## Exponential Growth

June 8th, 2009Living cells multiply. Their number grows exponentially. The more there are the faster they increase. An exponentially growing population has a doubling time: the time it takes for the population to double. Having a doubling time is a characteristic of exponential growth. In the time it takes each individual cell to divide into two cells, the whole population of cells doubles. The population doubles because each of its members doubles.

Anything that is growing will eventually double in size and then double again and then again if one waits long enough. But only exponential growth has a single characteristic period of time that is the *same* *for every doubling*.

Things that do not grow exponentially are the distance the train carries you away from the train station or the amount of coffee in the cup you are filling. These increase only linearly with time. The increase has a rate of growth but no characteristic doubling time. The time for the second doubling is not the same as that for the first doubling. Rather it is twice as long. The third doubling takes four times as long. So, in linear growth, no single period of time characterizes a doubling.

An amount of money invested at a compounded interest rate of, say, 7%/year, grows exponentially. It has a doubling time. Ten years. After 10 years the return on the investment will be as much as the original investment itself. The original investment will have doubled in value. Each of the dollars in it will have doubled. In the *next ten years the money will have doubled again* – to four times the original investment. The next doubling – to eight times the original amount – again takes ten years. The rate of growth, r, is related to the doubing time, T, by the simple formula: rT = Ln 2 = 0.7 (approx). At a growth rate of 10%/yr the doubling time is 7 years.

The motion diagram shows exponential growth through four doubling times at the rate of 7% per second. The exponentially growing brown bar doubles in size every 10 seconds. The green bar increases in size linearly at 7% per second. Clicking on GO starts the growth. Clicking on STOP freezes time. You can assess the doublings by stopping at 10 seconds (first doubling), 20 seconds (two doublings) and 30 seconds (three doublings) etc.

No matter what mathematics governs its increase, any physical quantity must eventually stop increasing. Nothing goes on increasing forever. Eventually the mathematics of growth fails to describe the phenomenon. Growth is never sustainable.

See Can growth be sustainable?