Euler's Number

Summary: Euler’s number $e\approx 2.71828$ is defined by the property that $e^t$ is (not only proportional to, but equal to) its own derivative; every other exponential $a^t$ is just a horizontally-scaled version of it.

In practice, everything is written as $e^{ct}$ instead of $a^t$ because $c$ is directly readable as the growth rate:

• $2^t=e^{\ln (2)\cdot t}\approx e^{0.693t}$, telling you immediately that the quantity grows at $69.3\%$ per unit time.
• $e^{0.05t}$, where $c=0.05$ tells you directly the quantity grows at $5\%$ per unit time

What’s special about $e$

From exponential-function, every $a^t$ is proportional to its own derivative with a base-specific constant $k_a$: $\frac{d}{dt}{a}^t=k_a\cdot a^t$

For base 2, $k_2\approx 0.69$; for base 3, $k_3\approx 1.10$. Somewhere between 2 and 3 the constant must equal exactly 1. That number is $e$:

$\boxed{\frac{d}{dt}{e}^t=e^t}$

This isn’t a coincidence about $e$ — it’s the definition of $e$. Asking “why does $e$ have this property?” is like asking “why does $\pi$ happen to be the ratio of a circle’s circumference to its diameter?” The property is what picks out the number / the number emerges from the property).

A geometric restatement: at every point on the graph of $e^t$, the slope of the tangent line equals the height of the curve at that point.

Intuition

Think of $e^t$ as your position on a number line at time $t$, starting at $1$. The condition “derivative = self” says your velocity always equals your position — the further from 0 you are, the faster you flee from it. This is the cleanest possible runaway-growth law.

The horizontal-rescaling fact

All exponentials are horizontally-scaled versions of each other. $4^t$ grows twice as fast as $2^t$ because $4^t=2^{2t}$. More generally, choosing a different base just means feeding the input through a constant scaling.

So we can always rewrite $a^t$ in the form $e^{c\cdot t}$. The chain rule then gives the derivative for free:

$$\frac{d}{dt}{e}^{ct}=c\cdot e^{ct}$$

The proportionality constant moves out of “mystery limit” status and becomes whatever number $c$ we put in the exponent.

This works because $e^t$ is its own derivative — the one base where the proportionality constant is exactly $1$. Rewriting $a^t=e^{\ln (a)\cdot t}$ then lets the chain rule do the rest: $\frac{d}{dt}{a}^t=a^t\cdot \ln (a)$. See below.

All exponentials in terms of $e$

Given any $a^t$, the rescaling constant ($c$, above) is the natural-logarithm of $a$:

$$a^t=e^{\ln (a)\cdot t}$$

So:

$$\frac{d}{dt}{a}^t=a^t\cdot \ln (a)$$

This finally explains the mystery constants from exponential-function: $k_2=\ln (2)\approx 0.6931$, $k_3=\ln (3)\approx 1.0986$, $k_8=\ln (8)=3\ln (2)$ (which is why $k_8=3k_2$).

Rewriting $4^t=e^{\ln (4)t}$ to find its derivative easily (via chain rule)

$$\begin{array}{rl}\frac{d}{dt}{4}^t& =\frac{d}{dt}{e}^{\ln (4)\cdot t}\\ & =e^{\ln (4)\cdot t}\cdot \ln (4)\\ \text{hence, }\frac{d}{dt}{4}^t& =4^t\cdot \ln (4)\end{array}$$

Why we always write exponentials this way

In applied calculus, almost nobody writes $2^t$ or $3^t$. Everything is written as $e^{ct}$, because:

• The derivative is trivial: $\frac{d}{dt}{e}^{ct}=c{e}^{ct}$.
• The constant $c$ in the exponent is the proportionality constant between the quantity and its rate of change. For a population growing at 5% per unit time, $c=0.05$. The number you want to read off is right there.
• Differential equations of the form $\dot{f}=cf$ have solutions $f(t)=f_0{e}^{ct}$ — no rewriting needed.

Connection to the complex case

The differential-equation characterisation (“derivative = $c$ times self, starting at 1”) is what extends $e^x$ beyond the reals — to complex numbers, matrices, and operators. The “repeated multiplication” intuition does not extend; only the dynamical one does. See complex-exponential-rotation.

There is also a group-theoretic restatement of what makes $e$ special: viewing $\exp$ as a homomorphism from the additive to the multiplicative group of complex numbers, $e$ is the unique base for which a vertical slide of 1 unit on the imaginary axis maps to a rotation of exactly 1 radian on the unit circle. (For base 2, you’d get $\ln 2\approx 0.693$ radians per unit; for base 5, $\ln 5\approx 1.609$.) See group-theory-intro and complex-exponential-rotation.

Sources

• What’s so special about Euler’s number e?
• e(iπ) in 3.14 minutes, using dynamics
• Euler’s formula with introductory group theory