HALF-LIFE
N(k·t½) === N₀/2ᵏ bitwise — the law of halves as an identity, no dice anywhere.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
A thousand unstable nuclei and a stopwatch. Each real nucleus decays at random — but the population obeys a law with no randomness in it at all. This counter draws the law, not the dice: deterministic, like the muons, and disclosed as such.
The law of halves
N equals N₀ times two to the minus t over t-half: every half-life, exactly half. The clock variable x counts half-lives — and in x, iodine’s eight days and uranium’s four and a half billion years are the SAME curve. The nuclide only names the units.
Activity
The activity A equals lambda N — and it is identically minus dN/dt, the outflow equal to the bookkeeping rate. This is the circuits wing’s node law running on nuclei instead of charge. In SI it reads in becquerels; the curie is exactly 3.7 times ten to the ten of them, by definition.
Eight half-lives
The sweep runs x from zero to eight. Watch the staircase on the curve: at each integer the population is exactly half its last tick, and the grid extinguishes in its fixed order down to one part in 256.
Audit
Audited: the half-life ticks are exact — N at k half-lives equals N₀ over two to the k, bitwise, for k up to forty; activity matches minus dN/dt at three parts in ten billion; becquerel to curie round-trips bitwise at the defaults; and two to the minus x matches e to the minus x ln two at seven parts in ten to the sixteen — the rc capacitor and the damped spring, wearing a nuclear name tag.