On this humble brown planet, we’re used to things growing at a steady pace. Trees add a ring every year. Families expand by one child or marriage at a time. Even in their most extreme months of food-gobbling growth spurts, teenagers will sprout at most a few inches. All of these are examples of **linear growth** (or something close to it). It’s modest, approachable – something the human brain has no trouble grasping.

But not all growth is like this. Take the old story of the sultan and the beggar. The beggar comes before the sultan, pleading for some rice to eat. When the sultan asks how much he needs, the beggar cleverly points to a nearby chessboard. He asks the sultan to put 1 grain on the first square, 2 on the next, 4 on the next, 8 on the next, and so on, doubling the number of grains for each successive square.

The sultan agrees, not realizing that on the 64^{th} and final square, he’ll need to stack 600 trillion pounds** **of rice – enough to cover Rhode Island to a depth of 1400 feet. That’s **exponential growth**. It may start slow, but it quickly reaches dizzying heights.

This kind of growth occurs surprisingly often. Look at the human population, or the number of Facebook users, or even the amount of money you make on a brilliant investment. Most of these growth patterns eventually hit some kind of ceiling (the planet will hold only so many humans; Facebook has begun to max out its user base; and your investment can climb only so high), but as long as exponential growth is at work, change proceeds far faster than our linear minds are accustomed to.

To explain this idea to my students, I start with a graph of the function f(x) = 2^{x}, using a one-inch scale. Continue reading