The Exponential Function

The exponential function in its general form describes any process whose rate of change is proportional to its current value. Given a coefficient a and growth rate k, it takes the form:

f (x) = a \cdot e^{k x}

y = e^1.0x

Exponential Function

a = 1.0coefficient

k = 1.0growth rate

At x = 0: y = 1.00Slope at x = 0: 1.00Growing exponential

1. Bernoulli, 1683

Jacob Bernoulli in 1683 asked: if you compound interest more and more frequently, what happens in the limit? Start with principal 1, rate 100%, compounded n times per year:

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

He computed this for large n and noticed it converges to something between 2 and 3. He had no name for it. He simply proved the limit existed.

2. Logarithms as Areas (1640s)

Grégoire de Saint-Vincent was computing the area under the hyperbola y = 1/x between x = 1 and x = t. He noticed something remarkable: the area from 1 to ab equals the area from 1 to a plus the area from 1 to b.

\int_{1}^{ab} \frac{1}{x} d x = \int_{1}^{a} \frac{1}{x} d x + \int_{1}^{b} \frac{1}{x} d x

ln(2.0) + ln(1.8) = ln(3.60) = 1.281

Logarithm as Area

a = 2.0upper bound of first area

b = 1.8multiplier

∫₁^a 1/x dx

= ln(2.0)

0.693

∫_a^ab 1/x dx

= ln(1.8)

0.588

∫₁^ab 1/x dx

= ln(3.60)

1.281

Alphonse Antonio de Sarasa recognized this as the defining property of logarithms — that log(ab) = log(a) + log(b). So the area under 1/x is a logarithm. But a logarithm to what base? The base whose logarithm equals 1 is the point where the area from 1 to that point equals 1. This is historically significant: logarithms were defined before their base was identified.

3. A Function That Is Its Own Derivative (Newton & Leibniz, 1660s–1680s)

When Newton and Leibniz developed calculus, they needed functions satisfying a particular equation:

\frac{d y}{d x} = y

This is the equation of something whose rate of growth equals its current size. The solution is eˣ. Being its own derivative is arguably the most fundamental definition of eˣ. They constructed it from scratch using power series.

The tool they had was the idea that a function might be expressible as an infinite polynomial:

f (x) = a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + a_{4} x^{4} + \dots

They didn't know what f(x) was. They just asked: if dy/dx = y holds, what must the coefficients be?

Step 1 — Differentiate term by term:

f^{'} (x) = a_{1} + 2 a_{2} x + 3 a_{3} x^{2} + 4 a_{4} x^{3} + \dots

Step 2 — Impose f'(x) = f(x) and match coefficients of each power of x:

constant: x : x^{2} : x^{3} : a_{1} = a_{0} 2 a_{2} = a_{1} 3 a_{3} = a_{2} 4 a_{4} = a_{3}

Step 3 — Choose f(0) = 1, so a₀ = 1. Everything follows by force:

a_{1} = 1, a_{2} = \frac{1}{2}, a_{3} = \frac{1}{6}, a_{4} = \frac{1}{24} ⟹ a_{n} = \frac{1}{n !}

The pattern forces every coefficient. The result:

f (x) = 1 + x + \frac{x ^{2}}{2 !} + \frac{x ^{3}}{3 !} + \frac{x ^{4}}{4 !} + \dots = e^{x}

This function was constructed purely from the requirement that its derivative equals itself. The choice f(0) = 1 gives the fundamental solution. If you chose f(0) = C, you'd get Ceˣ — the general solution is y = Ceˣ.

Euler

Euler inherited three disconnected things : Bernoulli's limit (1 + 1/n)ⁿ sitting unnamed between 2 and 3; the series 1 + x + x²/2! + ···; and the loose notion of exponential curves aˣ that Leibniz had named but not deeply analyzed.

His first move was to evaluate the series at x = 1:

1 + 1 + \frac{1}{2 !} + \frac{1}{3 !} + \frac{1}{4 !} + \dots = 2.71828 \dots = e

Same number as Bernoulli's limit. He then proved they are identical — the series and the limit converge to the same value. He called this number e, and showed the series is precisely eˣ.

Into the Complex Plane: Euler's Formula

By Euler's time, imaginary numbers — square roots of negatives — were used reluctantly and mistrusted. He took the series for eˣ and substituted x = iθ, where i = √(−1) and θ is a real number:

e^{i θ} = 1 + i θ + \frac{( i θ ) ^{2}}{2 !} + \frac{( i θ ) ^{3}}{3 !} + \frac{( i θ ) ^{4}}{4 !} + \dots

Now he expanded the powers of i carefully. The powers of i cycle with period 4:

i^{1} = i, i^{2} = - 1, i^{3} = - i, i^{4} = 1, i^{5} = i (cycle of period 4)

Substituting and separating real and imaginary parts:

e^{i θ} = c o s θ (1 - \frac{θ ^{2}}{2 !} + \frac{θ ^{4}}{4 !} - \dots) + i s i n θ (θ - \frac{θ ^{3}}{3 !} + \frac{θ ^{5}}{5 !} - \dots)

These were already known series, but from a completely different branch of mathematics.

e^{i θ} = cos θ + i sin θ

At θ = π, cos(π) = −1 and sin(π) = 0, which gives:

e^{iπ} + 1 = 0

Use the slider below to explore how e^(iθ) traces the unit circle as θ varies.

e^iθ = 0.707 + 0.707i

θ = 45.0°

θ = 45.0°0° → 360°

cos θ = 0.707sin θ = 0.707|e^iθ| = 1.000Rotating...

What It Means Geometrically

Think about what eˣ does on the real line — it converts addition to multiplication. Move x forward by a step and the output gets scaled. It stretches. Now ask: what does e^(iθ) do as θ increases? It doesn't stretch — the modulus is always 1:

e^{i θ} = cos^{2} θ + sin^{2} θ = 1

Instead, it rotates. As θ goes from 0 to 2π, e^(iθ) traces the unit circle exactly once. The exponential function, depending on whether its input is real or imaginary, either stretches or rotates. For a general complex input x + iθ:

e^{x + i θ} = e^{x} \cdot e^{i θ} = e^{x} (cos θ + i sin θ)

It simultaneously stretches and rotates. The real part controls magnitude; the imaginary part controls angle. Euler's formula revealed that any oscillating phenomenon can be written as a sum of exponentials. Every frequency is a rotation e^(iωt) in the complex plane.

What Else Is Exponential?

eˣ is the unique function that is its own derivative and equals 1 at x = 0.

Population grows exponentially — each individual reproduces independently of history.
Radioactive atoms decay exponentially — each atom has a fixed probability of decaying per unit time, regardless of age.
A capacitor discharges exponentially — the current depends only on the voltage now, not how long it has been charging.
Heat dissipates exponentially — Newton's law of cooling says the rate of heat loss is proportional to the current temperature difference.

The Algebraic Structure

There is a cleaner way to say what the exponential does. It is the unique map that converts addition into multiplication:

e^{a + b} = e^{a} \cdot e^{b}

Formally, it is a group homomorphism from (ℝ, +) to (ℝ⁺, ×) — from the real line under addition to the positive reals under multiplication. Move forward by a in the additive world, and you scale by eᵃ in the multiplicative world. This is exactly why logarithms were useful for computation before calculators: multiplication is hard, addition is easy, and the exponential is the dictionary between the two.

The complex numbers have a richer additive structure than the reals. Adding a real number moves you along a line. Adding an imaginary number rotates you. So when you ask what happens to the homomorphism with imaginary inputs — what converts imaginary addition into multiplication — the answer has to be rotation.

Assumptions Made

That the power series converges.
That a function satisfying dy/dx = y actually exists and is unique — this is an ODE result.
That term-by-term differentiation of infinite series is valid — this is Real Analysis.

More to Know

What the exponential does to structure beyond ℝ and ℂ.
The matrix exponential eᴬ.