Exponential growth and decay
Introduced| English | Meaning |
|---|---|
| exponential growth/ˌekspəˈnenʃl ɡrəʊθ/ | An increase in which a quantity is multiplied by the same factor greater than one over equal intervals. |
| compound/ˈkɒmpaʊnd/ | Describing interest calculated on both the original principal and interest previously added to it. |
| decay/dɪˈkeɪ/ | A decrease in a quantity over time, often modelled by an exponential relationship. |
| depreciation/dɪˌpriːʃɪˈeɪʃn/ | A reduction in the value of an asset over time, often modelled as a percentage decrease. |
| half-life/hɑːf laɪf/ | The time for the number of undecayed nuclei in a radioactive sample to halve. |
One bacterium becomes a billion
- A bacterium divides every 20 minutes. After 10 hours (30 divisions), one cell becomes $2^{30} \approx 1\,073\,741\,824$ — over a billion.
- That's exponential growth 指数增长: the same percentage increase every period, compounding 复利.
- Linear growth adds a fixed amount; exponential growth multiplies — and it always wins in the end.
The compound formula
- When a quantity changes by the same percentage each period:
- $P$ = starting value, $r$ = percentage rate per period, $t$ = number of periods.
- Growth: $r > 0$; decay 衰减: $r < 0$ (or use a multiplier below 1).
Exponential growth (curve) starts slowly but eventually pulls far ahead of linear growth (straight line).
Compound interest
Money grows by (1 + r) every year, so compound interest curves above simple interest. Drag the rate and the number of years.
Exponential growth & decay
y = a·bˣ
Change the base b: b > 1 grows, 0 < b < 1 decays — useful for interest and populations.
A car loses 15% of its value each year. What yearly multiplier do you use?
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Losing 15% leaves 85%, so the multiplier is 0.85.
Depreciation of a car's value each year is an example of:
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Losing a fixed percentage each year is exponential decay (multiplier < 1, applied repeatedly).
The compound growth formula is: final = P × (1 + ______)^t, where r is the percentage rate.
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The rate r% must be converted to a decimal: r/100. So the multiplier is (1 + r/100).
Exponential decay and depreciation 折旧
- Decay works the same way with a multiplier below 1.
- A car worth 20 000 loses 15% each year → multiplier $0.85$.
- After 4 years: $20\,000 \times 0.85^4 = 20\,000 \times 0.5220 \approx 10\,440$.
Half-life 半衰期. A radioactive isotope decays 10% per year. Its "half-life" (time to halve) is $\dfrac{\ln 2}{\ln(1/0.9)} \approx 6.6$ years — you can find it by solving $0.9^t = 0.5$.
A car worth 20000 depreciates 15% per year. Its value after 4 years (to the nearest 10) is:
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20000 × 0.85⁴ = 20000 × 0.5220 ≈ 10440.
Worked example
- Population of 5000 grows 8% per year. After 6 years:
- $5000 \times 1.08^6 = 5000 \times 1.5869 \approx 7934$.
Compound ≠ simple. 8% per year for 6 years is not $8 \times 6 = 48\%$ total (that would give $5000 \times 1.48 = 7400$). Compounding gives $58.7\%$ total — almost 11% more, because each year's growth earns growth too.
A population of 5000 grows 8% per year. What is the population after 6 years (nearest whole number)?
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5000 × 1.08⁶ = 5000 × 1.5869 ≈ 7934.
8% growth per year for 6 years gives a total increase of exactly 48%.
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Compounding gives 1.08⁶ ≈ 1.587, a 58.7% increase — more than 48% because growth earns growth.
Where exponential growth shows up
- Epidemics: early spread is often exponential (each infected person infects others).
- Investments: compound interest is exponential growth of your money.
- Technology: Moore's Law predicted computing power doubling every ~2 years for decades.
You've got it
- exponential change uses the compound formula $P(1 + r/100)^t$
- growth = multiplier $> 1$; decay/depreciation = multiplier $< 1$
- a 15% yearly loss → multiplier $0.85$; after 4 years: $20\,000 \times 0.85^4 \approx 10\,440$
- exponential always beats linear eventually — compounding earns growth on growth