Complex Exponential and Rotation

Summary: When $c=i$, the rule “velocity = $c\times$ position” forces $e^{it}$ to move on the unit circle, since multiplying by $i$ rotates 90°; one half-revolution gives $e^{i\pi }=-1$.

Reframing (extending) the definition of exponentiation

For non-real exponents, "$e^{it}$" no longer means repeated multiplication — that interpretation breaks down. What survives is the defining differential equation of the exponential (see eulers-number):

$$\frac{d}{dt}{e}^{ct}=c\cdot e^{ct},e^0=1$$

We extend $e^x$ to complex inputs by demanding this equation continues to hold. Everything else (including $e^{i\pi }=-1$) follows.

Multiplication by $i$ is rotation by 90°

A quick reminder of the geometry of the complex plane (Recall $i=\sqrt{-1}$ by definition, so $i^2=-1$):

$$i\cdot (a+bi)=ai+b{i}^2=-b+ai$$

The point $(a,b)$ becomes $(-b,a)$ — that’s a 90° counterclockwise rotation about the origin. Multiplying any complex number by $i$ rotates it by a quarter-turn.

Setting up the dynamical system

Treat $e^{it}$ as a position in the complex plane at time $t$. The differential equation says:

velocity = $i\times$ position = position rotated 90° counterclockwise

The position at time $t$ is $e^{it}$. The velocity is the derivative $\frac{d}{dt}{e}^{it}=i\cdot e^{it}$ — which is just the position multiplied by $i$, i.e. rotated $90°$ counterclockwise. So the velocity arrow always points perpendicular to the radius, with the same magnitude — the signature of uniform circular motion.

At every point in the plane, the velocity arrow is perpendicular to the radius arrow, with equal length. This is the velocity field of uniform circular motion.

Why it traces the unit circle

Starting position: $e^{i\cdot 0}=1$, i.e. the point $(1,0)$. The velocity there is $i\cdot 1=i$, i.e. straight up, magnitude 1.

A particle whose velocity is always perpendicular to its position, with magnitude equal to the radius, undergoes uniform circular motion at angular speed 1. Starting at radius 1, it stays on the unit circle forever.

After $t$ seconds:

• Distance travelled along the circle: $t$ (since speed = 1)
• That distance equals an angle of $t$ radians
• So $e^{it}$ is the point on the unit circle at angle $t$

$$e^{it}=\cos (t)+i\sin (t)$$

This is Euler’s formula — derived without any series expansions, purely from the dynamical reframing.

Plugging in $\pi$

After $t=\pi$ seconds, the particle has travelled half the circumference, ending diametrically opposite the start:

$$e^{i\pi }=-1$$

Equivalently $e^{i\pi }+1=0$ (Euler’s identity). After $t=2\pi$ it’s back to 1: $e^{2\pi i}=1$.

What the imaginary unit really did

Compare to the real cases:

$c$	velocity rule	motion
$1$	along the radius, outward	exponential growth
$-1$	along the radius, inward	exponential decay
$i$	perpendicular to the radius	circular orbit

A real $c$ stretches or shrinks the position vector — moving you along a line. An imaginary $c$ rotates it — moving you around a circle. A complex $c=a+bi$ does both simultaneously, producing logarithmic spirals.

Group-theory perspective review

The dynamical view above is one of two complementary lenses. The other comes from group theory: $\exp$ is the structure-preserving map (a homomorphism) between two ways of viewing the complex numbers as a group.

Side	Group	Elements are…
Input	additive $(\mathbb{C},+)$	sliding actions on the plane
Output	multiplicative $({\mathbb{C}}^{\ast },\times )$	stretch + rotate actions fixing 0

The exponential identity $e^{x+y}=e^x\cdot e^y$ is the homomorphism property: adding inputs (composing slides) corresponds to multiplying outputs (composing stretch+rotates).

Both groups split into a “real part” and a “complex-specific part,” and $\exp$ matches them up:

Input (slide)	Output (stretch / rotate)
Horizontal slide by $x$	Stretch by $e^x$ (positive real)
Vertical slide by $y$ (i.e. $iy$)	Rotation by $y$ radians (unit circle)

This is why imaginary inputs produce pure rotation — the imaginary axis maps to the unit circle, which is the rotational (modulus-1) subgroup of $({\mathbb{C}}^{\ast },\times )$. And $e^{i\pi }=-1$ is just: vertical slide by $\pi$ ↦ rotation by $\pi$ radians (half-turn) ↦ the multiplicative action of the number $-1$.

Why $e$ specifically?

Different bases differ only in the rate at which a unit of vertical slide converts to radians of rotation: $2^{ix}$ gives $\ln 2\approx 0.693$ rad per unit, $5^{ix}$ gives $\ln 5\approx 1.609$ rad per unit. The base $e$ is the unique one that converts 1 unit of slide into exactly 1 radian — making the imaginary-axis-to-unit-circle map an isometry. The deeper “why” lives in calculus (see eulers-number).

Visualising the full $z\mapsto e^z$ transformation

Group-theoretically, $\exp$ takes the entire complex plane to the punctured plane ${\mathbb{C}}^{\ast }$:

• Vertical lines (constant real part, varying imaginary) → concentric circles of radius $e^{\text{Re}(z)}$ centred at the origin.
• Horizontal lines (constant imaginary part, varying real) → rays from the origin at angle $\text{Im}(z)$.

A geometric mnemonic: roll the plane up into a cylinder (wrapping vertical lines into circles), then squish that cylinder onto ${\mathbb{R}}^2$ around 0 — the imaginary axis becomes the unit circle, and everything to its right gets stretched outward, everything to its left gets squished toward 0.

What to remember

• The ”$e^x$ as repeated multiplication” picture is dead in the complex case. Two pictures survive: $\exp$ as the solution to $\dot{f}=cf$ (dynamical), and $\exp$ as the homomorphism $(\mathbb{C},+)\to ({\mathbb{C}}^{\ast },\times )$ (group-theoretic). They are the same fact viewed from two angles.
• $e^{i\pi }=-1$ is not mysterious — it’s the position of a particle that has moved around the unit circle for a half-revolution, or equivalently, the multiplicative action that a vertical slide of $\pi$ on the additive side maps to.
• This is the gateway to seeing why exponentials show up in oscillation, wave mechanics, Fourier analysis, and quantum mechanics.
  • Real exponents → pure growth/decay (scaling only, no rotation);
  • Imaginary exponents → pure oscillation (rotation only, no scaling);
  • Complex exponents → both at once (scaling and rotation).

Sources

• Strongly recommended: e(iπ) in 3.14 minutes, using dynamics
• Euler’s formula with introductory group theory — the homomorphism perspective