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This is something I became aware of while reading a textbook about computer arithmetic (specifically the parts about 2's complement). The book was very non-formal about it, being a hardware textbook, but I conjectured what I think if a reasonable formalization of the idea. For the sake of argument I'll call them hypernatural numbers. I'll stick to the binary version but I'm pretty sure you could define them for an arbitrary base.

Definition: A (base 2) hypernatural number $A$ is a sequence $(A_0,\ A_1,\ ...)$ of binary digits such that for some natural $n$, all $A_i$ with $i \geq n$ are equal. ie, a hypernatural either has infinite leading 1s or infinite leading 0s.

Operations: For the sake of brevity I'll just say: imagine the hypernatural represents a binary number in positional notation, and use the elementary school algorithms to do hypernatural addition and multiplication. You could make this more rigorous by defining a rule to get $(A + B)_i$ and $(A \times B)_i$ as a function of the first $i + 1$ elements of $A$ and $B$. But hopefully you get the idea.

Isomorphism to the relative integers: Basically, the hypernaturals with leading 0s are isomorphic to the natural numbers (obviously), and the hypernaturals with leading 1s are isomorphic to the strictly negative integers. Let the bijection $B$ from leading-0 hypernaturals to naturals mean "interpret the sequence as a binary number". Then $B'(A) = -(B(\bar A) + 1)$ from leading-1 hypernaturals to negative integers, where the overbar is a logical NOT. The union of $B$ and $B'$ is an isomorphism from the hypernaturals to the relative integers under addition and multiplication.

(To convince yourself of this, let $A$ be the infinite sequence of 1s. If you apply the standard algorithm to add one to it, you carry the 1 infinitely and end up with an infinite sequence of 0s. So $A + 1 = 0$, and $A = -1$).

Anyway. This must have a name. What is it?

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How is this not the usual integers? – Zhen Lin Dec 23 '12 at 13:03
Just $\mathbb Z$ – Hagen von Eitzen Dec 23 '12 at 13:11
I suppose you could call that two's complement with infinite number of bits. – Karolis Juodelė Dec 23 '12 at 13:15
It becomes interesting once you remove the requirement that the sequence becomes eventually constant. In that case then what you have is the 2-adic numbers, $\mathbb{Z}_2$. – Zhen Lin Dec 23 '12 at 13:35
@JackM The p-adics can be defined for any natural $p \geq 2$. It's just that if $p$ is composite, there would be zero-divisors in $\mathbb{Z}_p$. – Shaun Ault Dec 23 '12 at 13:55
up vote 1 down vote accepted

Normally one does not give different names to isomorphic algebraic structures when they are constructed in different ways. So we call "complex numbers" a field that can be defined either by defining a multiplication on the $\def\R{\mathbf R}\R$-vector space $\R^2$, or as the quotient ring $\R[X]/(X^2+1)$, or as the centraliser of $\left({0\atop1}~{-1\atop0}\right)$, or in whatever other way that leads to an isomorphic field*. For the same reason one would not give a separate name to this particular manner of defining a ring isomorphic to $\mathbf Z$.

However in computer science one could describe this as unbounded length two's complement representation of the integers (or two's complement bigints). I think (though I haven't actually studied many implementations) that this representation is less popular than sign-magnitude bigints, due to the belief that multiplication and division are harder to define in two's-complement than with sign-magnitude. But having once implemented two's complement bigints in assembly language, has convinced me that this belief is mistaken; it actually requires fewer case distinctions than using sign-magnitude would.

*If however (rather unlikely) one could come up with a definition of the field $\mathbf C$ in a manner that avoids having a distinguished subfield to be identified with $\R$ (and so no distinguished automorphism of complex conjugation), then I would consider this a really different construction, probably worthy of a separate name.

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