Half-Life
Functions & Advanced AlgebraHalf-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.
Example
If a radioactive substance has a half-life of $10$ years: start with $80$g. After $10$ years: $40$g. After $20$ years: $20$g. After $30$ years: $10$g. It keeps halving every $10$ years.
Key Insight
Half-life is constant: it takes the same amount of time to go from $80$g to $40$g as from $10$g to $5$g. The half-life does not change based on how much you start with.
Definition
For exponential decay $A(t) = A_0 \cdot e^{-rt}$, the half-life $T_{1/2}$ satisfies $A(T_{1/2}) = A_0/2$, giving $T_{1/2} = \ln(2)/r \approx 0.693/r$. The model can also be written $A(t) = A_0 \cdot (1/2)^{t/T_{1/2}}$.
Example
Carbon-14 has $T_{1/2} = 5730$ years. After $11{,}460$ years (two half-lives): $(1/2)^2 = 1/4$ of original remains. If a sample has $25\%$ of original C-14, it is about $11{,}460$ years old.
Key Insight
Half-life applies in medicine (drug clearance), nuclear physics, chemistry (reaction kinetics), and finance (depreciation). The formula $T_{1/2} = \ln(2)/r$ connects the observable half-life to the underlying continuous decay rate $r$.
Definition
Half-life $T_{1/2} = \ln(2)/\lambda$ where $\lambda$ is the decay constant in $N(t) = N_0 e^{-\lambda t}$. The mean lifetime is $\tau = 1/\lambda = T_{1/2}/\ln(2)$. Quantum mechanically, unstable states have widths $\Gamma$ related to the lifetime by the energy-time uncertainty: $\Gamma \tau \sim \hbar$ (natural units), making half-life measurable via spectral line widths.
Example
Nuclear decay chains: U-238 ($T_{1/2} = 4.5$ billion yr) decays through a chain to Pb-206. The secular equilibrium of intermediates can be computed from their respective half-lives using systems of linear ODEs.
Key Insight
The memoryless property of exponential decay (equivalent to a constant hazard rate) distinguishes it from Weibull or log-normal failure distributions used in reliability engineering, where the failure rate changes over time.