Classic formulae for entropy and cross entropy contain operations 0x and log2 x that are 1 not defined on all inputs. This can lead to calculations with problematic subexpressions such as 2 0 log2 0 and uncertainties in large scale calculations; partiality also introduces complications in logical 3 analysis. Instead of adding conventions, or splitting formulae into cases, we create a new algebra 4 of real numbers with two symbols ±∞, for signed infinite values, and a symbol named ⊥ for the 5 undefined. In this resulting arithmetic, entropy, cross-entropy, Kullback-Leibler divergence, and 6 Shannon divergence can be expressed without any further conventions concerning. The algebra may 7 form a basis for probability theory more generally.