Rate of Decay

The rate of decay of a radioelement -- also called its activity -- is defined as the number of nuclei of its atoms that decay per unit time. If $dN$ is the (small, negative) change in the number of undecayed nuclei over a short time interval $dt$, the rate of decay is written $-\frac{dN}{dt}$.

The minus sign is essential, not decorative: since the population of undecayed nuclei is shrinking with time, the raw quantity $\frac{dN}{dt}$ is itself negative. But 'rate of decay' is meant to be a positive, physically sensible quantity (you cannot have a negative number of disintegrations per second), so the minus sign is inserted in the definition specifically to flip that negative $\frac{dN}{dt}$ into a positive activity. …