Why do some people have difficulty grasping exponential growth?

In My Humble Opinion
1

There’s a brain teaser that goes something like this:
Imagine you have a very large piece of very thin paper. Let’s say the paper is one thousandth of an inch thick. (The example works better with inches rather than with millimeters.) So, that means that if you have a stack of one thousand sheets of this paper, the stack will be one inch high.
Okay, now imagine that you cut the sheet of paper in half and put one half on top of the other. That means you now have two thousandths of an inch, right? Do this again, and you’ve got four thousandths, then eight, then 16, and so on. Got it? Got it.

Let’s say we do it 50 times. Now, answer quickly without doing any calculations: How high will the stack of paper be?

Everyone that I’ve asked will offer an answer that is not even close to the right answer. So, I’ll say something like this:
Let’s count how many sheets we have after each time that we cut the sheets in half.
(I start counting on my fingers, starting with my left thumb.) First time, 2, then 4, 8, 16, 32. (Next hand) 64, 128, 256, 512, 1024.
So, after ten times, we have about 1,000 sheets, which means the pile is one inch high. We’re going to do this 40 more times. How high will it be after 40 more times?

The answer that I usually get is 6 or seven feet, which makes no sense because that’s about 80 inches, so all they’ve done (kind of) is multiply one inch times forty and then double it because … uh … we’re doubling here, right?

Then I say (and you can imagine that the person is starting to get irritated by now) “We double the height every time we cut the sheets in half. Right? So, let’s see what happens if we do this 10 more times. We were at one inch, then 2, 4, 8, 16, 32, which is about 3 feet, then 64, 128 (with is about 10 feet), then 20 feet, then 40, 80 feet. And we have 30 more times to go.”

At this point, they’re pissed off and they don’t want to guess any more. Already the answer is way bigger than they thought, they’re feeling stupid, and they don’t know what to say next. I reassure them by saying that practically nobody gets the answer right because the problem is deceptively hard. If they do volunteer another answer for 50 repetitions, it’s maybe a thousand feet, but they find it hard to understand how a pile of thin paper can be that high after doubling the height only 50 times.

Over the years, I’ve posed this brain teaser to several people and I’ve never got an answer that was within a few orders of magnitude of the correct answer, even with the prompting. Okay, there are people who understand exponents and logarithms and could provide a close approximation fairly quickly but I’ve never posed the question to someone like this. Also, it seems that using really thin paper makes the problem more difficult even though it’s only about one order of magnitude thinner than thick paper.

It really is amazing how difficult it is for some people to grasp the nature of exponential growth. Well, there is a quote that is often misattributed to Einstein: “Compounding interest is the most powerful force in the world.”

FWIW, I didn’t come close to the correct answer when I heard it, but my excuse is that I was 10 years old.

2

People forget (or never learned) math. 2^50 is a pretty simple concept for anyone who works in one of the physical sciences or other, similar, fields. Not sure I could do it in my head, though. That’s why we have calculators.

3

If I’m doing my math right (on a calculator) I’m coming up with nearly 18 million miles (is that right? I think it is, I checked it like 4 times.)…are you really expecting people to figure that out in their heads?

ETA
2^50=1125899906842624
1125899906842624/1000/12~93824992237 feet
93824992237/5280~17769885 miles

4

I get the concept to some degree, but I would have trouble with coming up with a decent answer. For some reason, a lot of math concepts are difficult for me.

It’s not that I’m dumb, it’s just that math was never my strong point.

5

I didn’t seem to have any trouble with the problem, but perhaps I was given a heads-up by the title. My first thought was 2^50, and I would’ve answered you “I have no idea. Can I use a calculator?” But even if I did have a brain fart on my first guess, I’d realize my error at some point in the conversation, long before I got frustrated.

BTW, I hate that exponential growth and geometric growth are the same thing. 2, 4, 8, 16, 32, …, should be called geometric growth and 2, 4, 16, 256, 65536, …, should be called exponential growth. But that’s just me.

6

Why would you expect an answer within a few orders of magnitude when 2^50 is an order of magnitude different from 2^49?

7

The problem with the particular example you describe is that people are prone to incorrectly imagine a piece of paper that can be folded in half 50 times. 50 sounds like a realistically low number at first glance.

I think people are quicker to come up with the right answer if the correctly realize that it’s just a math problem and that no paper could ever be folded that many times. If you don’t think of it abstractly, anybody is going to be confused by the prospect of a standard piece of paper being miles and miles high.

Maybe another way to make the point more clearly, with the same example, is to imagine a 10" x 10" paper and illustrate how unbelievable small the remaining “area” is as its folded in half again and again.

I think examples other examples of exponential growth are much more accessible. For instance, if you ask people about a simple bacterial growth model matching your paper exactly, I predict most people would correctly predict it being astronomically high very quickly.

8

Nope. But not only are they way off on their first guess, their revised estimates are still very low, even after being given some indication that the rate of increase is quite a bit greater than they thought.

9

Well, I was thinking the same thing when I was composing the post but I think that geometric growth is a subset of exponential growth, and despite my specific example, I am interested in how people grasp other types of exponential growth.

FWIW, the Wikipedia article on Exponential growth agrees with me.

10

I thought about it for a minute and the answer I came up with was a trillion inches.

After 10 iterations, you increased the thickness about a thousandfold. So every 10 iterations, it will increase another thousandfold.

Start = 1/1000th inch
10 = 1 inch
20 = 1000 inches
30 = 1 million inches
40 = 1 billion inches
50 = 1 trillion inches.

Easy peasy. Your sample set happened to be people who don’t like or know math.

11

You don’t need to be great at mental arithmetic, as long as you know that 2^10 is roughly 1000, therefore 2^50 is roughly 1000^5, which is a 1 followed by 15 zeroes. So you would have that many sheets of paper at the end. Since they’re a thousandth of an inch, divide by 1000 to get the answer in inches: the stack is roughly a trillion inches tall.

12

In common usage, order of magnitude refers to powers of 10, so 2^50 is not an order of magnitude different from 2^49. But, I can accept your interpretation as well.

In simple terms, the guesses are not even close to the correct answer. I’ve never had someone say “1000 feet” or “1 mile” as their first guess.

13
  • because some people may not have learned it that well,
  • because some people may have forgotten the concepts through lack of use
  • because it’s not always easy to imagine “big” numbers visually
  • because some people may not bother to put much thought into a “math” question when there’s no real reason to do so (some guy giving me a “brain teaser” isn’t really worth my time)

it’s like that stupid shit that went around (and leaked out of) facebook a while back, the “1x1x1x1x1x1x1x1x1x0=?” thing. I absentmindedly nerfed it the first time because I wasn’t arsed to look at it carefully and remember order of operations. But there was no shortage of people hooting and hollering about how “stupid” the people who got it wrong were. It just turned into a massive circle-jerk of perceived self-importance, and that’s the first impression I got from this thread.

14

You’re right about my sample set.

There’s a more direct way of getting the height in inches, if you know that the number of sheets is 2^50 and you know that log(2) is approx. 0.3.
Number of sheets = 2^50 = 10^(50*0.3) = 10^15.
Height in inches =10^(15-3) = 10^12 inches.
(I see that Manduck did something similar.)

Then height=10^(12-1)=10^11 feet
And height=10^(11-4)=10^7 miles.

15

I estimated about a billion miles on reading the problem, without doing any calculations. When I calculate it out, I get the same answer as Joey P, so I was off by less than two orders of magnitude. What do I win?

16

Yeah, those are some possible reasons, but I think that it’s something more basic, and has little to do with someone’s knowledge of math.

Yes, if you know the math, you can calculate the answer, but, I suggest that, for most people, the answer doesn’t make sense. It’s like there’s something about the way our brains function, and it’s not just difficulty in understanding large numbers. It’s something about non-linear growth.

17

Easier example might be using the old tale of chess inventor. Persian King asked him to come up with an interesting game and when he showed to King 64 square board and pieces King was so thrilled he wanted to reward game inventor. He thought for a second and then asked King to put two rice grains on 1st square and then go all the way to the 64th doubling the quantity from previous square.

So the King thought that’s easy and started… 2, 4, 8, 16, 32,64, 256… and obviously stopped before realizing that the final tally is 18,446,744,073,709,551,616 and, apparently, there was not enough rice in world at that time. In today’s numbers that would be 2,6 million grains of rice for each of 7 billion people on Earth. Probably still not enough rice…

18

Galileo,

“answer quickly without doing any calculations”
Bolding mine. I see others have gotten the right answer but not how one could have guessed it involved a trillion (rather than a million, billion, more) without any calculations.
Are you surprised that people don’t understand the concept of exponential growth or that people can’t answer a question that requires calculation to be anywhere near the ballpark without doing calculations? I think plenty of people understand the concept of exponential growth but your question doesn’t test that.

As for my guess as to why we don’t have an intuitive understanding of exponential growth involving extremely high numbers: same reason we don’t see in ultraviolet or infrared and can’t fly: It wasn’t that useful in the last few million years.

19

It’s not intuitive because few things in everyday life exhibit exponential growth. Most things that could exhibit exponential growth (reproduction of organisms, for example) quickly hit a limit.

We don’t grasp it because it’s not really all that useful.

20

Fair enough.

And (as I back myself into a corner …) I don’t think I’ve used those exact words, and I definitely have not emphasized “any”. I try to convey that I’m not looking for an elaborate calculation but a gut feel for the answer. Given enough time and effort, most people will get close to the right answer, but I’m looking for their “sense” of what the answer could be.

I agree that the first question is not a good test of their understanding of exponential growth, but I do follow up with quite a few calculations to give an idea of it, and they still don’t revise their answer up into the millions. (Also, in my preamble, I set the stage with 2,4,8,16 thousandths of an inch.)

In any case, you can try variations of the question yourself on anyone who is good-natured enough to put up with it.

I agree. :slight_smile: