What is Euler's formula actually saying? | Ep. 4 Lockdown live math

Euler’s formula stops being a mysterious “imaginary exponent” once the exponential function is treated as a specific power series (exp), not as repeated multiplication of the constant e. With that shift, the identity e^{iθ} = cos(θ) + i sin(θ) becomes a concrete claim about what the series does in the complex plane: the terms rotate by powers of i (quarter-turn steps) while their lengths shrink according to θ^n/n!, and the infinite vector sum lands exactly on the unit circle at angle θ.

The lesson begins by anchoring complex numbers on the unit circle: a complex number one unit from the origin at angle θ has real part cos(θ) and imaginary part i sin(θ), which is why cos(θ) + i sin(θ) is the natural “polar-to-Cartesian” form. The familiar shorthand e^{iθ} is then introduced—but immediately challenged. Many people interpret e^{x} as “multiply e by itself x times,” which breaks down for non-integer and especially imaginary inputs. Instead, exp(x) is defined as the infinite series 1 + x + x^2/2! + x^3/3! + …, where the factorial in the denominator makes the series converge and gives exp(x) a precise meaning for all real x.

From that series, a key structural property emerges: exp(a+b) = exp(a)·exp(b). This isn’t obvious from the notation “e^{x},” but it follows from how the factorial-weighted terms expand and recombine. The instructor uses a live multiple-choice reasoning exercise to show that this functional equation forces strong consequences—such as exp(5) = exp(1)^5 and exp(1/2)^2 = exp(1)—and highlights a subtle point: the value exp(0) = 1 is not automatic for every function satisfying exp(a+b)=exp(a)exp(b); it’s a specific feature of the exp function defined by the series.

Only after that foundation is laid does the discussion move to complex inputs. Powers of i cycle every four steps: i^1 = i, i^2 = −1, i^3 = −i, i^4 = 1, and then repeat. Plugging iθ into the exp series produces a sum of vectors whose directions follow that four-step rotation pattern, while magnitudes are governed by θ^n/n!. A visualization for θ = 1 (i.e., exp(i)) and a computational check in Python illustrate that the real and imaginary parts land where cos(θ) and sin(θ) predict.

The famous special case θ = π yields e^{iπ} = −1. The explanation emphasizes why this is less about the constant 2.71828 “showing up” directly and more about exp’s defining series: the computation of exp(iπ) never needs to insert 2.71828 explicitly. What matters is that exp(x) behaves like a function whose real inputs match the conventional e^{x} shorthand, while complex inputs trace circles. The takeaway is that Euler’s formula is a statement about exp’s geometry—its complex values move around the unit circle with angle θ—setting up the next lecture’s deeper reason for the circular motion.