Скачать или смотреть Visualization and Application of Euler's Identity

Euler’s formula basically tells us how to move around a circle using a special kind of exponential number.

Imagine you start at the point (1, 0) on a circle drawn on a flat surface (like the unit circle in the complex plane).
Now, as you slowly increase the angle θ, the expression e^(iθ) gives you the exact coordinates of where you are on that circle at that moment.

When θ = 0, you’re at the rightmost point of the circle.

As θ increases, you move smoothly counterclockwise along the circle.

When θ = 90°, you’re at the top;

180°, you’re at the left;

270°, you’re at the bottom; you’re at the bottom;

360° (or 2π radians), you’re back where you started.

So the value of e^(iθ) marks your position while you move around the circle—it’s like a continuous “circle navigator.”

Now, when θ = π (half a circle), you reach the leftmost point, which corresponds to –1 on the number line.

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