As the title says, how come the value e is special? it has functions and rules just dedicated to it, but what sets it apart from another random number?
In: 397
It’s a constant related to a quantity that grows at a rate proportional to its own size, like a mythical dragon that eats constantly and that gets bigger when it eats more, and eats more when it gets bigger. In particular the dragon grows M times faster than normal when its size is M, so by the time it’s double in size it’ll eat twice as fast which means it’ll grow twice as fast and be able to eat even faster, and then grow even faster, and on and on in a positive feedback loop.
Let y be the size of the dragon (say in metric tons), and x be how much time has passed (say in years). For the rate of growth we’ll use metric tons per year, and the rate of growth updates continuously (the split moment the dragon gets bigger, it also eats faster). If we chart the size of the mythical dragon as
y = 2^(x)
then it grows at a rate ‘less than’ its own size.
In the case of
y = 3^(x)
it grows at a rate ‘greater than’ its own size.
However, when
y = e^(x)
it grows at a rate exactly ‘equal’ to its own size.
(Equal meaning the number parts are equal, so 4 metric tons and 4 metric tons per year are ‘equal’)
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