Complex Numbers: Rotations, Dilations and Logarithmic Spirals

A complex number z = x + iy packages two real coordinates. It can represent a point, a vector, an amplitude and phase, or the state of an oscillating system. Its power does not come from a mysterious imaginary symbol, but from an arithmetic rule that respects plane geometry.

MIT notes derive the polar form and Euler identity connecting Cartesian and angular coordinates. ^[1] In this representation, addition remains vector addition; multiplication becomes a particularly simple combination of rotation and scaling.

Magnitude is a distance and cannot be negative. Argument is an angle defined modulo 2π: the same point can be described by θ, θ + 2π, or θ − 2π. The atan2 function is preferable to arctan(y/x) because it preserves the quadrant and handles the vertical axis correctly.

Choosing a principal argument imposes a branch cut. This matters for the complex logarithm and roots: one cannot make a full circuit around zero while keeping the argument everywhere continuous and single-valued.

Magnitudes multiply and angles add. Multiplying by $q=\rho e^{i\varphi}$ therefore transforms every point by a scale factor ρ followed by a rotation φ about the origin. If ρ = 1, it is a pure rotation; if φ = 0, a pure scaling.

3.1 The same operation as a matrix

This matrix shows what complex notation compresses. Two coupled real equations become one multiplication. The gain is not merely aesthetic: compositions are computed by multiplying magnitudes and adding phases.

This formula separates two dynamics. The factor $e^{at}$ controls radius; $e^{ibt}$ controls angle. For a = 0, the trajectory is a circle. For b = 0, it remains on a ray. When both a and b are nonzero, it is a logarithmic spiral.

The University of California, Irvine curve-geometry page writes the logarithmic spiral directly in complex form as x + iy = exp((growth + i)t). ^[2] This equation does not imply that every complex exponential is a spiral: the real part of the coefficient is essential for radial evolution.

At each step, argument gains φ and magnitude is multiplied by ρ. If 0 < ρ < 1, points converge to zero; if ρ > 1, they diverge; if ρ = 1, they remain on a circle. When φ/2π is rational, directions repeat after finitely many steps. When it is irrational, directions never close exactly and become dense on the circle when ρ = 1.

The discrete sequence and continuous curve can share the same sample points without being the same object. The first is an orbit of a discrete dynamical system; the second is a continuous parametric trajectory.

In coordinates (θ, ln r), a logarithmic spiral ln r = ln a + bθ becomes a straight line. The complex logarithm therefore performs a deep geometric operation: it turns multiplication into addition and multiplicative growth into translation.

But Log is multivalued before a branch is chosen: adding 2π to the argument adds 2πi to the logarithm. The origin is singular because neither angle nor logarithm is defined there. A spiral can approach it indefinitely without reaching a point where phase remains defined.

In a linear differential equation, a conjugate pair of eigenvalues a ± ib produces a combination of growth or decay $e^{at}$ and oscillation at frequency b. In a phase plane, the solution is a spiral focus when a ≠ 0 and an ideal circular center when a = 0.

Here the spiral describes the evolution of two state variables, not necessarily a path in physical space. A damped electrical circuit, a small mechanical oscillation, or a population model may draw similar foci while relying on different causal mechanisms.

• Phase rotation is not necessarily material rotation.
• A phase circle is not a spiral.
• A spiral in state space may not be visible in physical space.
• Two systems sharing local eigenvalues may differ far from equilibrium.
• Elegant complex notation does not establish a common origin between phenomena.

Complex numbers are the natural language of rotations because multiplication directly composes angle and scale. A logarithmic spiral appears when rotation is accompanied by multiplicative growth or decay. The limiting case ρ = 1 marks the essential conceptual boundary: rotating is not spiraling.