelcb21 Apr 23 '22 at 00:02
  • Not sure this is the question, but when people speak of "a complex number $z$" it does not refer to the Cartesian $z$-axis, which represents only a real number, not a complex number. – Andrew D. Hwang Apr 23 '22 at 00:19
  • The diagram and edits don't help, for me at least. In this setting, $x + iz$ and $y + iz$ are complex, but is that the question? – Andrew D. Hwang Apr 23 '22 at 12:17
  • It's just that I always see the Cartesian Plane and Complex Plane shown separately, but in my understanding, they can be presented together. I just want to know if my understanding is correct. – rafaelcb21 Apr 23 '22 at 14:49
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    They can be presented together. But your presentation is incorrect. The "together presentation" is that the complex number $z = x+iy \in \mathbb C$ is identified with the ordered pair $(x,y) \in \mathbb R^2$. If you had drawn a picture with no $z$-axis, and with one glowing red point on the $x$--$y$ plane labelled something like this $$z=x+iy \leftrightarrow (x,y)$$ then that would have been perfect. – Lee Mosher Apr 24 '22 at 12:46
  • Thank you for the explanation – rafaelcb21 Apr 24 '22 at 16:03

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