The Math Behind Everything: Why e Rules the Universe

In 1683, Jacob Bernoulli was trying to figure out how to get rich. He was working through compound interest problems—practical math for practical goals. But somewhere between calculating loan payments, he stumbled onto something that would turn out to be far more valuable than any bank account: a number that describes how literally everything in the universe changes.

That number is e, approximately 2.718281828..., trailing off into infinity. And according to a new video from STEM in Motion, it's not just another mathematical curiosity—it's the fundamental language the universe uses whenever anything grows, decays, spreads, or transforms.

The strange thing about e is how it keeps showing up in places that seem to have nothing to do with each other. Your savings account. Viral pandemics. Cooling coffee. Radioactive decay. The expansion of the universe itself. What connects all of these? They're all systems where the rate of change depends on the current state.

The Accident That Changed Mathematics

Bernoulli's thought experiment was straightforward: What happens if you compound interest not yearly, not monthly, but continuously—every millisecond, every infinitesimal moment? Could you become infinitely wealthy?

Start with $1 at 100% annual interest. Compound once: you get $2. Compound twice a year at 50% each time: $2.25. Monthly: $2.61. Daily: $2.71. As you add more and more compounding periods, the total doesn't explode toward infinity. It settles at a specific value: 2.71828...

This was e making its first appearance in Western mathematics, though Bernoulli didn't fully understand what he'd found. It took Leonhard Euler, working decades later, to figure out why this number kept appearing.

The Only Function That Is Its Own Derivative

Euler's insight was geometric. Most mathematical functions are messy—their rate of change produces a different, uglier equation. But e raised to any power has a unique property: its derivative equals itself. As the video explains, "The rate of change, the slope of the curve is exactly equal to the value of the curve itself."

In practical terms: if you have some quantity growing exponentially with base e, the speed at which it's growing at any moment equals the amount you currently have. This sounds abstract until you realize how many real systems work exactly this way.

Rabbit populations don't reproduce at a fixed rate—the more rabbits you have, the faster they multiply. Viruses don't spread linearly—the more infected people exist, the faster the disease propagates. The video describes this as "a self-feeding loop," and that's precisely right. These aren't just curves; they're systems where the present state directly determines the rate of change.

The mathematical expression is deceptively simple: the rate of change depends on how much you already have. "Whenever the rate of change depends on the current state," the video notes, "e is going to be in the answer."

The Mirror Image: Why Everything Dies

But continuous growth isn't the whole story. Everything that builds up eventually breaks down, and e describes that process too—just running in reverse.

Radioactive decay follows this pattern. An atom doesn't disappear on a schedule; its probability of decaying depends on how much radioactive material remains. Half as many atoms means half as much decay. The equation looks nearly identical to exponential growth, except with a negative exponent: n(t) = n₀e^(-λt).

This is why your coffee cools the way it does. When it's much hotter than room temperature, heat flows out rapidly. As it approaches equilibrium, the cooling slows asymptotically. It never quite reaches room temperature—it just gets exponentially closer. Same mathematics, different context.

Sound fading in a concert hall. Atmospheric pressure dropping as you climb a mountain. Signal loss through concrete walls. They're all variations on the same geometric theme: a system where each incremental change takes a fixed percentage of what remains, not a fixed amount.

The Unexpected Places e Hides

Some of e's appearances feel almost sneaky. Take the bell curve—the normal distribution that describes everything from human heights to measurement errors. That familiar hump-shaped curve? It's built on e raised to negative x². The video describes this as "the geometry of pure variation," and it's a good description. The bell curve is what you get when random factors pile up in every direction, and e is the only function that can smoothly bridge "perfectly central" to "vanishing at the edges."

Then there's artificial intelligence. Every time ChatGPT or similar models choose the next word, they're using something called the softmax function—a calculation that runs all the probability scores through e. Why? Because "e to the x grows so aggressively, it stretches the differences. It pushes the likely winners to the top and crushes the noise to zero." Without e amplifying those distinctions, AI language models would produce incomprehensible mush.

And perhaps most elegantly: Euler's formula, e^(ix) = cos(x) + i sin(x). This equation connects exponential growth to circular rotation, which sounds like mathematical poetry until you realize it's the foundation of how we process every digital signal. WiFi. MP3s. MRI scans. They all rely on this property of e to wrap signals around circles and extract hidden frequencies.

From Butterflies to the Big Bang

The same mathematics that describes your bank account also explains why weather forecasts fall apart after a week. In chaotic systems, tiny errors grow exponentially—measured by something called the Lyapunov exponent. That's e again, quantifying how quickly the present slips beyond our ability to predict the future. The butterfly effect isn't just a metaphor; it's a statement about exponential error propagation.

And at the largest scale imaginable: cosmic inflation. In the first tiny fraction of a second after the Big Bang, the universe didn't just expand—it doubled, then doubled again, then doubled again, hundreds of times over. "That wasn't just an explosion," the video explains. "It was an exponential explosion. The very fabric of space was stretching according to e."

Our entire observable universe, in this view, is the result of e running wild for an infinitesimal moment.

The Question That Remains

So why this number? Why 2.718281828... and not some other value?

The video offers an answer: "e is the only number that satisfies the most fundamental rule of the universe. The rate of change is proportional to the state."

But that raises a deeper question—one the video doesn't fully address. Is e special because the universe happens to work this way, or does the universe work this way because e is somehow mathematically inevitable? Did Bernoulli discover e, or did e discover us?

Pi describes space—where things are, how they're arranged. Euler's number describes time—how things change, how they become. If the universe is fundamentally a story about transformation rather than stasis, then maybe e isn't just a useful constant. Maybe it's the most important number we've ever found.

By Nadia Marchetti, Unexplained Phenomena Correspondent