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In the Argand diagram (complex plane), numbers on the horizontal axis represent real numbers.
No, it could be a qualitative variable.
If you mean in regard to organizational charts, the vertical lines represent the hierarchy who reports to whom, while the horizontal lines represent lateral or peer associations rather than chain of command.,
Yes. It is the function f(x) = c where c is a constant.
It depends on the information being plotted. It is often the independent variable.