Half-Life
Half-life is the time required for half of the atoms of a radioactive isotope in a sample to decay into a different isotope.
Formula
N(t) = N_0 \left(\tfrac{1}{2}\right)^{t/t_{1/2}}
Definition
Half-life is the amount of time it takes for half of the radioactive atoms in something to change into a different kind of atom: after one half-life, half are left, and after two, a quarter are left. Written $t_{1/2}$, it is constant for each isotope and unaffected by temperature, pressure, or chemical state, and because the parent-to-daughter ratio changes predictably, it is the basis for radiometric dating; after $n$ half-lives, the fraction of parent left is $\left(\tfrac{1}{2}\right)^n$. Half-life follows from first-order decay kinetics, $N(t) = N_0 e^{-\lambda t}$, giving $t_{1/2} = \frac{\ln 2}{\lambda}$. Decay is a quantum-mechanical process with a fixed probability per atom per unit time, so individual decays are random while the bulk behavior is exactly exponential. Half-lives span from fractions of a second to more than $10^{15}$ years, and the useful range for dating a sample is roughly $0.1$ to $10$ half-lives, outside which the parent or daughter becomes too scarce to measure precisely.
Example
Carbon-14 has a half-life of $5{,}730$ years, so if a bone starts with $100$ carbon-14 atoms, about $50$ remain after $5{,}730$ years and about $25$ after $11{,}460$ years. A rock sample with $1$ parent atom for every $3$ daughter atoms has one quarter of the parent remaining, so two half-lives have passed; if the isotope is potassium-40 with a half-life of $1.25$ billion years, the rock is $2.5$ billion years old. Carbon-14 decays by beta emission to nitrogen-14 with $t_{1/2} = 5{,}730$ years, corresponding to $\lambda = 1.21 \times 10^{-4}\ \text{yr}^{-1}$. A sample retaining $12.5\%$ of its original carbon-14 is $3$ half-lives, or about $17{,}190$ years, old.
Key Insight
Half of the remaining atoms decay each half-life, so the amount never quite reaches zero; it just keeps getting halved. Pick the isotope to fit the age: carbon-14 is useless for dinosaur bones because after $66$ million years (over $11{,}000$ half-lives) not a single original atom would remain. Half-life is a statistical statement, not a promise about any single atom. One carbon-14 atom might decay tomorrow or in $50{,}000$ years, but a sample of $10^{20}$ atoms halves on a schedule you could set a clock by.