concept

Euler's formula

Euler's formula states that eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos\theta + i\sin\theta: the exponential of an imaginary number is the point on the unit circle at angle θ\theta. The statement raises two questions: why the growth constant ee appears in a statement about circles, and why placing ii in the exponent produces rotation. This page answers both.

This page builds on complex numbers, in particular the fact that multiplying by ii is a quarter-turn rotation.

What exe^x is

The exponential function is defined by a growth rule, independent of any particular base: it is the function whose rate of change equals its current value.

ddx ex=ex\frac{d}{dx}\,e^{x} = e^{x}

Money earning 100% continuously compounded interest follows this rule: the larger the balance, the faster it grows, in exact proportion. The number e≈2.71828e \approx 2.71828 is this function's value at x=1x = 1. An equivalent construction reaches xx in nn small steps, each step multiplying by (1+x/n)(1 + x/n):

ex  =  lim⁡n→∞(1+xn) ⁣ne^{x} \;=\; \lim_{n \to \infty}\Big(1 + \frac{x}{n}\Big)^{\!n}

Each factor applies one small increment of growth, and the limit is continuous compounding. Nothing in either definition involves circles: ee arises from growth in proportion to current value. The circle appears in the next step.

Replacing xx with iθi\theta

Putting an imaginary number in the exponent keeps the same growth rule, with the constant ii now inside it. This step is a domain extension, not an ordinary substitution parameterization ↗:

ddθ eiθ=i⋅eiθ\frac{d}{d\theta}\,e^{i\theta} = i \cdot e^{i\theta}

The equation says that the velocity of the moving point equals ii times its position. The complex-numbers page established that multiplying by ii is a quarter turn, so the velocity is always perpendicular to the position and equal to it in length.

That fact forces circular motion. A point whose velocity is perpendicular to the line back to the origin never moves toward or away from the origin; its distance cannot change, so it stays on the circle through its starting point. The point starts at 11, at distance 1 from the origin, so it remains on the unit circle, and because the velocity equals the position in length it moves along the circle at speed 1. After the parameter has advanced by θ\theta, the point has traveled an arc of length θ\theta, and an arc of length θ\theta on a unit circle subtends exactly the angle θ\theta. The point therefore sits at angle θ\theta, whose coordinates are cos⁡θ\cos\theta horizontally and sin⁡θ\sin\theta vertically:

eiθ=cos⁡θ+i sin⁡θ.e^{i\theta} = \cos\theta + i\,\sin\theta.

This resolves both questions from the introduction. The constant ee appears because its defining property, growth equal to current value, is what makes the velocity match the position exactly. The rotation appears because ii redirects that growth perpendicular to the position, and growth that is always perpendicular to the position is rotation.

The same argument in discrete steps

The compound-interest construction makes the same argument visible in finite steps. Applied to iθi\theta, it computes (1+iθ/n)n(1 + i\theta/n)^n: the product starts at 11 and is multiplied nn times by (1+iθ/n)(1 + i\theta/n), and each multiplication adds to the current value a perpendicular increment of size θ/n\theta/n. When the steps are few the path is a coarse spiral that overshoots the circle, because each finite increment carries the point slightly outward. As nn grows, the increments shrink, the overshoot vanishes, and the path converges onto the unit circle, ending at the point at angle θ\theta.

In the widget, pick an angle and raise nn from 1 toward 64. The two readouts track the two claims: the final length approaches 11 and the final angle approaches θ\theta.

The chain shows the nn partial products, each step adding a perpendicular increment to the previous point. The marked target is cos⁡θ+isin⁡θ\cos\theta + i\sin\theta on the unit circle. At n=1n = 1 the single step lands well off the circle; by n=64n = 64 the chain is indistinguishable from the arc.

Why the courses use it

Setting θ=2πft+φ\theta = 2\pi f t + \varphi parameterization ↗ makes ei(2πft+φ)e^{i(2\pi f t + \varphi)} a point rotating at ff revolutions per second from starting angle φ\varphi. This is the rotating-point picture of a sinusoid, written as a formula. Its real and imaginary parts are the cosine and sine, which is why a single correlation against e−i2πfte^{-i 2\pi f t} measures both the cosine and the sine at once, and why a sinusoid's amplitude and phase are carried together in one complex number.

Check your understanding

In the web

Builds on Complex numbers · Parameterization and substitution.