Infinite Slices, Finite Truth: A Journey to e

Under certain constraints, numbers reveal consistent, elegant, and often surprising behavior.

Constants compress infinite behavior into finite truths. When you put simple rules on infinite things, you discover patterns that feel designed, but are not — they just are.

The constant **e** emerges when something is continuously compounded by itself. As the number of compounding intervals increases from once a year, to once a day, to once a second, to infinitely many times per unit time, the final amount converges to **e**. It is the natural result of 100% growth, infinitely divided, but also infinitely frequent. The tiniest pieces of 1, added infinitely, become the most elegant function of all — one that grows by the measure of its own existence.

Take 1, break it into tinier and tinier pieces, and compound it over and over — you get e.

Suppose you have growth rate _r_ = 1 (i.e., 100% per unit time). Instead of growing all at once at the end, what if you split that growth into smaller intervals — say, twice a year, or ten times, or a thousand?

Let **_n_** be the number of times you divide the growth and apply it throughout the year. Then, the formula for the compounded amount becomes:

$(1+\frac{1}{n})^n$

As _n_ approaches infinity (meaning you're compounding continuously) the value of this expression approaches a mysterious constant:

$e = \lim_{n \to \infty} \left(1 + \frac{1}{n} \right)^n$

This constant, **_e_ ≈ 2.718**, is what you get when growth is applied **not in steps**, but **continuously**. It's the purest form of compounding — the mathematical fingerprint of systems that are always growing by what they already are.

Imagine a plant growing. If it grows with discrete jumps (like once a day), it’s artificial. But if it’s always stretching toward the sun, instant by instant, every fiber expanding based on how much it already is, that’s e. That’s continuous time

**e is the limit** of how fast something grows if it's always growing, not in steps. Nature doesn’t tick in steps — it flows. If you could zoom in to the growth of a tree, a bank account, or even the infection of a virus, you’d see that it’s not adding in lumps… it’s _compounding. In **every infinitesimal moment**, the investment (or quantity) is growing by a fraction of itself.

Among all the functions in math, only one has the magical property that its rate of change is equal to itself. That function is $e^x$.

$$

\frac{d}{dx} e^x = e^x

$$

Let’s see it quickly:

$$

\frac{d}{dx} e^x = \lim_{h \to 0} \frac{e^{x+h} - e^x}{h} = \lim_{h \to 0} \frac{e^x(e^h - 1)}{h} = e^x \cdot \lim_{h \to 0} \frac{e^h - 1}{h} = e^x

$$

No matter where you look along its curve — at ( x = 1 ), ( x = 100 ), or ( x = -5 ) — its steepness always matches its current height. It’s as if the function is chasing its own shadow, always moving as fast as it already is.

If you prefer to build it not through compounding, but through pure arithmetic, _e_ also arises as the sum of an infinite series:

$e = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \dots$

Each term gets smaller, but the pattern never ends — like a breath that never fully fades

And just as $e^x$ tells you how fast something grows when growth is fueled by itself, its inverse, $ln(x)$, tells you how long it takes to reach a value when you're growing like that. If $e^x$ is the accelerator, then $ln(x)$ is the odometer. One is a rising curve, the other a slow unraveling. Together, they form a perfect pair: one builds forward by what is, the other traces backward to where it began.

What does $e^{-x}$ mean?

It describes a process where, at every instant, the quantity is shrinking in proportion to how much is left — just like compounding, but in reverse.

For example, $e^{-1}$ means about 36.8% of the original remains after time unit of continuous decay.

$e^{-5}$ doesn’t mean “a -500% rate” — that doesn’t make sense. It means the decay is happening so aggressively that only 0.67% of the original is left after 1 time unit. The exponent simply tells you how strongly the decay operates over time. The more negative it is, the faster the decay — not because you're subtracting, but because you're multiplying by smaller and smaller fractions.

| r | $e^{-r}$ | % of original left |

| --- | -------- | ------------------ |

| 0.5 | 0.6065 | ~60.7% |

| 1 | 0.3679 | ~36.8% |

| 3 | 0.0498 | ~5.0% |

| 5 | 0.0067 | ~0.67% |

So if $A_0$=100 then:

At 100% decay rate: $A_{1}$ =36.79

At 300% decay rate: $A_{1}$ =4.98

At 500% decay rate: $A_{1}$=0.67

You’re not saying “subtract 500% of the value” (which makes no sense).

You’re saying “**decay so rapidly** that only 0.67% remains after 1 unit of time.”

There’s this wonderful moment when someone realizes that a number isn’t just a number. That’s when you stop doing math and start feeling it. That’s when you know you’re onto something. e makes time fluid and truly continuous. _e_ isn’t just a number. It’s the **limit of self-compounding**, the **heartbeat of exponential systems**, and the **only function whose rate of change is equal to its current state**. _e_ arises not from human design, but from the geometry of nature itself. It’s what the universe arrives at when change is left to unfold uninterrupted