Local rings are often characterized as rings that have the property that for every element x, either x or 1 − x is a unit.
This is equivalent to saying that for every element x, either x or 1 + x is a unit. The characterization in terms of addition seems more intuitive to me. Is there any reason that everyone seems to prefer the subtraction characterization?
Because $x + (1 - x) = 1$ so if $x \in \mathfrak{m}$ and $1 - x \in \mathfrak{m}$ then $1 \in \mathfrak{m}$ (which would be a contradiction). The addition happens between $x$ and $1 - x$ rather than between $x$ and $1$.
Two other reasons are that $x \mapsto 1 - x$ is an involution and that the formula for $(1 - x)^{-1}$ (say in a complete local ring) is much nicer:
$$ \frac{1}{1 - x} = 1 + x + x^2 + \dots.$$