*Wikipedia* defines exponential growth correctly, if not clearly, as “a specific way that a quantity may increase over time. It occurs when the instantaneous rate of change of a quantity with respect to time is proportional to the quantity itself.”^{1} Let me approach this definition somewhat tangentially by comparing two sorts of numerical progressions.

The numbers 1, 11, 21, 31, 41, 51 represent the first five steps of an arithmetic series for which each entry after the first equals its predecessor plus 10, while the numbers 1, 2, 4, 8, 16, 32 represent the first five steps of a geometric series for which each entry after the first equals twice its predecessor. Nothing remarkable yet, but compare what happens in the next five steps:

51, 61, 71, 81, 91, 101 versus 32, 64, 128, 256, 512, 1,024.

Both series start off at one, but in only ten steps our geometric series beats its rival by more than a factor of ten.

Arithmetic series increase linearly. Our example is described by a linear function of *n*: 1 + 10 × *n*, with *n* = 0, 1, 2, … . Pythagorean philosophers of the sixth century BCE were familiar with such series. They knew the sum of the integers from 1 to *m* is equal to *m*(*m* – 1)/2. Those sums were called triangular numbers, because only these numbers of dots, 3, 6, 10, etc., can be arrayed symmetrically to form equilateral triangles.

Geometric series increase exponentially, with the entries of our example given by 2* ^{n}*, again with

*n*= 0, 1, 2 ... . Euclid understood such series. He showed the sum of the first

*n*entries of our example to be 2

^{ (n + 1)}– 1.

^{2}

A far more dramatic demonstration of the power of exponential growth is provided by the legend of the chessboard, as described in 1256 CE by the Kurdish historian Ibn Khallikān. Once upon a time, the game-playing tyrant king Shihram sought a challenging new board game. One of his subjects, a penniless mathematician, Sissa ibn Dahir, struggled for months to invent the game we call chess. The king was so delighted with the new game that he asked Sissa what reward he might wish for his marvelous creation. The mathematician requested neither gold nor silver, but simply one grain of rice for the first square of the chessboard, two for the second, four for the third, and so on as a geometric series. The king thought Sissa quite foolish, but agreed to his terms. His ministers soon discovered that not even all the nation’s rice could fulfill Sissa’s request. “You are a genius,” said the king to Sissa, thereupon appointing him his most senior advisor.

Sissa’s request amounted to 2^{64} – 1 rice grains, or about 18 billion billion of them. This corresponds to about 500 billion tons of milled rice, or a thousand times more than today’s annual global rice harvest.

The truly exponential growth of geometric series can pose great dangers, as Thomas Robert Malthus would argue centuries later:

Population, when unchecked, goes on doubling itself every 25 years, or increases in a geometric ratio. … The means of subsistence, under circumstances the most favorable to human industry, could not possibly be made to increase faster than in an arithmetic ratio.^{3}

Today the world population is growing less rapidly, yet it is poised to double in the next 66 years. The threat of overpopulation remains with us.

To conclude, let us return to the definition of exponential growth I mentioned at the start: a growth rate proportional to the quantity that is growing. Mathematically, the challenge is to find a function of time that equals its own time derivative, which is to say, to solve the simplest-seeming differential equation: *df*/*dt* = *f*. The solution emerged over the course of a century, from the work of the Scottish mathematician John Napier in 1614 to that of the Swiss mathematician Leonhard Euler in 1728. The general solution to the equation is *f*(*t*) = *ae ^{t}*, where

*a*is an arbitrary constant and

*e*is called Euler’s (or Napier’s) constant: a transcendental number approximately equal to 2.718.

^{4}

An exponentially growing quantity, such as the number of bacteria of a certain sort multiplying by dividing in a medium with enough food and space, is given by N(*t*) = N(0)*e*^{(t/}^{τ}^{)}, where τ is the mean interval between individual bacterial fissions. The doubling time of N(*t*) equals τ times the natural logarithm of two (approximately 0.69). The increase of N(*t*) over a time interval Δ*t* that is much smaller than τ is given approximately by (Δ*t*/τ) × N(*t*).

As a bonus, the exponential function describes exponential decline, such as the declining radioactivity of a sample of a radioactive isotope. Each and every radioactive isotope has its own characteristic half-life. That of uranium-238 is about 4.5 billion years, which is approximately the age of the earth. Thus, there was twice as much of this isotope on our newborn planet than there is today. The half-life of radium-226 is about 1,600 years. I shall conclude this essay with a puzzle. The earth’s age is about equal to three million half-lives of radium-226. The result of that many halving is a number unimaginably tiny. Every atom of radium that was present in the newborn earth would have decayed long ago. How then was it possible for Marie Curie to isolate a tenth of a gram of radium-226 from a few tons of pitchblende?