Intuitions about exponential growth

Suppose you fold over a sheet of paper, you'll then have something twice as thick as the original. Fold it again and you'll have something 4 sheets thick. With each fold the thickness doubles. This is an exponential growth and unlike linear growth, it accelerates.
Well, what if we start with paper 0.001 meters thick, how high would it be if it were folded 100 times? It turns out the answer is a thickness which is a bit more than the diameter of the universe. When written in meters, the number has 28 digits (before the decimal point).

Initially that may seem rather counter intuitive. However with a little bit of training, it can become intuitive. We actually do have lots of experience in dealing with this kind of dramatic exponential growth. Think about how we represent regular integers. Adding a zero will increase the number by a factor of 10. That's an example of exponential growth. The value of the number increases exponentially with the number of digits. We all know that a number with 4 digits, say 1,000 is much much smaller than a number with 8 digits, say 10,000,000. In this case when we have doubled the number of digits and the value increased by a factor of 10,000.
So, the moral of the story is that our experience with representing numbers using the standard decimal digits, gives us a good understanding of exponential growth.

Coming back to our folding example, we note that 3 foldings, causes 3 doublings, which increases the thickness by a factor of 8, which is pretty close to 10. So as a very rough rule of thumb, we'll almost be adding a new digit to the thickness after every three foldings.
If you want to be a bit more precise, then note that 2 to the power of 10 is 1,024 which is close to 1,000. So ten doublings (2 to the power of ten), causes an increase of just over 1,000, which means we add 3 digits. When we have 100 doublings, that is 10 times 10 doublings: \[2^{100} = (2^{10})^{10} = (1,024)^{10} \approx (10^3)^{10} = 10^{30} \] We started with something \(0.001 m\) thick and ended with something approximately \(10^{27} m\) thick.

Is exponential growth always very fast? Actually no. It depends on how long it takes to double. Suppose you are paid an annually compounding interest rate of 0.1% on your deposits in a bank. That is an form of exponential growth. But it will take 693 years for your money to double. That's the bad news. But the good news is that if you start with one euro it will grow to over a billion after it has doubled 30 times. \[ 2^{30} = (2^{10})^3 = (1,024)^3 \approx (10^3)^3 = 10^9\] But that will take more than 20,000 years.