The Mathematics of Growth and Decay

The Curve That Shapes Everything

If there is one mathematical idea that every thinking person should internalize, it is the exponential function. It governs the growth of populations, the spread of ideas, the decay of atoms, and the accumulation of knowledge.

The general form is deceptively simple:

$$f(t) = A_0 \cdot e^{kt}$$

Where $A_0$ is the starting quantity, $k$ is the rate constant, and $t$ is time. When $k > 0$, we get growth. When $k < 0$, decay. The same equation, mirrored.

Compound Growth: The Eighth Wonder

In the human world, the most familiar exponential is compound interest — which Einstein may or may not have called “the eighth wonder of the world,” but which deserves the title regardless.

The discrete version:

$$A = P \left(1 + \frac{r}{n}\right)^{nt}$$

As $n \to \infty$, this converges to the continuous form:

$$A = Pe^{rt}$$

Here, Euler’s number $e \approx 2.71828$ appears — a constant as fundamental to mathematics as $\pi$, arising from the question: what happens when growth is always happening?

Feel the Curve

Numbers on a page are abstractions. Move the sliders below and watch the curve respond. Notice how small changes in rate produce enormous changes in outcome over time. This is the power — and the danger — of exponential processes.

Interactive: Compound Growth Explorer

Principal ($)$1,000

Annual Rate (%)5%

Years10

Future Value

$1,628.89

Total Interest

$628.89

The Rule of 72

A beautiful approximation: divide 72 by the growth rate to estimate the doubling time.

$$t_{\text{double}} \approx \frac{72}{r}$$

At 6% growth, doubling takes ~12 years. At 2%, ~36 years. The difference between civilizational timescales and human ones often comes down to a few percentage points.

Why This Matters

Exponential intuition is not natural to us. Our ancestors evolved in a world of linear processes: one more step, one more berry, one more danger. We are poorly equipped to feel the difference between 2% and 4% growth over decades.

Yet the most important processes in the modern world — technological progress, ecological collapse, compound learning — all follow exponential curves. Understanding them is not merely mathematical literacy. It is a survival skill for a species navigating exponential times.