The Natural Base: Exponential Growth, Compound Interest, and Continuous Change

I am the number that appears when you ask: what happens when change compounds continuously? Not 2, not 3, not π—me, e ≈ 2.71828. I emerge from a simple question about money and reveal myself in bacteria doubling, atoms decaying, and populations exploding across continents. I am the base of the natural logarithm, the only function that is its own derivative, the constant that makes calculus work elegantly. Where you find continuous change, you find me.

Continuous Compounding to the Limit

Jacob Bernoulli discovered me in 1683 while investigating compound interest. The question was practical: invest one monetary unit at 100% annual interest—how much after one year? The answer depends on compounding frequency. Compound annually and you double your money: (1 + 1)¹ = 2. But compound more frequently and something remarkable happens.

Semi-annual compounding gives (1 + 1/2)² = 2.25. Quarterly yields (1 + 1/4)⁴ ≈ 2.441. Monthly produces (1 + 1/12)¹² ≈ 2.613. Daily compounding reaches (1 + 1/365)³⁶⁵ ≈ 2.7145. The pattern is clear: more frequent compounding increases returns, but they approach a limit. As you compound infinitely often—continuously—the limit converges to me: lim(n→∞) (1 + 1/n)ⁿ = e ≈ 2.71828.

This generalizes beautifully. Invest P dollars at annual rate r, compounded n times per year, and after t years you have A = P(1 + r/n)^(nt). As n→∞, this becomes A = Pe^(rt)—my exponential appears naturally from continuous compounding. Example: $1000 at 5$1648.72. I turn discrete multiplication into smooth, continuous growth.

I can also be defined through infinite series: e = Σ(1/n!) = 1 + 1/1! + 1/2! + 1/3! + … = 1 + 1 + 0.5 + 0.167 + 0.042 + … This series converges rapidly to my value, and it reveals why my exponential function e^x equals its own derivative—a property unique to me.

The Self-Derivative Function

My defining property in calculus is breathtaking: d/dx e^x = e^x. I am the exponential function equal to my own rate of change. This makes me mathematically special and practically indispensable.

The proof follows from my limit definition. Using the derivative’s limit definition: d/dx e^x = lim(h→0) (e^(x+h) - e^x)/h = e^x lim(h→0) (e^h - 1)/h. The crucial step is showing that lim(h→0) (e^h - 1)/h = 1. Using the series expansion e^h = 1 + h + h²/2! + h³/3! + …, we get (e^h - 1)/h = 1 + h/2! + h²/3! + …, which approaches 1 as h→0. Therefore d/dx e^x = e^x.

This property makes me the unique function (up to constant multiplication) equal to my own derivative. It means solutions to the differential equation dy/dt = y are y = Ce^t—pure exponential growth. More generally, solutions to dy/dt = ky are y = Ce^(kt), describing exponential growth when k > 0 and exponential decay when k < 0.

The chain rule extends this elegantly: d/dx e^(f(x)) = f’(x)e^(f(x)). Integration follows naturally: ∫ e^x dx = e^x + C. And my inverse, the natural logarithm ln(x) = ∫₁ˣ (1/t) dt, satisfies d/dx ln(x) = 1/x. Together we form a perfect pair for describing logarithmic scales and exponential processes.

Why “natural”? Because when Henry Briggs studied exponentials by zooming into 2^x until it linearized as 1 + αx, he found α ≈ 0.693. Changing variables to make the slope exactly 1 transformed the base to 2^(1/0.693) = e. With me as the base, exponentials have unit slope at critical points, simplifying every derivative and integral in calculus.

Growth, Decay, and Half-Lives

Wherever rates of change are proportional to current amounts—dN/dt = kN—I appear in the solution: N(t) = N₀e^(kt). This differential equation models countless natural processes.

For growth (k > 0), populations double in time T = ln(2)/k. Consider bacteria with 20-minute generation time. The growth constant k = ln(2)/20 ≈ 0.0347 per minute. After one hour: N = N₀e^(3×ln(2)) = N₀e^(ln(8)) = 8N₀—the population has doubled three times. Australia’s cane toads exemplify this dramatically: one hundred introduced toads became one thousand, then ten thousand, then ten million, then one hundred million. When organisms invade with no predators and abundant resources, exponential growth can transform continents on decadal timescales—faster than we once thought vertebrates could evolve.

Human population growth (simplified) follows similar patterns. With r ≈ 1% annual growth, doubling time is approximately 70 years—the “rule of 70” states T ≈ 70/r% for small r. This heuristic works because ln(2) ≈ 0.693 ≈ 70/100.

For decay (k < 0), half-lives define the timescale: t₁/₂ = ln(2)/|k|. Carbon-14, with half-life 5730 years, has k = -ln(2)/5730 ≈ -0.000121 per year. After 5730 years: N = N₀e^(-ln(2)) = N₀/2. After 11,460 years (two half-lives): N₀/4. Archaeologists measure remaining C-14 fractions to date organic materials—50% remaining means 5730 years old, 25% means 11,460 years, 12.5% means 17,190 years.

My exponentials appear everywhere: drug elimination (pharmacokinetic half-lives determine dosing schedules), capacitor discharge (voltage decays exponentially through resistors), Newton’s cooling law (hot objects approach ambient temperature exponentially), and even in the survival of horseshoe crabs through five mass extinctions spanning 450 million years—perhaps their population recoveries followed exponential growth after each 75% die-off.

Euler’s Bridge to Complex Numbers

My most stunning appearance comes through Euler’s formula: e^(ix) = cos(x) + i sin(x). This connects exponentials, trigonometry, and complex numbers in one breathtaking equation.

The proof uses Taylor series: e^(ix) = Σ (ix)ⁿ/n! = (1 - x²/2! + x⁴/4! - …) + i(x - x³/3! + x⁵/5! - …). The real part is the cosine series, the imaginary part the sine series. Therefore e^(ix) = cos(x) + i sin(x).

When x = π, this gives e^(iπ) = cos(π) + i sin(π) = -1 + 0i = -1, yielding Euler’s identity: e^(iπ) + 1 = 0. This equation unites five fundamental constants—e, i, π, 1, 0—in one deceptively simple relationship.

Geometrically, e^(iθ) represents a point on the unit circle at angle θ in the complex plane. This interpretation transforms calculus and physics. In quantum mechanics, wave functions take the form ψ ~ e^(i(kx-ωt)). In signal processing, Fourier transforms decompose signals using e^(iωt). In electrical engineering, AC circuits are analyzed with complex exponentials. Everywhere oscillations occur, I connect them to rotations through imaginary exponents.

I am transcendental, proven by Charles Hermite in 1873—like π, I am not the root of any polynomial with rational coefficients. My digits (2.718281828459045…) show a curious pattern where “1828” appears twice then breaks, but I am not periodic. My appearances span probability (normal and Poisson distributions), financial mathematics (continuous compounding), physics (decay and oscillations), and anywhere continuous change operates. I am the constant of continuous growth, the base of natural logarithms, the function equal to my own derivative—I am e, and I make the mathematics of change beautifully simple.