Covid-19 cases are now “exponentially” growing – but what does that mean?

The prime minister has warned that Covid-19 infections are now increasing “exponentially”. That sounds technical, but should we be frightened? The Latin exponere means “to place out”; so an “exponent”, from the 16th century in English, is a proposition (or later a person) that sets forth some idea. It was adopted in algebra for expressions such as “bⁿ”: here the “exponent”, n, sets forth the number of times b (the “base”) should be multiplied by itself. As n increases, b increases exponentially.

“Exponential growth” is often loosely used to mean “large” or “fast”, but it needn’t be either: it simply means that the increase is proportional to the total quantity. Thus, the “complicated exponential-horned gramophone” owed by Professor Welch in Kingsley Amis’s Lucky Jim is one in which the horn gets wider the further it is from the base, not one that keeps growing indefinitely until it swallows the Earth.

In computer science, problems requiring “exponential time” take exponentially longer to solve the bigger they are, until some will need longer than the lifetime of the universe. No doubt Boris Johnson hopes, if only for his own sake, that Covid-19 is not such a problem.