Scope and assumptions
Each half-life halves the amount; decay per half-life = 50%. Illustrative.
Something loses half its amount, then half of what's left, then half again — three half-lives in and you're down to one-eighth, not the 150% you'd guess from adding up. The math is just start times one-half to the power of n, and it shows up anywhere a thing decays by repeated halving.
The core relation
Remaining = Start × (1/2)^n (n = number of half-lives elapsed)
Each step removes 50% of whatever is present at that moment. Because you're always halving the current remainder, the amount left after a few steps looks like this:
| Start | Half-lives | Math | Remaining |
|---|---|---|---|
| 100 g | 1 | 100 × 1/2 | 50 g |
| 100 g | 2 | 100 × 1/4 | 25 g |
| 100 g | 3 | 100 × 1/8 | 12.5 g |
| 100 g | 4 | 100 × 1/16 | 6.25 g |
Why it only approaches zero
Each step cuts the remainder in half, so the value gets arbitrarily close to zero but never lands on it in a finite number of steps. After 3 half-lives, 12.5% remains — the substance isn't "150% gone," it's one-eighth still here. That asymptotic tail is what makes half-life useful for dating and dosing: you can always say "this much is left," never "it's exactly finished."
A rate is not a half-life
Half-life is a clock (e.g., 5 years); the rate depends on that clock, and the two don't convert by simple division. Suppose a material loses 10% of its mass every year. The naive guess is "10% a year, so half in 5 years" — but that's linear thinking. The real half-life solves:
0.5 = (1 − 0.10)^n → n = ln 0.5 ÷ ln 0.9 ≈ 6.58 years
Because each year's loss is taken from a smaller base, reaching 50% gone takes noticeably longer than five years. To flip it around, a 5-year half-life implies a yearly loss of only 1 − (1/2)^(1/5) ≈ 12.9%, not a flat 20%.
A drug's half-life on the clock
A medicine with a 4-hour half-life: take 100 mg, after 4 hr 50 mg remain, after 8 hr 25 mg, after 12 hr 12.5 mg. Dosing schedules are built so the next dose arrives before the level drops too low — half-life math, not guesswork. The same relation answers "when is it mostly cleared?": after about 5 half-lives (here 20 hours) roughly 97% is gone.
Half-Life Conversion Traps
- Adding the 50%s — three half-lives is 87.5% gone (12.5% left), never 150%.
- Subtracting a flat amount — decay is multiplicative, not linear; a fixed "−50 g" breaks down.
- Treating half-life as a rate — a 5-year half-life is a clock, not "20% per year"; the true yearly rate is ~12.9%.
- Forgetting the exponent — remaining uses (1/2)^n, not 1 − n/2.
- Mixing time units — the half-life period must match the elapsed-time unit (hours vs years).
Related growth & decay guides
For the rest of the growth-and-decay cluster, start at the hub: Population Growth Percentage. Adjacent: Percentage Decay Calculation and Percentage Change Over Time.
FAQ
How do I get the half-life from a decay rate? If a substance loses 10% per year, the half-life isn't 5 years (that would be linear). Solve 0.5 = (1 − 0.10)^n, so n = ln 0.5 ÷ ln 0.9 ≈ 6.58 years. The multiplicative path is longer than the naive divide.
What's the difference between half-life and mean lifetime? Mean lifetime τ = t½ ÷ ln 2 ≈ 1.44 × t½. Half-life is the time to reach 50% remaining; mean lifetime is the average time a single atom survives. Two different clocks for the same decay.
Can I use a "third-life" or other fractions? Yes. Remaining = start × (1/f)^n for any fraction f after n such intervals, or more generally remaining = start × e^(−λt) with λ the decay constant. Half-life is just the f = 2 special case.
Does half-life apply to growing things too? Growth uses doubling time — the same math with a factor of 2 instead of 1/2. Radioactive decay halves; bacterial growth doubles. Both are exponential with a fixed fractional step.
Sources: Khan Academy, Half-life; Investopedia.
Disclaimer: Illustrative. Real half-lives are measured physically/chemically. Informational only.
Sources and references
- Khan Academy — Half-lifehttps://www.khanacademy.org/math
- Investopedia — Half-lifehttps://www.investopedia.com/
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