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Comment by Double_Cast | Hacker News Reader

         /
       (0)
      /   \
    0      (1)
   / \     / \
  0   1   0  (1)

  = 011 = 3 (in decimal)

a) Numeral Systems (e.g. ternary) are just trees, and specific numerals are just paths from root to leaf.

b) A 6-digit numeral roughly corresponds to a tree of length 6.

c) Base_10 corresponds to a tree with 10 possible children for each node.

d) e is the most efficient multiplier when trying to achieve compound growth in the fewest iterations of multiplication.

> Also, why does Euler's constant appear all over the place?

e is special because e^x is its own derivative. It also acts as a "bridge" between addition and multiplication. It often appears where growth or trees are involved.

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Thank you for taking the time to explain, with a diagram even.

Another comment mentioned "radix economy", which, together with your description helped me understand (generally) why Euler's constant is the most efficient base in terms of number of digits needed to express numbers.

https://en.wikipedia.org/wiki/Radix_economy

I don't understand point d)

What do you mean by efficient?

Any other literature you could reference? Cause this sounds interesting and I'd like to do some more research into this.

The cost of finding a leaf in balanced tree (with data only at leaves, which is how we represent integers in base B) of size N is on average proportional to its branching factor B multiplied by its height H. Height is (log N)/(log B), so cost is B * (log N)/(log B).

By derivatives, that's minimized when B satisfies

0 = (log N) (1 / (log B) + B(-1/((log B)^2)(1/B)

0 = 1/(log B) - 1/(log B)^2

1 = log B

Base of the logarithm = B.

This looks like you can pick any logarithmic base b you want, and so any B you want, but in fact the derivative I wrote assumes e is the base (hence the term "natural" logarithm.

Other bases b would yield a scaling factor of ... e/b, since d(e^x)/dx = e^x

and b^x = e^((ln b) x)

You can chase "deeper" reasons for this all day long by digging deeper into the the many definitions/properties of e and proving they are equivalent.

Playing with e is the most fun you can have in pre/calculus.

Frankly, it's just a pattern I've observed from playing with numbers myself. I'm unsure how to explain it properly, and I have no academic sources to point toward. You can probably find a better explanation somewhere in an article on optimization problems.

The intuition is that (e) is the optimal water-level when limited water is distributed among a variable amount of buckets where all the filled buckets multiply each other. Alternatively, it's like how volume() is maximized where

  volume(x, y, z) = (x y z)
  const = (x + y + z)

when (x = y = z). Except in our original situation, the number of dimensions is arbitrary instead of fixed.

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