For those who find the above so-called Grothendieck construction somewhat puzzling, here's a little motivation and background. It is the group completion of a monoid. The simplest case is the group completion of the free monoid on one generator. The monoid has as elements all the finite strings (including the empty string) built from the symbol x. Its operation is concatenation. Any such string is uniquely defined by the number of x symbols that occur, so this monoid is isomorphic to the natural numbers.

Its group completion must add an inverse for all elements. We will denote the formal inverse of x by the symbol x' with the relations xx' = x'x = 1 where 1 denotes the empty string. This cancellative concatenation is commutative because xx' and x'x both equal 1. Thus a string can be identified (though not uniquely) by a pair of natural numbers (m,n) that count the number of x and x' occurrences.

To see the connection with the sign representation of integers, add an orientation to the relations xx' = x'x = 1 to get the length-shortening reduction rules xx' -> 1, x'x -> 1. It's then easy to see that every string has a unique normal form that is either empty or consists entirely of x symbols or entirely of x' symbols, corresponding to the cases 0, +n and -n, respectively. For if a string is not of this form it must have at least one x and one x' element. But then there must be at least one adjacent pair of x and x' elements. Hence the string admits a reduction and cannot be a normal form.