appetizing analogy. For a long time, the standard interpretation of numbers was quantity. It is absurd to talk about the quantity of $3e^{i\pi/12}$. Formally the complex numbers allow us to “solve” equations like $x^2=-1$ but, one might argue, the term “number” loses its meaning.

The mathematical trick of introducing complex numbers paid off richly. Some hard number-theoretic problems have been solved using the methods of complex analysis. Eventually it even became possible to give a physical meaning to complex numbers, e.g., a complex number $a+bi$ can be interpreted as impedance where the real part $a$ is resistance and the imaginary part $b$ is the reactance [@W-EI], and complex numbers of absolute value $\le1$ can be interpreted as quantum amplitudes.

The original purpose of complex numbers was to solve certain algebraic equations with real coefficients. What is the corresponding purpose of negative probabilities? What plays the role of algebraic equations? These are the questions addressed in this paper. Our proposal is admittedly — and provably — limited. But it supports some of the usages of negative probabilities in the literature. One has to start somewhere.

For simplicity and to separate concerns, we restrict attention to finite spaces.

Signed probability spaces {#sec:prob}
