The short answer
Half-life is the time it takes for half of a quantity to decay. Remaining amount follows N(t) = N₀ × (1/2)^(t/t½) — after one half-life, 50% remains; after two, 25%; after three, 12.5%. This calculator solves that formula for whichever value you don't know: remaining amount, elapsed time, initial amount, or the half-life itself.
Key takeaways
- Decay is exponential, not linear — the quantity never reaches exactly zero, it just gets closer and closer with each half-life.
- Half-life and the decay constant (λ) describe the same rate two different ways: λ = ln(2) ÷ half-life.
- Doubling the elapsed time doesn't double the amount lost — after 2 half-lives, 25% remains, not 0%.
- The same math models radioactive decay, drug elimination from the body, and any process where a constant fraction disappears per unit of time.
How much remains after each half-life
| Half-lives elapsed | Percent remaining |
|---|---|
| 0 | 100% |
| 1 | 50% |
| 2 | 25% |
| 3 | 12.5% |
| 4 | 6.25% |
| 5 | 3.125% |
Worked example: carbon-14 dating
A sample retains 25% of its original carbon-14. Carbon-14's half-life is about 5,730 years. How old is the sample?
25% = (1/2)² — exactly 2 half-lives have passed
t = 2 × 5,730 years = 11,460 years
When the remaining percentage is a clean power of 1/2, you can skip the logarithm entirely and just count half-lives. For messier percentages, the general formula t = -ln(N/N₀) ÷ λ (used internally by this calculator) handles any remaining fraction.
Half-life vs. decay constant
λ = ln(2) / t₁/₂ ≈ 0.6931 / t₁/₂
Half-life (t½) is the more intuitive number — "time for half to disappear" — while the decay constant (λ) is what actually plugs into the underlying exponential formula, N(t) = N₀ × e^(-λt). They describe the exact same physical process; picking one over the other is purely a matter of which is more convenient for the problem at hand.
Common mistakes to avoid
- Treating decay as linear — assuming 100 units with a 10-year half-life means 0 units left after 20 years, when the real answer is 25 units (2 half-lives).
- Mixing time units between fields — entering half-life in years but elapsed time in days produces a meaningless result.
- Expecting the decayed quantity to reach exactly zero — exponential decay approaches zero asymptotically but mathematically never quite arrives.
- Confusing half-life with mean lifetime (1/λ) — mean lifetime is about 44% longer than half-life and is used in different contexts, mainly physics.
Related calculators
- Log Calculator — work directly with the natural logarithm behind the decay formula.
- Exponent Calculator — explore the exponential function that half-life is built on.
- Percentage Calculator — convert between remaining amount and percent remaining.
- Scientific Notation Calculator — work with the very small quantities left after many half-lives.