I'm unfamiliar with the literature in this area. May I ask why? (Also, why does Euler's constant appear all over the place?)
/
(0)
/ \
0 (1)
/ \ / \
0 1 0 (1)
= 011 = 3 (in decimal)
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.
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.
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.
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.
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)
That is just a natural consequence.