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I've made a schematic of the number systems, and I noticed it's not very systematic.

Do the categories labelled with a question mark below have a name?

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Specifically, imaginary integers, noninteger (real or imaginary) rationals, etc. I have not found any on Wikipedia.

Were these numbers studied in detail? E.g. imaginary number theory.

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    $\begingroup$ Gaussian integers (integers plus $i$) are extensively studied. $\endgroup$ – Arthur Aug 12 at 21:14
  • $\begingroup$ Also positive (or non-negative) integers and rationals. $\endgroup$ – marty cohen Aug 12 at 21:29
  • $\begingroup$ What about them? $\endgroup$ – curious_metazoan Aug 12 at 21:41
  • $\begingroup$ A somewhat related question is included here math.stackexchange.com/questions/20/… , but the main question is different. $\endgroup$ – curious_metazoan Aug 12 at 21:46
  • $\begingroup$ It's not clear what the lines are meant to demarcate. You are missing algebraic integers. $\endgroup$ – Bill Dubuque Aug 12 at 22:00
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It is not entirely clear what sets of numbers you are interested in, but a few sets which don't seem to appear in your diagram are:

  • $\mathbb{Z}[i]$, the Gaussian integers. The Gaussian integers consist of all (complex) numbers of the form $a + ib$, where $a$ and $b$ are integers. As a ring, the Guassian integers are obtained by adjoining $i$ to the integers, where $i$ is an object which satisfies the property that $i^2 = -1$. The Gaussian integers were first studied by Gauss (who would have guessed?) in connection with problems in number theory (or, rather, what we would now call number theory)
  • $\mathbb{Q}[i]$, the Gaussian rationals. The Gaussian rationals consist of all (complex) numbers of the form $p + iq$, where $p$ and $q$ are rational numbers. The Gaussian rationals are a field, which may be obtained by adjoining $i$ to the rational numbers. I am not aware of anything particularly interesting about the Gaussian rationals, but the Wikipedia page linked above gives an application to Ford spheres (whatever those are).
  • The algebraic integers. An algebraic integer is a (complex) number which is the root of a monic polynomial with integer coefficients.

The sets listed above have been studied (or, at least, are important enough to merit Wikipedia pages). What makes these sets interesting or worth studying is that they have some algebraic structure: $\mathbb{Z}[i]$ is a ring (and an integral domain, even); $\mathbb{Q}[i]$ is a field, and the algebraic integers form a ring. Other (unnamed) sets in the diagram include:

  • $i\mathbb{N}$, $i\mathbb{Z}$, $i\mathbb{Q}$, and $i\mathbb{R}$. These are the sets which consist of (complex) numbers of the form $ix$, where $x$ is an element of the appropriate set of real numbers (e.g. $i\mathbb{N} = \{ in : n\in\mathbb{N}\}$). As groups additive groups, each of these is isomorphic to the corresponding set of real numbers (e.g. $i\mathbb{N} \cong \mathbb{N}$). These sets do not form rings, as they are not closed under multiplication. As such, they are not terribly interesting, and (as far as I know) have not been extensively studied.
  • The computable and noncomputable numbers. A computable number is a number which can be computed to arbitrary precision in finite time. The computable numbers are a subset of the reals (though we could easily talk about computable complex numbers, as well), and are a superset of the algebraic reals.
  • The sets $\mathbb{C}$ with some large set removed (e.g. the Gaussian integers, the Gaussian rationals, etc). For example, the set $\mathbb{C} \setminus \mathbb{Q}[i]$ might be thought of as the "Gaussian irrationals". Such sets correspond to the irrational (real) numbers, the transcendental (real) numbers, the noncomputable (real) numbers, and so on. To the best of my knowledge, these sets are entirely uninteresting—they lack nice structures, and are defined by this lack. I can't imagine them appearing in any real applications.
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    $\begingroup$ Worth emphasis: the Gaussian integers are precisely the algebraic integers in the field of Gaussian rationals, and analogously for Eisenstein integers, etc. $\endgroup$ – Bill Dubuque Aug 13 at 3:46

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