Solve for remaining amount, half-life, or elapsed time.

Three questions, one equation

Exponential decay is N = N₀ × 0.5^t/T, where N₀ is the starting amount, T is the half-life and t is elapsed time. This calculator solves it in whichever direction you need, and the values it opens with check out: 100 units with a half-life of 10, after 20 elapsed — two half-lives — leaves 25.0000.

• Remaining amount (the default): N = N₀ × 0.5^t/T
• Half-life, from a measured decay: T = t × ln(0.5) ÷ ln(N∕N₀)
• Elapsed time, from how much is left: t = T × ln(N∕N₀) ÷ ln(0.5)

All three are the same relationship rearranged, which is why switching modes never changes the underlying answer — only which variable you are solving for.

Worked example: radiocarbon dating

Carbon-14 has a half-life of 5,730 years. A sample that retains 10% of its original carbon-14 is, by the time formula:

t = 5,730 × ln(0.10) ÷ ln(0.5) ≈ 19,035 years old.

That is the actual arithmetic behind radiocarbon dating headlines, and it explains why the method gets imprecise past roughly 50,000 years: by then the remaining fraction is so small — under 0.2% — that ordinary measurement noise swamps the signal.

Half-lives you meet outside a physics class

The same formula runs a 200 mg dose of caffeine (half-life roughly 5 hours) through your system:

Drug clearance, radioactive tracers in medical imaging, and capacitor discharge in electronics all follow the identical shape — only N₀ and T change.

Solving backwards: what the half-life mode is for

If you only have two measurements — say 100 units falling to 60 after 15 days — you can recover the half-life itself: T = 15 × ln(0.5) ÷ ln(0.6) ≈ 20.35 days. This is how half-lives get measured in the first place: nobody watches a full decay to zero, because exponential decay never actually reaches it (see below) — they measure the drop over a known interval and solve backwards.

Why the result never hits exactly zero

Each half-life removes half of what remains, not a fixed amount, so N in the formula shrinks toward zero without ever arriving there mathematically. In reality a physical sample is made of a finite number of atoms or molecules, so it does empty out once the count reaches one and that last unit decays — the smooth curve is an approximation that holds as long as the numbers involved are large.

What is half-life?

Half-life is the time it takes for a quantity undergoing exponential decay to fall to exactly half its starting amount. It's a fixed property of the decaying substance or process, no matter how much you start with — after one more half-life passes, whatever remains gets cut in half again.

What's the formula for radioactive or exponential decay?

N(t) = N0 × 0.5^(t / half-life), where N0 is the starting amount, t is the elapsed time, and half-life is the time it takes for the quantity to halve. The exponent t/half-life simply counts how many half-lives have gone by, and each one multiplies what's left by 0.5.

Does this only apply to radioactivity?

No — exponential decay with a well-defined half-life shows up anywhere a quantity shrinks by the same proportion over equal time intervals. Besides radioactive isotope decay, the same math describes how the body eliminates many drugs and medications over time, and capacitor discharge in electronics. Only the meaning of the quantity and the unit of time change; the formula itself stays identical.

Can I solve for time or half-life instead of remaining amount?

Yes — use the "Solve for" toggle above the inputs. Remaining Amount needs the initial amount, half-life, and elapsed time. Half-Life needs the initial amount, remaining amount, and elapsed time, solved as half-life = t × ln(0.5) / ln(remaining/initial). Elapsed Time needs the initial amount, remaining amount, and half-life, solved as t = half-life × ln(remaining/initial) / ln(0.5). All three modes are the same decay formula, just rearranged algebraically for the unknown you need.

Worked example: starting with 100 units at a half-life of 10 (days, years — any consistent unit), after 20 units of time (2 half-lives): remaining = 100 × 0.5^(20/10) = 100 × 0.25 = 25 units. Working backward, an initial amount of 100 that decays to 25 after 20 units of time implies a half-life of exactly 10, confirming the three modes agree with each other.