Half-Life & Decay Calculator
Radioactive decay from a half-life — where the average atom lives 44% longer than the half-life and ten half-lives is 1/1024, not a thousandth.
The half-life and the elapsed time just have to share a unit — hours, years, whatever you like. This is the continuous exponential law, which describes a large sample well and a handful of atoms not at all: decay is random, and with ten atoms left the curve is a probability rather than a prediction. Nothing is uploaded.
The average atom lives 44% longer than the half-life
Half the atoms are gone by the half-life — but decay has no memory, so the survivors are exactly as likely to last another one as the original sample was. The tail runs a long way, and the mean lifetime works out at 1/ln 2 = 1.4427 half-lives, about 44% longer than the figure everyone quotes.
| Measure | Time | In half-lives |
|---|---|---|
| Half-life | 12.00 | 1.0000 |
| Mean lifetime | 17.31 | 1.4427 |
| Time to 10% left | 39.86 | 3.3219 |
| Time to 1% left | 79.73 | 6.6439 |
| Time to 0.1% left | 119.59 | 9.9658 |
The decay constant underneath all this is ln 2 over the half-life, and its reciprocal is the mean lifetime — the same relationship read from the other end. Nothing here ever reaches zero, which is why every threshold is a choice of fraction rather than an end date.
And ten half-lives is 1/1024, not a thousandth
Every step is exactly one over a power of two, so ten of them is 1 in 1024 — 0.0977%. That is slightly less than a thousandth, so the familiar rule overstates what is left, and a true thousandth actually arrives at 9.97 half-lives, just before ten rather than after.
| Half-lives | Remaining | Exactly | Gone |
|---|---|---|---|
| 1 | 50.0000% | 1 / 2 | 50.0000% |
| 2 | 25.0000% | 1 / 4 | 75.0000% |
| 3 | 12.5000% | 1 / 8 | 87.5000% |
| 5 | 3.1250% | 1 / 32 | 96.8750% |
| 7 | 0.7813% | 1 / 128 | 99.2188% |
| 10 | 0.0977% | 1 / 1,024 | 99.9023% |
| 20 | 0.0001% | 1 / 1,048,576 | 99.9999% |
The discrepancy is under three percent, which is nothing in conversation and worth a glance when the number is a safety margin rather than a figure of speech. The powers of two are the exact thing; the round decimals are the approximation, which is the reverse of how the rule usually gets taught.
How to use
- Enter the half-life in any unit you like.
- Enter the time elapsed in the same unit.
- Optionally enter a starting amount.
- Read what is left, and how long each threshold takes.
Frequently asked questions
How do I calculate what is left after decay?
One half to the power of the elapsed time over the half-life. After three half-lives an eighth is left, after ten it is 1/1024 — every step is exactly one over a power of two.
Is the half-life the average lifetime?
No, and this is the thing most worth knowing. The mean lifetime is the half-life divided by ln 2, which is 1.4427 times longer — about 44%. Half the atoms are gone by the half-life, but the survivors keep going and the tail is long.
Why is the mean longer than the half-life?
Because decay has no memory. An atom that has survived ten half-lives is exactly as likely to last another one as a fresh atom was, so there is no ageing to bring the tail in. That memorylessness is what stretches the average.
Is ten half-lives really a thousandth?
Not quite — it is 1/1024, or 0.0977%. That is slightly LESS than a thousandth, so ten half-lives has already gone a little past the mark and a true thousandth arrives at about 9.97 half-lives.
Does that difference matter?
Under three percent, so almost never in conversation. It is worth a glance when the figure is a safety margin rather than a figure of speech — the powers of two are the exact thing and the round decimals are the approximation.
What is the decay constant?
ln 2 divided by the half-life — the fractional loss per unit time, which is what the physics actually contains. Its reciprocal is the mean lifetime, so the two quantities are the same fact from opposite ends.
Does anything ever fully decay?
Not on this curve. Exponential decay approaches zero without reaching it, which is why every practical threshold is a choice of fraction rather than an end date. With a handful of atoms left the curve is a probability, not a prediction.
Does this send anything anywhere?
No. Every figure is computed in your browser, and nothing is uploaded or stored.
🔒 This tool runs entirely in your browser. Nothing you enter is uploaded, logged, or stored.