In standard computer arithmetic, there are two sets of numbers.
N-bit unsigned numbers. The elements are natural numbers in $(0, 2^N]$. Arithmetic operations is defined as for the natural numbers except all operations are modulo $2^N$.
N-bit signed numbers. The elements are integers in $(-2^{N-1}, 2^{N-1}]$. Arithmetic operations rules for complement [1] [2] arithmetic are more complex than a simple modulo.
- For example, $-2^{N-1} = --2^{N-1} = 2^{N-1}-1 + 1$.
What are these two sets (and their elements) called?
I understand these sets of numbers and their corresponding operational semantics completely, but I want to know how these sets and their elements are correctly named from a mathematical perspective. For example: the generic concept "unsigned numbers" and their arithmetic operations is more correctly (or usually) called something like "the field of natural numbers".