This paper introduces the concept of e-variables and the numeraire e-variable, which are nonnegative random variables that satisfy specific properties in the context of statistical testing. The main result establishes that under no conditions on the null hypothesis $\mathcal{P}$ or the alternative $\mathbf{Q}$, there exists a unique strictly positive e-variable called the numeraire, denoted as $X^*$. This numeraire has the property that $\mathbb{E}_{\mathbf{Q}}[X/X^*] \leq 1$ for every other e-variable $X$. The numeraire is log-optimal, meaning $\mathbb{E}_{\mathbf{Q}}[\log(X/X^*)] \leq 0$. The paper also introduces the concept of the reverse information projection (RIPr), which is a sub-probability measure $\mathbf{P}^*$ defined via the density $d\mathbf{P}^*/d\mathbf{Q} = 1/X^*$. The RIPr is characterized by minimizing the relative entropy between $\mathbf{Q}$ and $\mathbf{P}^*$ over the effective null hypothesis $\mathcal{P}_{\text{eff}}$. The paper provides tools for finding the numeraire and RIPr in concrete cases and discusses their interpretations, including their role in optimal betting strategies and the absence of a reference measure. The results are extended to power utilities and reverse Rényi projections, providing a more general optimality theory.This paper introduces the concept of e-variables and the numeraire e-variable, which are nonnegative random variables that satisfy specific properties in the context of statistical testing. The main result establishes that under no conditions on the null hypothesis $\mathcal{P}$ or the alternative $\mathbf{Q}$, there exists a unique strictly positive e-variable called the numeraire, denoted as $X^*$. This numeraire has the property that $\mathbb{E}_{\mathbf{Q}}[X/X^*] \leq 1$ for every other e-variable $X$. The numeraire is log-optimal, meaning $\mathbb{E}_{\mathbf{Q}}[\log(X/X^*)] \leq 0$. The paper also introduces the concept of the reverse information projection (RIPr), which is a sub-probability measure $\mathbf{P}^*$ defined via the density $d\mathbf{P}^*/d\mathbf{Q} = 1/X^*$. The RIPr is characterized by minimizing the relative entropy between $\mathbf{Q}$ and $\mathbf{P}^*$ over the effective null hypothesis $\mathcal{P}_{\text{eff}}$. The paper provides tools for finding the numeraire and RIPr in concrete cases and discusses their interpretations, including their role in optimal betting strategies and the absence of a reference measure. The results are extended to power utilities and reverse Rényi projections, providing a more general optimality theory.