1. Introduction

Complex numbers were invented to solve equations like x² = −1. But their true power lies in their geometric interpretation: in the complex plane, multiplication is a simultaneous rotation and dilation. This property makes complex numbers the natural tool for describing spirals.

2. The complex plane

A complex number z = a + bi is represented in the complex plane by the point with coordinates (a, b). The modulus |z| = √(a² + b²) is the distance to the origin. The argument arg(z) = arctan(b/a) is the angle with the real axis.

The polar form of a complex number is z = r·e^(iθ), where r = |z| and θ = arg(z). This representation is particularly suited to multiplication.

3. Multiplication as rotation

Multiplying two complex numbers in polar form is simple:

r₁e^(iθ₁) · r₂e^(iθ₂) = r₁r₂ · e^(i(θ₁+θ₂))(multiplication complexe)

Moduli multiply and arguments add. Multiplying by e^(iθ) is a pure rotation of angle θ (the modulus remains 1). Multiplying by r·e^(iθ) is a rotation of angle θ combined with a dilation of factor r.

4. Euler's formula

Euler's formula is one of the most beautiful equations in mathematics:

e^(iθ) = cos(θ) + i·sin(θ)(formule d'Euler)

For θ = π, we obtain the famous identity e^(iπ) + 1 = 0. This identity is not a mysterious coincidence — it is the geometric description of a half-rotation: e^(iπ) = −1 means that rotating 180° in the complex plane transforms 1 into −1.

5. The complex logarithmic spiral

The logarithmic spiral can be elegantly described in the complex plane. The sequence of points z_n = r^n · e^(inθ) traces a logarithmic spiral: at each step, we multiply by r·e^(iθ), performing a rotation of angle θ and a dilation of factor r.

If r > 1, the spiral moves away from the origin. If r < 1, it approaches it. If r = 1, we get a circle. The logarithmic spiral is therefore the natural trajectory of iterated multiplication in the complex plane.

6. Conclusion

Complex numbers reveal that rotation and the spiral are fundamental concepts of arithmetic, not just geometry. Complex multiplication is a rotation-dilation, and the logarithmic spiral is the natural trajectory of this iterated operation. This unification of algebra and geometry is one of the most beautiful achievements of mathematics.