How it works
Half-life describes decay that halves at a constant rate: after one half-life, half the original amount remains; after two, a quarter; after three, an eighth. One equation captures this, and it rearranges four ways depending on what you already know.
remaining = initial × 0.5^(elapsed time ÷ half-life)
initial = remaining × 2^(elapsed time ÷ half-life)
elapsed time = half-life × log₂(initial ÷ remaining)
half-life = elapsed time ÷ log₂(initial ÷ remaining)
This works for anything that decays exponentially at a fixed rate — radioactive isotopes, drug concentration in the body, or any other quantity that loses a constant fraction over each equal time step.
Worked example
100 units of a substance with a 10-day half-life, after 20 days.
- Divide elapsed time by half-life: 20 ÷ 10 = 2 half-lives.
- Halve the initial amount that many times: 100 × 0.5² = 100 × 0.25.
The remaining amount is 25.
Common questions
Does the amount ever reach exactly zero?
Mathematically, no — each half-life only ever removes half of what's left, so the curve approaches zero without ever quite touching it. In practice, once the amount is small enough it's treated as negligible or undetectable.
Can I use this for radiocarbon dating?
Yes — solve for elapsed time using carbon-14's half-life of about 5,730 years, the estimated initial amount, and the amount remaining today.
Why must remaining be less than initial?
This calculator models decay only, where the amount shrinks over time. If the remaining amount isn't smaller than the initial amount, there's no valid decay time or half-life to solve for.