On Wikipedia > Number > Classification there is this diagram:
A. This diagram implies that the irrational numbers are a subset of solely the real numbers. But afaik, that's not true, because any number, which is not rational, is an irrational number, and thus all imaginary numbers are irrational, because they are not rational.
I think the diagram suffers from the fact that some irrational numbers are real numbers and some are complex numbers with non-zero imaginary part. So irrational numbers are neither a subset of solely the real numbers nor solely of $\mathbb{C}\setminus\mathbb{R}$.
B. Moreover, the diagram seems to imply that the complex numbers consist solely of real numbers and purely imaginary numbers. Afaik, all complex numbers which have both, a non-zero real part and a non-zero imaginary part, are neither real nor imaginary, so they are missing in the diagram.
Is this diagram misleading? Or is it even wrong?
Best Answer
Does ‘insensitive’ refer to every entity that isn't sensitive, or is this adjective understood to apply only to humans? This is how I persuade myself that it's not illogical to define the irrationals as $\mathbb{R} {\setminus} \mathbb{Q}.$
Similarly, ‘nonpositive’ is conventionally defined on the reals (so, $i$ isn't considered a nonpositive number). Even so, however, when it's not perfectly clear that the context is strictly real, to avoid ambiguity, I find myself writing “nonpositive real number”.
The diagram is a little misleading, but it's technically correct: the labels are merely category names, and the illustration does not assert that Imaginary numbers and Real numbers form a partition of the Complex numbers, only that the former two are disjoint subsets of the latter.