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. 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.

There’s a great kid’s book about this! It’s called One Grain of Rice and it tells basically the same story of the sultan and the beggar (the raja and a girl this time). It has awesome illustrations–on the first day, a bird brings the girl a grain of rice, and by the 30th day, hundreds of elephants deliver all the rice in the kingdom. I read it with first graders last year!

Of course, when asked to draw a picture of the problem and solution in the story, one kid drew a starfish and a shark. So the nuances may have been lost on them.

Aw, that sounds like a cute book! Of course, as with most great literature, it takes the 6-year-old mind a few years to soak in the nuances.

Or – maybe – it’s the 25-year-old mind that’s missing the subtleties. Go back to it in a few years, and you might say to yourself, “Ah yes, the starfish and the shark! How could I have missed it?”

I have some math friends who would absolutely love this, and love to show it to their middle school students… :) Thank you, be well~

Thanks for reading – I hope they find it fun/helpful!

You know, if you turned the Yao Ming series into one of those animated movies that show the pictures being drawn, you’d have a cute fun alternative to the “Powers of Ten” movie (http://www.powersof10.com/film). And I actually like the idea of “powers of 2” better than “powers of 10”, because the effect of exponential growth is more of a surprise when you start with 2, 4, 8 than when you start with 10, 100, 1000.

Cool, thanks for linking to that film – I hadn’t seen it before. I think you’re right that the powers of 2 pack a more surprising punch than the powers of 10, which is part of why I like the “rice on the chessboard” story.

Of course, a Yao Ming video would be better if Yao himself would agree to do it… hmm…

I will have to share this with my math students when discussing exponential growth! Can Yao Ming come to San Diego?

You’ll have to try him on Twitter!

What is the Twitter name? http://www.segmation.wordpress.com

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This is how Moore’s law work. Consider the recent NSA scandal: we use the same amount of data as we did before (okay maybe 2 or 3x more) but their capacity to store and decipher grows along the lines of Moore’s law: this is why I tend to disagree …….. actually I’m going to do a new post about this ( see http://lostinmist.wordpress.com for more)

I love the reference you put to “A Trip to the Moon” by the time we reach Yao’s thighs. Not many people would understand the reference to that classic film. We at the Disorder, though, really aren’t what many would call normal. Funny, yes. Normal? Eh, overrated. I’m actually the in-house nerd (writing posts about stuff like Superman’s Split-Personality Disorder [link for the curious:http://thedisorderpodcast.wordpress.com/2013/06/16/supermans-identity-crises/%5D and the like), so I might be the only one who appreciates the awesome of this. But still, great post!

Yes…I literally almost laughed out loud when I saw the moon with the rocket in his eye. I haven’t really seen the original “Trip to the Moon” movie, but I’ve seen the movie “Hugo,” which deals with the history of that old film.

Glad you enjoyed that reference! (I picked it up from Hugo, too.) It was one of the few ways, with my limited drawing skills, that I could make the moon actually look like the moon…

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exponential growth is so powerful, thanks for writing this post!

Thanks for reading!

This is definitely what I’ll want on the return on my next financial investment. :-) Just give me so much cash I can carpet Rhode Island, please. :-)

Well, if you’re using $1 bills, that’s only $300 billion! Totally attainable, right?

I’ll talk to financial advisor and explain my expectations. It should go well. I’m kicking in a whole buck, right? Ah, the intersection of math and imagination is bliss!

My daughter had a couple of homework problems related to this. Where n is between 1 and 64, find the formula for how many grains of rice on square n, and the weight of rice on square n. When she had finished I asked her to put that amount of rice into perspective (just how much rice is that?). She figured out that for n = 64, it was more rice than has been grown on earth since the year 1400. Some cheeky beggar in that story!

That’s a cool way of conceptualizing it – I hadn’t thought about comparing it to the amount of rice grown over the centuries. How did she make that calculation?

We made one rather simple assumption, that the amount of rice grown increases as science and technology of agriculture advances. So we just divided the total rice by the world rice production in 2011 (Google “world rice production”). The point for her was not to get an accurate figure but to familiarise her with making and acknowledging an assumption, and putting the answer into a context that makes it mean something. Maths becomes pretty abstract later on so I think at this point it helps to relate the numbers to something; to give her something to anchor her understanding to.

I do worry though about how we transition to the more abstract concepts later on. Having said that, just last week we were discussing the problem of determining the water temperature at the bottom of a lake, and we talked about the thermocline which required interpreting a chart and considering what the gradient might be telling us. When we eventually introduce functions so that a chart can be considered the output of a function those ideas will, I think, carry over nicely.

That sounds great – I might steal that for a future assignment. (“Take these absurdly large numbers, and find a way to put them in context.”) You’re absolutely right that some experience with inspecting her assumptions and finding concrete anchors for abstract ideas will pay off down the road.

And that’s a cool experience with thermoclines, too. The surest path to an abstraction is often through one well-studied example.

“Functions” are the single deepest, toughest concept for the Precalc students I’ve taught. They often claim they grasp the idea, but their work says otherwise. So it’s good to hear that your daughter is getting pretty adept at manipulating information, and considering “input data” and “output data,” so I bet that functions will feel less like a big leap than like a logical next step.

Wow, that is really cool! :D

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Being able to describe complex things in an understandable way is definitely an elegant thing to do… especially since it deals with math.

Glad you enjoyed it – cool blog, btw.

Thanks for posting this. keep posting.

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Thick though my head may be, I had no sweat grasping this. Any body who can get past the impenetrability of my head so easily is deserving of the highest praise.

Hey, glad you enjoyed it! I think a lot of smart people wind up with the mistaken impression that they can’t “do math,” in large part because explanations of math often come in overly technical or obscure language… so it means a lot to me that this felt accessible!

Ben loved this. You are so right that people do not get compounding.

And you surely make up for bad drawing with the way you tell a story :)

Could you draw this out for my business model please?! :)

Facebook’s growth is not exponential. Exponential growth requires a constant percentage change in Y for constant changes in X. If you look at the yearly user data for Facebook, then there is no constant percentage change whatsoever in Facebook’s users.

I could be wrong. If i am, please correct me. This is for a math assignment.

You’re quite right–Facebook’s growth is not strictly exponential. In fact, no real-world quantity will show perfect exponential growth–there will always be an error term.

With my students I showed them that exponential growth is a fairly accurate model of Facebook’s growth from about the time it had 5 million users up to roughly 400 million. Then the growth slowed, compared with exponential growth.

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Reminds me of Prof. Bartlett’s, “Arithmetic, Population and Energy” video:

http://albartlett.org/presentations/arithmetic_population_energy_video1.html

Got here from HP — Wonderful content! Reminds me of Prof. John Allen Paulos, ABC column and his books (Innumeracy: Mathematical Illiteracy and Its Consequences, A Mathematician Reads the Newspaper). Simon Singh also has a way of teaching us things (Fermat’s Last Theorem, The Simpsons and Their Mathematical Secrets).

Maybe it was logarithmic graph paper that fooled our eyes and made exponential growth “look” not so bad.

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A wonderful illustration about linear vs exponential growth for all network marketeers. For those who are still in la la land, this is a wake up call. A job won’t make you rich. Your boss will make sure of that even if you live to work to 100 years. All it will do is make sure you are Just Over Broke (JOB). Sadly, this is the fate of many lulled to false security.

I will definitely share in my group, great idea. Also requesting to check muy post at https://shareofheartblog.wordpress.com/

Please can you give me some feedback on this – it’s for a school project and I’ve not got a clue how it’s going – thank you so much :)

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So amazingly made that even a dummy can understand linear and exponential growth. Really creative drawings, i must say!

Just LOVE this post …

Laughed my head off …and YES!!! I ‘got’ it … read and understood by a simple mind like me …..I’d say that’s a RESULT!

Really great post, particularly enjoyed the illustrations :) I don’t know if it’s just how its displayed on my tablet but it seems you mention the function is f(x)=2x which would not give you the doubling you’re after but a straight line, 1:2, 2:4, 3:6 4:8 5:10 etc. I believe you mean f(x)=x^2 as in x squared I.e 1:1, 2:4, 3:9, 4:16, 5:25, 6:36, 7:49, 8:64, 9:81, 10:100

I believe he meant f(x)=2^x i.e. 0:1, 1:2, 2:4, 3:8, 4:16, 5:32, 6:64, 7:128, 8:256, 9:512, 10:1024. This gives you the doubling

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no words!!…really awesome

So amazingly made that even a dummy can understand linear and exponential growth. Really creative drawings, i must say!

It’s my topic in maths :D but now it seems to be more interesting to me!

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Haha, you are demonising his height.