3 ms·

I really don't like this way of thinking about it. e isn't important, the exponential function is. e shows up so often because we've chosen to write exp(x) as

• Denvercoder9 5y ago
  The fact that e = 2.718... is a fundamental property of the exponential function, though. It's not an arbitrary choice.
  • ianai 5y ago
    Indeed. That a member of the real number line has this important relationship to the differential operator, the complex plane and number systems, and thus all of trig, calculus, and quantum mechanics is pretty impressive to put it lightly. (Trig through the many relationships of e^x with cosine and sine functions.) The GP comment reads as either a grab at elite character at best or flat out anti-intellectual at worst. No need to bring it in here.
    • jstx1 5y ago
      I feel like you've missed the point of my comment. I said that the exponential is important and you've repeated that here so we don't disagree about that. My point is to distinguish between the exponential function in general and particular value of the exponential function when evaluated at 1.
      • ianai 5y ago
        Which just so happens to be the one power of a number that helps the most if you want to do any actual, decimal calculations with a number without any other decimal expansions of it at hand.
    • abnry 5y ago
      The point being made is that the _function_ is different than the _constant_ producing that function through exponentiation. I think that's kind of fair. Take this headline: The function exp(x) = 1 + x + x^2/2 + x^3/6 + ... is the most beautiful function in mathematics. It is its own derivative, has "product linearity", i.e. exp(x+y) = exp(x) exp(y), and is related to trig functions through complex numbers. The number e isn't doing the heavy lifting, it is the function. The number e comes from the function, not the other way around. Even the famous equation with pi and e is a consequence of the function. And the Taylor series is the easiest way to see the relationship with trig functions. To be fair, there might be a difference in dispensation at play. Those who prefer a more causal or "active" feel to mathematics would prefer the function framing while those who prefer a more platonic or "mystical" feel would prefer the constant framing.
      • ianai 5y ago
        Idk feels pretty arbitrary to say the Fourier expansion of a function matters more than any other expression of the function when the whole point of the Fourier transformation is precisely its ability to express any function in terms of an orthonormal set of functions.
• naasking 5y ago
  > the fact that exp(1) = 2.718.. and we call that e is pretty insignificant and boring. The constant itself is still pretty interesting. Using e as a base for all numbers yields optimal information density IIRC. Binary (base 2) is close to e so it's information density is not bad, but this also tells us that trinary (base 3) would be even better on this metric since it's closer. There are lots of interesting properties like this that end up linked to e.
  • Qem 5y ago
    Related: https://web.williams.edu/Mathematics/sjmiller/public_html/105Sp10/addcomments/Hayes_ThirdBase.htm https://web.williams.edu/Mathematics/sjmiller/public_html/10...
  • hinkley 5y ago
    I wonder sometimes if when Dennard scaling finally grinds to a halt, some desperate and clever individuals will switch us to trinary circuitry, for that last 37% theoretical limit.
• neantherpi 5y ago
  Sincerely, you completely changed the way I look at e. Interesting that ln(x) does follow such a notational style; to change the base one can either divide by ln(b) or use a separate notation (log vs ln). I also had put a lot of weight into e being transcendental but it seems like as long as x is rational the value will also be transcendental, so not that special (if someone could confirm).