How this calculator works
Half-life describes exponential decay: a quantity that loses a fixed fraction of itself over each fixed span of time, so the amount remaining keeps halving at regular intervals. It shows up in radioactive decay, drug elimination from the bloodstream, and any process where the rate of decrease is proportional to the amount left.
This calculator solves the half-life equation for whichever piece you don't already know. Give it the initial amount, elapsed time, and half-life to find what remains; give it the initial and remaining amounts plus the half-life to find how much time has passed; or give it the initial and remaining amounts plus the elapsed time to find the half-life itself.
The formula
N = N₀ × (1/2)^(t / t₁ᐟ₂)Solved for elapsed time: t = t₁ᐟ₂ × log₂(N₀ / N)Solved for half-life: t₁ᐟ₂ = t × log(2) / log(N₀ / N)N₀ is the initial amount, N is the amount remaining, t is elapsed time, and t₁ᐟ₂ is the half-life. Elapsed time and half-life must be in the same time unit (both in years, both in days, and so on) — the calculator does not convert between units for you.
Worked example: carbon-14 dating
- Carbon-14 has a widely published half-life of about 5,730 years.
- Start with 100 units. After exactly one half-life (5,730 years): N = 100 × (1/2)^(5730/5730) = 100 × 0.5 = 50.
- After exactly two half-lives (11,460 years): N = 100 × (1/2)^(11460/5730) = 100 × 0.25 = 25.
- Working backward, if a sample has decayed to 25% of its original carbon-14, roughly two half-lives — about 11,460 years — have passed.
Frequently asked questions
Does the amount ever reach exactly zero?
No. Exponential decay approaches zero but never mathematically reaches it — after enough half-lives the remaining amount becomes vanishingly small but stays positive. In practice, physical or measurement limits eventually make the remaining quantity undetectable.
Why is carbon-14 dating only reliable up to about 50,000 years?
After roughly eight to nine half-lives (around 50,000 years), so little carbon-14 remains that it becomes very hard to measure precisely, and the margin of error grows too large for confident dating. Other isotopes with longer half-lives are used for older samples.
Can half-life be used for things other than radioactivity?
Yes. The same math describes any quantity that decays at a rate proportional to how much is left, including drug concentration in the bloodstream (elimination half-life) and the decline of a capacitor's charge in some circuit models.
What happens if I enter a remaining amount larger than the initial amount?
The calculator rejects it. This equation models decay, so the remaining amount can never exceed the initial amount — a larger 'remaining' value would imply the quantity grew, which this formula does not represent.
Do the time units matter for the answer?
The unit itself doesn't matter, but elapsed time and half-life must use the same unit. If your half-life is in years, enter elapsed time in years too; mixing years with days will give a meaningless result.