We are always oblivious of daily phenomena. For example, when we boil a cup of tea and then put it at normal room temperature, how fast is the water cooling down?
We don’t know. We intuitively think it should be proportionate to the temperature difference between the hot tea – close to 100 degrees versus the room – say about 25 degrees in Celsius. Experiments are conducted and verified our intuition is true, which is unusual in the scientific world.
So basically, this equation that the “Integral’s derivative equals itself” needs to be valid:
If f(x) is a value 2 powered by x, applying calculus formula as
h = 0.6931…
Now to mimic the situation of cooling down the hot water in a room environment, we want to force h = 1, and reversely compute the value c =
2.71828 … , it is thereafter named as e.
Certainly, if we try on equations other than the format of a value powered by a variable x, we can find other constant values. However, e is commonly and universally accepted now similar to the way we accept pie = 3.1415926…
Now take another angle, inspired from this article by this article, where it gives a real life scenario, where a person is saving money into a bank, the bank promise to return 100% interest in one year, that means, if he deposit one dollar in the beginning of a year, he will collect 1*(1+1)^1 = 2 dollar at the end of that year. So the question here is what if he can withdraw the money everyday with the same interest deal – one dollar in one year? Sure, he should get a lot more, but how much more? it is 1*(1+1/365)^365 = 2.714567. In this vein, what if you can withdraw by hours, by minutes, to unlimited minute scale, the equation is
This is the natural constant e.