Exponential Decay (Half-Life) Calculator
Half-life is the natural language for anything that fades by a constant fraction over time, from radioactive isotopes to medicines clearing the body to signals dropping off. This calculator tells you how much of a decaying quantity remains after a given time. Enter the starting amount, the half-life (the time for the quantity to fall by half), and the elapsed time, and the tool computes how many half-lives have passed, raises one-half to that power, and multiplies by the starting amount to give the amount remaining. The key insight is that decay is multiplicative, not subtractive: each half-life removes half of whatever is left, so after one half-life half remains, after two a quarter, after three an eighth. The elapsed time need not be a whole number of half-lives; the formula handles partial half-lives exactly. Use the same time unit for the elapsed time and the half-life so their ratio is unitless, and the remaining amount comes out in the same units as the starting amount. The quantity approaches zero but never quite reaches it. Every figure is computed deterministically from the formula shown below, with a worked example that reconciles exactly to the calculator's defaults so you can follow each step.
The amount remaining after exponential decay is N = N0 x (1/2)^(t / T). Starting from 100 with a half-life of 15 and an elapsed time of 30 (two half-lives), the amount remaining is 25.00.
Half-life decay formula
N = N0 x ( 1/2 )^( t / T )
N0 = starting amount at time zero
T = half-life (time to fall by half)
t = elapsed time (same unit as T)
t / T = number of half-lives that have passed
Divide the elapsed time by the half-life to get the number of half-lives, raise one-half to that power, and multiply by the starting amount. The result is what remains.
Worked example
Suppose you start with 100 units, the half-life is 15, and 30 units of time have passed.
- Half-lives elapsed = t / T = 30 / 15 = 2
- Fraction remaining = (1/2) raised to the power 2 = 0.25
- Amount remaining = 100 x 0.25 = 25.00
The amount remaining is 25.00. These are the calculator's default inputs, so the result above matches the widget exactly.
Fraction remaining by half-lives
How much of the starting amount remains after each number of half-lives.
| Half-lives | Fraction remaining |
|---|---|
| 1 | 0.5000 |
| 2 | 0.2500 |
| 3 | 0.1250 |
| 4 | 0.0625 |
Each half-life leaves half of whatever amount remained before it.
Exponential decay half-life calculator: frequently asked questions
What is half-life?
Half-life is the time it takes for a decaying quantity to fall to half its value. After one half-life, half remains; after two, a quarter; after three, an eighth, and so on. It applies to radioactive isotopes, drug concentrations in the body and any process that decays by a constant fraction over equal time intervals.
What is the half-life decay formula?
The remaining amount is N = N0 times (1/2) raised to the power (t divided by T), where N0 is the starting amount, t is the elapsed time, and T is the half-life. The exponent t over T is the number of half-lives that have passed. The calculator raises one-half to that power and multiplies by the starting amount.
Why does the amount never reach zero?
Exponential decay halves the remaining amount each half-life, so it approaches zero but never exactly reaches it in this model. After ten half-lives less than 0.1 percent remains, which is negligible in practice. The mathematics describes a smooth curve that gets ever closer to zero without crossing it.
Can I enter a fractional number of half-lives?
Yes. The elapsed time and half-life can be any positive values, and their ratio need not be a whole number. If t is 1.5 times T, then 1.5 half-lives have passed and the formula raises one-half to the 1.5 power. The calculator handles partial half-lives exactly using the power function.
What units should I use?
Use the same time unit for the elapsed time and the half-life, whether seconds, hours, days or years, so their ratio is unitless. The starting amount can be in any unit (grams, milligrams, counts, percent) and the remaining amount comes out in the same unit. Only the ratio of time to half-life affects the fraction remaining.
Official sources
- Exponential functions and decay reference: US National Institute of Standards and Technology (NIST). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.