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

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. Continue reading

The Graph Factory

I want my students to see graphing as a subtle, meaningful craft. But when I mess up and assign too many graphs for homework, they just sprint through them, cranking them out like cheap factory products. It goes something like this…

Me: How’s that graphing going?

Student: No time, man! I’ve got sixty logarithms that need to ship to customers tonight, and the assembly line’s been down for hours. I’m cranking out asymptotes by hand over here – I’ve got no time for your funny business!

Me: But why? What’s the point of these graphs?

Student: Hey, not my place to ask questions. I just hit my graph quotas, and try to make it home for dinner with the wife and kids.

Me: But you’re making mistakes. Sine curves don’t have sharp corners.

Student: So slap a warning label on ‘em, for all I care! Continue reading

Stupid Graphs!

Dear Students Who Think Graphing is Stupid,

Right on! Graphs are stupid. Cosmically stupid. Deliberately stupid. In fact – and I hate to pull this rhetorical trick on you, but you leave me with no choice – that’s kind of the point.

A graph is not an end product. It’s more like a map – a simplified picture of something big and complex, a schematic diagram that shows important features and omits distracting details.

Take the function f(x) = x2. This relationship includes a dizzying number of input-output pairs, more than you could ever hope to list. Even with every atom in the cosmos at your disposal, you could never create a “complete” graph of it: to fill in the details, you’d need particles smaller than quarks, and to reach the extremities, you’d have to extend far past the most distant galaxies.

So when you graph f(x) = x2, you’re not depicting the whole function. You’re drawing a simplified version, a pocket-sized map. “Function for Dummies,” you might call it, or “Idiot’s Guide to This Function.” Continue reading