← ELI5 · Nestor G Pestelos Jr

Mathematics

What is exponential growth?

It's when something grows by a percentage of itself each step, instead of by the same fixed amount every time, so the bigger it gets, the bigger each step's increase gets too.

Doubling means the increase itself keeps growing

Six bars, each exactly double the height of the one before: 1, 2, 4, 8, 16, and 32 1 2 4 8 16 32 +32 this step +1

Going from 1 to 2 adds one. Going from 16 to 32 adds thirty-two. Same doubling rule, much bigger jump, because it's applied to a much bigger number.

Looks slow, then suddenly isn't

A bottle that fills as bacteria double every minute; a slider moves the clock from 11:00 to 12:00 11:55 3.1% full

Bacteria that double every minute fill an empty bottle in exactly one hour. Drag the slider. Watch how long it stays looking almost empty.

Why this trick works on almost anyone

Physicist Albert Bartlett used this exact bottle to argue that people misjudge exponential growth (a population, a debt, a resource running out) because most of the curve looks calm right up until the last couple of doublings, and by then there's no time left to react.

Nothing grows like this forever

A dashed line climbing ever faster off the top of the chart, labeled pure exponential growth assuming no limits, next to a solid line that climbs the same way at first but then bends flat, labeled real growth once it hits a limit keeps compounding forever limit reached levels off at the limit

A resource-limited population, a spreading illness, a market (anything real) runs into a ceiling: not enough food, not enough susceptible people left, not enough buyers. Growth slows and flattens instead of climbing forever.

Sources: Albert A. Bartlett, "Arithmetic, Population and Energy" (lecture transcript). "45.3 Environmental Limits to Population Growth," Biology 2e, OpenStax. Longer, more technical version: Reference: Exponential Growth.

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