Is this extension a field? Or perhaps some other structure?
The extension depends on two basic ideas:
- A definition for a unit of infinity, the same as one given by Roger Penrose, and
- The infinity unit, like the imaginary $i$, forming a new number when paired with a real number. These pairs may be referred to as array numbers. They do not, therefore, follow rules similar to transfinite or hyperreal arithmetic.
Definition: A unit of infinity is represented by the infinity symbol "$\infty$" and indicates a single line array of the real numbers. $\infty^{n}$ indicates an $n$-dimensional array. $\infty$ is not a number and is unsigned. The exponent $n$ is either an integer or $\infty$.
Comment: Penrose uses this as a notation (not a number system) for $n$-real-dimensional space where $\infty^{n}$ "expresses that this continuum of points is organized in an $n$-dimensional array" (Road to Reality, 2004, chapter 16.7, p. 379). Here, unlike with Penrose, $n$ can be a non-positive integer.
Arithmetic rules: Since the $\infty$ unit always has a real part, arithmetic rules for the array numbers are the same as for polynomials, and not, as mentioned above, transfinites or hyperreals. So for example,
- $x$ + $x{\infty^{1}}$ $\cdots$ + $x{\infty^{n}}$ is irreducible
- $x{\infty^{n}}$ + $y{\infty^{n}}$ = $(x$ + $y){\infty^{n}}$
- $x{\infty^{n}}$ $\cdot$ $y{\infty^{p}}$ = $(x$ $\cdot$ $y){\infty^{n+p}}$
- and so on
Note that the reals are a subset of the array numbers when $n = 0$.