The Number e

Algebra 2 · Axiom Academy

How the sequence climbs to — the constant behind continuous growth. Take the expression and let n grow. Each larger n nudges the value upward, yet it never pushes past one fixed ceiling. That ceiling is e . 2. Where e Comes From: Continuous Growth The limit first surfaced in compound interest . Invest 1 at a 100% annual rate and compound it n times a year, and the year-end balance is exactly (1+ n )^ n . Compounded n times a year, at rate r for t years: The more often you compound, the closer the balance creeps to e . Compound continuously — infinitely often — and the discrete steps smooth into a single curve that lands exactly on e . e earns the name natural through a property no other base shares: the function y = e^ x is its own rate of change. Its slope at every point equals its height at that point. This self-matching is why e^ x appears in every proportional process — one where the rate of change is proportional to the current amount: More individuals → more offspring → faster growth. More atoms → more decay events → faster decrease. A larger temperature gap → faster cooling. More drug in the body → faster elimination. Because e sits between 2 and 3 , the curve y = e^ x grows faster than 2^ x and slower than 3^ x — the natural rate right in the middle. All three pass through (0,1) . Passes through (0,1) : e^ 0 = 1 Grows faster than any polynomial Its rate of change equals itself : the natural logarithm, the inverse of e^ x e^ kx : general growth ( ) or decay ( )

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