Euler’s identity, eiπ + 1 = 0, brings together several important mathematical constants. It becomes easier to understand when connected to three ideas: complex numbers, angles on the unit circle, and Euler’s formula.
Start with Euler’s formula
For a real angle θ, Euler’s formula is eiθ = cos θ + i sin θ. Here, i is the imaginary unit, defined by i2 = −1. On the unit circle, cos θ is the horizontal coordinate and sin θ is the vertical coordinate of the point at angle θ.
Set θ = π. Since cos π = −1 and sin π = 0, the formula gives eiπ = −1 + i(0) = −1. Add 1 to both sides and the result is eiπ + 1 = 0.
Read the formula on the unit circle
The unit circle has radius 1, and the point at angle θ has coordinates (cos θ, sin θ). An angle of π radians is 180 degrees, which places the point at (−1, 0). At π/4, or 45 degrees, the coordinates are (√2/2, √2/2).
Why multiplying by i rotates a point
Write a complex number as z = a + bi. Multiplying by i gives iz = −b + ai, so the coordinates change from (a, b) to (−b, a). This is a 90-degree counterclockwise rotation in the complex plane. Applying the rotation four times returns to the starting point because i4 = 1.
Connect the formula to power series
The power series for eiθ can be separated into real and imaginary terms. The real terms form the series for cos θ; the terms multiplied by i form the series for sin θ. Together, they give Euler’s formula.
The equation and the animation are different layers
The source video interprets scenes from Animation vs. Math using Euler’s identity, the unit circle, and complex numbers. The mathematical relationships above can be checked independently. A character, color, or scene is part of the video creator’s storytelling and is not itself a mathematical proof. Calling the identity “the most beautiful equation” is an expression of taste, not a formal ranking.