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# Exponential growth

In applied mathematics, a population that grows exponentially grows at a rate proportional to the size of the population. For example, when there are three million individuals then the population grows three times as fast (in individuals per unit of time) as when there are one million. Exponential growth is so called because the population is an exponential function of time. The "individuals" need not be persons; they can be bacteria or dollars per share of a certain stock, or anything whose growth can be expressed in numbers. More generally, any function, not necessarily a function of time, grows exponentially if its value is proportional to its growth rate, i.e., to its first derivative.

The phrase exponential growth is also a misnomer used by persons ignorant of quantitative matters to mean merely surprisingly fast growth. In fact, a population can grow exponentially but at a very slow rate, and can grow surprisingly fast without growing exponentially.

In the long run, exponential growth of any kind will overtake linear growth of any kind (the basis of the Malthusian catastrophe). Exponential growth also eventually overtakes polynomial growth, and there is a whole hierarchy of conceivable growth laws that are sub-exponential and also super-linear, while faster than exponential growth is also possible. The linear and exponential models are merely rather simple candidates.

• 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 sufficent, the number of neturons increases exponentially.

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