e Approaches
e is what you measure with when the thing being measured is busy eating itself. Two domains, two limits, same number — watch them converge.
Compounding to a limit
limn→∞ (1 + 1/n)n = e
Bernoulli's interest problem. Every time you halve the compounding period, the limit creeps closer to e.
Primes thinning to a rate
π(x) ~ x / ln(x)
The Prime Number Theorem. The number of primes ≤ x divided by x/ln(x) approaches 1 as x grows.
why this matters
Both panels are doing the same job: chasing e from a different angle. On the left, money compounds itself — each payout grows the base it's calculated against, and the answer in the limit is e. On the right, primes thin themselves out — each new prime sieves its own multiples, and the density at scale x is 1/ln(x), controlled by the same e. Self-referential growth on the left; self-referential decay on the right. e is the ruler for both.