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
A lot of people here are giving examples without really explaining what the common thread between them is, why e appears in all of them.
e is kind of in the same situation as pi. There are thousands of formulas that have pi appearing in them despite not being trigonometry. It turns out that while the formulas aren’t about circles, they’re way more often related to angles, or sine and cosine waves, or areas of a circle or volume of a sphere, etc. Even in areas that have nothing to do with circles, we can still find ways to use those trigonometry bits to make it more manageable. As a consequence, pi appears everywhere.
e isn’t to do with circles, but it’s the situation of ‘what if the next state of a thing is based on the current state?’. e is kind of a magic number for that situation, because the line for e^(x) has a slope (rate of change) equal to its value (it’s its own derivative, in technical terms). That makes it the ‘neutral’ case. It turns out a lot of formulas get easier if you put them in terms of the neutral case, rather than some other number. Just like with pi and circles, even in formulas that aren’t explicitly about, say, population growth, putting things in terms of growing functions and derivatives can make things easier, and so e appears.
After a while, you start to realise that it’s not just easier, the underlying mechanisms really are more about growth than you thought!
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