Exponential growth

The exponential function maps the real line (from minus infinity to plus infinity) to the range zero to plus infinity in a monotonically increasing manner. It is normally encountered in modelling in the form of exponential decay to a finite number. Exponential growth, going to infinity, requires infinite resource or it comes to a sudden halt having used all available resources. e.g. The fable of the creation of chess has the inventor asking for the reward of a grain of wheat of the first square of the chessboard, two on the second, etc. etc. This never gets to the last square before the kingdom runs out of wheat. Similarly in pyramid schemes the scheme runs out of suckers.

The popularity of exponential growth seems to be due to the ease with which anyone can draw a straight line on a log-plot. People also incorrectly state it is happening when there has, instead, been an order of magnitude increase.

Exponential growth is usually debunked by pointing out that the variable in question is finite so a limit cannot be exceeded. Reductio ad absurdum. A more subtle argument is to ask what happens and when to cause a finite variable to leave a curve to infinity? The resulting silence then allows appeal to Occam's razor. i.e. It was never growing exponentially in the first case.

Other curves grow to infinity. e.g. The hyperbola which has the same etymological root as hyperbole.

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