Self-Reinforcement and Exponential Functions

Special relativity is really kind of mean. Not only it prohibits anything from going faster than the speed of light (therefore destroying our Star Trek-inspired dreams of interstellar travel) but also threatens with extreme adverse effects should anyone dare to even come close to the impenetrable barrier of c. Assuming we can deal with the steady increase of mass as the speed goes up, there is always this issue of time dilation. While you are taking your short (i.e. few years-long) trip to nearby star, time passed on Earth could very well be measured in centuries. Having a millennium to catch up might prove cumbersome, and rather frustrating. Just think of all the iPhone models you would have missed!

As a solace, though, you could get quite a pile of cash waiting for you to pick up. Let’s say you’ve put 10,000 dollars (or euro, or your favorite currency) into investment with a yearly interest rate of 10 percent. Every year, this deposit will therefore increase by one tenth, and this will happen continuously over the next 1000 years. Could you quickly tell how big the final amount will be, compared to the initial one? How many times will it increase?…

You shouldn’t be very hard on yourself if you answered instinctively with e.g. 100 times or something similar. I mean, such figures are totally, utterly wrong by many orders of magnitude because the actual value is bigger than 10⁴⁰. But it’s absolutely common to have problems with grasping exponential functions intuitively. In many ways this is quite pitiful, for they accurately describe many phenomenons that occur in nature, civilization, technology and culture. Yet they often escape understanding, leading to unfulfilled predictions, incorrect extrapolations, and plain old cognitive biases.

What is so bizarre about these functions that they tend to confuse a significant fraction, if not the majority of people?…