Math with Bad Drawings

How Fast is Exponential Growth? (Or, Yao Ming Confronts the Vastness of the Universe)

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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 64th 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) = 2x, using a one-inch scale. In other words, for every inch we move to the right, the graph will double in height.

Now, the curve quickly rises beyond what I can draw. But let’s imagine I continue the graph, even as it punches a hole in the classroom ceiling and continues to rise. To make it easier to picture, suppose that retired basketball star Yao Ming (7’6” tall) is lying on the horizontal axis.

By Yao’s ankles (about 7 inches), the graph is as tall as Yao himself.

By Yao’s shins (12 inches), the graph is over 300 feet tall – a little higher than the Tribune Tower in Oakland.

Just before Yao’s knee (after another six or seven inches), we reach the cruising altitude for an airplane.

At Yao’s thighs (34 inches), we’ve reached the moon.

Still below Yao’s belt (42.5 inches), we’ve reached the sun.

Around Yao’s elbows (61 inches), we’ve reached the next star system – far further than any manmade object has ever gone from earth.

By Yao’s shoulders (75 inches), we’ve reached the far end of the galaxy – a place so far away that even Star Trek treats it as unimaginably removed from home.

By 8 feet – just a few inches past the top of Yao’s head – we’ve reached the edge of the known universe. Yao’s eyes gaze into distances heretofore probed only by the most powerful space telescopes; he finds himself staring down the unfathomable limits of the physical realm.

This, my friends, is exponential growth. It is so blisteringly fast that it outruns the human imagination, leaping quickly beyond the scale of our daily existence, and then continuing to leap and leap, until things that are unimaginably large to us (like the distance to the moon) become unimaginably small in comparison to the new levels we’ve attained (like the distance to the edge of the galaxy).

Keep this in mind next time the phrase “exponential” pops up in conversation, whether it’s describing a company’s profits, the rate of climate change, or a basketball star’s exploration of the universe’s deepest mysteries.

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