Half-Life
Half-life is the time required for an exponentially decaying quantity to decrease to half of its original value.
Formula
T_{1/2} = \ln(2) / r
Definition
Half-life is the time it takes for a decaying quantity to shrink to exactly half of what it was; after each half-life you have half as much as before, and this is constant, it takes the same time to go from $80$g to $40$g as from $10$g to $5$g regardless of the starting amount. For exponential decay $A(t) = A_0 \cdot e^{-rt}$, the half-life satisfies $A(T_{1/2}) = A_0/2$, giving $T_{1/2} = \ln(2)/r \approx 0.693/r$, and the model can be rewritten $A(t) = A_0 \cdot (1/2)^{t/T_{1/2}}$. In terms of the decay constant $\lambda$ in $N(t) = N_0 e^{-\lambda t}$, $T_{1/2} = \ln(2)/\lambda$ and the mean lifetime is $\tau = 1/\lambda = T_{1/2}/\ln(2)$; quantum mechanically, unstable states have widths $\Gamma$ related to lifetime by the energy-time uncertainty $\Gamma \tau \sim \hbar$, making half-life measurable via spectral line widths.
Example
A radioactive substance with a $10$-year half-life starting at $80$g drops to $40$g after $10$ years, $20$g after $20$ years, and $10$g after $30$ years, halving every $10$ years. Carbon-14's half-life is $5730$ years, so after $11{,}460$ years (two half-lives) only $1/4$ of the original remains, meaning a sample with $25\%$ of its original C-14 is about $11{,}460$ years old. Nuclear decay chains, like U-238 ($T_{1/2} = 4.5$ billion years) decaying through a chain to Pb-206, can have the secular equilibrium of intermediates computed from their respective half-lives using systems of linear ODEs.
Key Insight
Half-life applies far beyond nuclear physics, in medicine (drug clearance), chemistry (reaction kinetics), and finance (depreciation), connecting the observable half-life to the underlying continuous decay rate $r$ via $T_{1/2} = \ln(2)/r$. The memoryless property of exponential decay (a constant hazard rate) distinguishes it from Weibull or log-normal failure distributions used in reliability engineering, where the failure rate changes over time.