# Exponential growth

In mathematics, a quantity that**grows exponentially**is one that grows at a rate proportional to its size. Anything that grows by the same percentage every year (or every month, day, hour etc.) is growing exponentially. For example, if the average number of

*offspring*of each individual (or couple) in a population remains constant, the growth is proportional to the number of individuals. Such an exponentially growing population grows three times as fast in individuals per year when there are six million individuals, as it does when there are two million. Bank accounts with fixed-rate compound interest grow exponentially provided there are no deposits, withdrawals or service charges.

Table of contents |

2 Technicalities 3 Examples of exponential growth |

### Misnomer

The phrase *exponential growth* is also incorrectly used by persons not versed in quantitative matters to mean merely surprisingly fast growth (a potential malapropism). In fact, a population can grow exponentially but at a very slow rate (as the fission process in a nuclear power plant), and can grow surprisingly fast without growing exponentially. Indeed, the logistic function grows approximately exponentially when it is growing very slowly, but nowhere near exponentially when it is growing fastest.

## Technicalities

If we call *x* this quantity, the rate of change *dx/dt* obeys by definition the differential equation:

*x(t)*=

*C*exp(α

*t*), whence the name of the associated growth.

*C*here is an arbitrary constant, determined by the initial size of the population.

In the long run, exponential growth of any kind will however overtake linear growth of any kind (the basis of the Malthusian catastrophe) as well as any polynomial growth, i.e., for all α:

## Examples of exponential growth

- Investing. The effect of compound interest over many years has a substantial effect on savings and a person's ability to retire.
- Biology.
- Bacteria in a culture dish will grow exponentially until the available food is exhausted.
- A new virus (SARS, West Nile, smallpox) will spread exponentially. Each infected person can infect multiple new people.
- Human population.

- An atomic bomb. Each uranium atom that undergoes fission produces neutrons, which in turn split more uranium atoms. If the mass of uranium is sufficient, the number of neutrons increases exponentially.
- A nuclear power plant. Same as above but this time the fission process (also called
*divergence of the reactor*) is controlled so that the growth, while exponential, is very slow. - Processing power of computers. See also Moore's law.