# What do mathematicians call the Two's Complement on 8-bits group?

It is isomorphic to $\mathbb{Z}_{2^8},$ only difference is the symbols usually identifying the elements of the set are from $\{-128, \ldots, 127 \}$ and not $\{0, \ldots, 256\}.$

What is an elegant way to say this in math?

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One says that one is using the least absolute value residues. Not a particularly elegant phrase! – André Nicolas Mar 23 '12 at 20:37
It's also known as a balanced complete system of residues. – Bill Dubuque Mar 23 '12 at 21:03
I heard the phrase symmetric representation, specifically, in the Maple command mods which represents $\mathbb{Z}_m$ using $\{-\lfloor \frac{m - 1}{2} \rfloor, \ldots, +\lfloor \frac{m}{2} \rfloor\}$ rather than $\{0, 1, \ldots, m-1 \}.$ – user2468 Mar 24 '12 at 0:48

A group needs to satisfy the group axioms, so you need to define the group operation. Presumably you are using addition modulo 256. There is a natural correspondence between the two sets that takes $n \in [-128,127]$ to $\begin {cases} n+256 & n \lt 0 \\n & n \ge 0 \end {cases}$ and you can show that the group structure is maintained.
I think we're all in agreement that it is a group and that it's strongly related to $Z_{2^8}$. The question seems to be what to call this group. – Keith Irwin Mar 23 '12 at 20:42