Enter an initial amount, a half-life duration, and an elapsed time to see the remaining amount and the number of half-lives elapsed update as you type.
| Half-Lives | % Remaining | Amount (x0 = 800) |
|---|---|---|
| 1 | 50% | 400 |
| 2 | 25% | 200 |
| 3 | 12.500% | 100 |
| 4 | 6.250% | 50 |
| 5 | 3.125% | 25 |
What the table shows: starting from 800 units, one half-life brings the amount down to 400 (50%), and each additional half-life cuts what remains in half again, 200 (25%) after two, 100 (12.5%) after three, 50 (6.25%) after four, and 25 (3.125%) after five. The default calculator settings above land exactly on the third row: a 6-unit half-life applied for 18 units of elapsed time is 3 half-lives, leaving 100 of the original 800 units.
Half-life is the fixed amount of time it takes for a quantity to fall to exactly half of its current value, no matter what that current value happens to be.
A sample that starts at 640 grams reaches 320 grams after one half-life. That same 320-gram sample then reaches 160 grams after another half-life of exactly the same length. The timer resets against whatever amount is left, not against the original 640 grams. This is what separates half-life from a countdown to a fixed date: it measures a proportional drop, and proportional drops behave the same way regardless of scale.
Because half-life is independent of the starting amount, it works as a fixed, reusable number for describing a decaying process. Whether a radioactive isotope sample weighs a milligram or a kilogram, the same half-life applies to both, since each is simply losing a fixed proportion of whatever mass currently remains.
Half-life has a mirror-image counterpart on the growth side of the same mathematics: doubling time. Where half-life asks how long it takes a shrinking quantity to fall to half its value, doubling time asks how long it takes a growing quantity to reach twice its value. Both are fixed time constants derived from the same underlying rate, just read in opposite directions.
Half-life is calculated from a decay constant with t½ = ln(2)/λ, and the remaining amount at any time is x(t) = x0 × 0.5(t/t½). The two formulas cover opposite directions of the same relationship: one finds the half-life itself, the other finds how much is left after any elapsed time.
Use this when a continuous decay constant λ is already known and the half-life itself needs to be found.
Use this when the half-life is already known and the goal is the amount left after a given elapsed time.
| Symbol | Name | What it represents | Example |
|---|---|---|---|
| x(t) | Remaining amount | The quantity left after t time units have passed. | 100 units |
| x0 | Initial amount | The quantity at t = 0, before any decay. | 800 units |
| t½ | Half-life | The fixed time needed for the quantity to fall by half. | 6 time units |
| λ | Decay constant | The continuous per-unit-time rate the half-life is derived from. | 0.1155 per unit |
| t | Elapsed time | How much time has passed since t = 0. | 18 time units |
What the table shows: only x0, t½, and t are ever typed in by hand. x(t) is the computed result, and λ is derived from the half-life through λ = ln(2)/t½ rather than entered directly. A 6-unit half-life corresponds to a decay constant of ln(2)/6 ≈ 0.1155 per unit, and that same constant would in turn produce back a half-life of ln(2)/0.1155 ≈ 6 if run through the formula the other way.
To calculate the remaining amount from a half-life, divide the elapsed time by the half-life to get the number of half-lives, raise 0.5 to that power, then multiply by the initial amount.
Identify x₀, the quantity present before any decay has taken place.
Identify t½, the fixed amount of time the process takes to halve, in whatever unit the elapsed time will also be measured in.
Compute t/t½ to get the number of half-lives that have passed. This value does not need to be a whole number.
Compute 0.5^(t/t½) to get the fraction of the original amount remaining.
Multiply x₀ by the result of step 4 to get x(t), the remaining amount.
A manufacturing plant tracks the activity of a catalyst sample rated at 900 activity units, with a known effectiveness half-life of 25 process-days. The half-life formula gives the remaining activity after 40 process-days.
After 40 process-days the catalyst retains 33.0% of its original activity. 1.6 half-lives have elapsed, which is more than one and a half full halvings but not yet two, exactly matching a remaining fraction between 25% and 50%.
Doubling time and half-life are mirror-image time constants: one measures how long a growing quantity takes to double, the other measures how long a shrinking quantity takes to halve. Both come from the same exponential formula, just applied with rates of opposite sign.
Because they share one underlying equation, converting between them is direct: a process with a decay constant λ has a half-life of ln(2)/λ, and a process with the equivalent growth rate has a doubling time of ln(2) divided by that growth rate. The doubling-time counterpart tool handles the growth-side version of this same relationship, including converting a periodic growth rate straight into a doubling time.
The practical difference shows up at the far ends of the timeline. Doubling time describes a quantity with no ceiling. Each doubling produces a larger absolute increase than the one before it. Half-life describes a quantity approaching a floor of zero that it never quite reaches. Each halving produces a smaller absolute decrease than the one before it. Anyone comparing the two directly can see this asymmetry using the site's main exponential growth calculator, which produces a rising doubling curve for a positive rate and a falling half-life curve the moment the rate turns negative.
Half-life is the fixed amount of time it takes for a quantity to fall to exactly half of whatever value it currently has. It is a property of the process itself, not of any particular starting amount, so the same half-life applies whether the quantity begins at 10 units or 10 million units.
After one half-life, half the original amount remains; after two half-lives, a quarter remains; the pattern keeps compounding rather than resetting.
The half-life formula is x(t) = x₀ × 0.5^(t/t½), where x₀ is the starting amount, t½ is the half-life, and t is elapsed time. Dividing elapsed time by the half-life gives the number of half-lives that have passed, and raising 0.5 to that power gives the fraction remaining.
Half-life also connects to a decay constant λ through t½ = ln(2)/λ, so either value can be derived from the other.
A quantity governed by exponential decay never reaches exactly zero, no matter how many half-lives pass. It only gets closer and closer to zero. After 10 half-lives less than 0.1% of the original amount remains, and after 20 half-lives less than 0.0001% remains, but the mathematical model never lands on a true zero.
In practice, once the remaining amount drops below one countable or detectable unit, it is treated as effectively gone even though the formula still returns a small positive number.
Half-life and the decay constant are reciprocally linked through t½ = ln(2)/λ, and equivalently λ = ln(2)/t½. A larger decay constant means faster decay and therefore a shorter half-life; a smaller decay constant means slower decay and a longer half-life.
The decay rate calculator converts between a periodic decay rate, a continuous decay constant, and a half-life in either direction.
No. Half-life does not depend on the starting amount at all. A sample of 1,000 grams and a sample of 10 grams of the same decaying substance both take exactly the same amount of time to fall to half their own respective starting masses.
This is what makes half-life useful as a fixed descriptive number for a process, rather than a value that has to be recalculated for every new quantity.
Divide the elapsed time by the half-life to get the number of half-lives, then raise 0.5 to that power and multiply by the starting amount. For 40 days elapsed against a 25-day half-life, that is 40 ÷ 25 = 1.6 half-lives, so the remaining fraction is 0.5^1.6.
The elapsed time does not need to be a whole multiple of the half-life. The formula works for any positive value of t, including fractional numbers of half-lives.
No. Half-life is not the same as an expiration date, because a decaying quantity never reaches zero on a fixed schedule the way an expiration date implies. Half-life describes a continuous, ongoing proportional loss, while an expiration date is a chosen threshold set by someone deciding when too little of the original quantity remains.
A product label might use several half-lives as a rule of thumb for setting a shelf date, but the two concepts measure different things.
Real-world examples of half-life include the radioactive decay of isotopes like carbon-14, the loss of a drug's concentration in the bloodstream over time, and the fading strength of a wireless signal's stored charge. Each case involves a quantity that loses a fixed proportion of whatever remains over each equal time interval.
The exponential decay calculator models the same falling curve directly from a percentage rate instead of a half-life value.