Complex Plane Arithmetic

The Argand plane

Plotting z = a + bi at coordinates (a, b) turns the algebra of complex numbers into a picture. Real numbers sit on the horizontal axis; pure imaginaries on the vertical. Every complex number is a vector from the origin to its point.

          imaginary
              │
            3 ┤      • z = 2 + 3i
            2 ┤    ╱
            1 ┤  ╱   |z| = √13 ≈ 3.61
   ─────────┼╱──┼──── real
           −1┤   2
           −2┤
              │

Once you accept this picture, the four operations stop looking like symbol-pushing and start looking like motion.

Addition — translation

Adding w = c + di shifts every point z by the vector (c, d):

z + w = (a + c) + (b + d)i

Geometrically, place the tail of vector w at the head of vector z. The diagonal of the resulting parallelogram is z + w. This is identical to vector addition in ℝ². Subtraction is addition of −w, which flips w through the origin first.

          imag
            │  z + w
          5 ┤        •
            │      ╱ |
          3 ┤    ╱z  | w
            │  ╱     |
          1 ┤╱       •
   ────────┼─────────── real
            │   2  4

Multiplication — rotation and scaling

The key derivation. Take z = a + bi and w = c + di. Distribute and use i² = −1:

z·w = (a + bi)(c + di)
    = ac + adi + bci + bdi²
    = ac + adi + bci − bd
    = (ac − bd) + (ad + bc)i

That formula is unmemorable on sight. The polar version is much cleaner. Write z = r₁e^(iθ₁), w = r₂e^(iθ₂):

z·w = r₁e^(iθ₁) · r₂e^(iθ₂)
    = r₁r₂ · e^(i(θ₁+θ₂))

So multiplication does two things simultaneously: multiply the moduli and add the arguments. Multiplying by w stretches the plane by a factor of |w| and rotates it by arg(w).

Special cases reveal the geometry:

• Multiply by a positive real r: pure scaling by r, no rotation.
• Multiply by e^(iθ): pure rotation by θ, no scaling. The unit circle acts as the rotation group SO(2).
• Multiply by i: rotation by π/2 (90° counterclockwise). 1 → i → −1 → −i → 1.
• Multiply by −1: rotation by π (point reflection through origin).

Conjugate — reflection

The conjugate of z = a + bi is z̄ = a − bi. Geometrically, it reflects z across the real axis.

          imag
            │
          3 ┤  • z = 2 + 3i
            │
          1 ┤
   ────────┼─────── real
         −1┤        2
         −3┤  • z̄ = 2 − 3i
            │

Three identities make conjugates indispensable:

z + z̄ = 2 Re(z)            (real part doubled)
z − z̄ = 2i Im(z)           (imaginary part doubled, times i)
z · z̄ = a² + b² = |z|²    (modulus squared, always real, ≥ 0)

The third identity is what lets us divide.

Division — the conjugate trick

How do you divide by a complex number? You can't put c + di in a denominator and walk away — you need a real denominator. The trick is to multiply top and bottom by the conjugate w̄ = c − di:

z/w = (a + bi) / (c + di)
    = (a + bi)(c − di) / ((c + di)(c − di))
    = ((ac + bd) + (bc − ad)i) / (c² + d²)
    = (ac + bd)/(c² + d²) + (bc − ad)/(c² + d²) · i

The denominator is now |w|², a non-negative real. The expansion of the numerator follows the multiplication rule with −di in place of di. In polar form, division is even cleaner:

z/w = r₁e^(iθ₁) / (r₂e^(iθ₂)) = (r₁/r₂) · e^(i(θ₁−θ₂))

Divide moduli, subtract arguments. Multiplication and division are mirror operations on the polar pair (r, θ).

Cartesian vs polar — when each form wins

	Cartesian (a + bi)	Polar (re^(iθ))
Addition / subtraction	Easy: componentwise	Painful: must convert
Multiplication	4 real multiplies + signs	1 multiply + 1 add
Division	Conjugate trick, divide by |w|²	1 divide + 1 subtract
n-th power	Repeated multiplication, error-prone	r^n · e^(inθ) — De Moivre
n-th roots	Solve a system, hard	r^(1/n) · e^(i(θ+2πk)/n)
Distance, modulus	√(a² + b²)	r directly
Best for	Sums, simple algebra, code	Powers, roots, rotations

Most working with complex numbers means switching forms as needed. Multiplying a sum of two complex numbers? Add in Cartesian, then convert. Computing z^17? Convert to polar first.

Where these operations show up

Phasor analysis in AC circuits

An AC voltage V₀ cos(ωt + φ) is represented as the complex phasor V₀ e^(iφ). Adding voltages from circuit branches becomes adding phasors. Capacitors and inductors contribute factors of 1/(iωC) and iωL — pure rotations of the phasor by ±π/2.

2D rotation in graphics

Rotating point (x, y) by angle θ is multiplying x + yi by e^(iθ). Two rotations compose by multiplying the phases — no matrix needed. This is the basis of the spinor representation later generalized to 3D quaternions.

Möbius transformations

Maps of the form z ↦ (az + b)/(cz + d) are conformal (angle-preserving) bijections of the extended complex plane. They take circles and lines to circles and lines. They underpin hyperbolic geometry models (Poincaré disk) and number-theoretic constructions (modular forms).

Fourier coefficients

For periodic functions, Fourier coefficients c_k = ∫ f(t) e^(−2πikt/T) dt are complex. Adding them shifts; multiplying them scales-and-rotates. The whole machinery of frequency-domain analysis is complex arithmetic on coefficients.

Quantum gates

A single-qubit quantum gate is a 2×2 unitary matrix with complex entries. Acting on a state amounts to complex linear combinations of |0⟩ and |1⟩ amplitudes. The Hadamard gate, Pauli X/Y/Z, phase gates — all written in terms of complex multiplication.

Counterexamples and cautions

• Distributing without expanding. Writing (a + bi)(c + di) = ac + bdi is the textbook error. You must expand all four terms and use i² = −1. The correct answer has both real (ac − bd) and imaginary (ad + bc) parts, and neither matches the naive guess.
• Division without conjugating. (2 + 3i)/(1 + i) is not 2 + 3i. Multiply by (1 − i)/(1 − i) to get (5 + i)/2.
• arg(zw) ≠ arg(z) + arg(w) on the principal branch. If both arguments are close to π, their sum can exceed π and wrap around. Arg is only additive modulo 2π; you may need to add or subtract 2π to land back in (−π, π].
• r = −1 is not the same point as (1, θ + π). Polar coordinates with negative radius are sometimes allowed by convention, but you must reflect through the origin: (r, θ) = (−r, θ + π). Mixing conventions silently leads to a 180° error.
• |z + w| ≠ |z| + |w| in general; only the triangle inequality holds: |z + w| ≤ |z| + |w|, with equality iff z and w lie on the same ray.

Common mistakes

• Forgetting the cross-term sign in multiplication. (a + bi)(c + di) = (ac − bd) + (ad + bc)i. The minus on bd comes from i² = −1; missing it is the most common bug.
• Conjugating only one term. conj(z + w) = z̄ + w̄, not z̄ + w. Conjugation distributes over both addition and multiplication.
• Treating Im(z) as a complex number. Im(z) is real — for z = a + bi, Im(z) = b, not bi.
• Picking the wrong arctangent quadrant. arg(z) = atan(b/a) only works in quadrant I. Use atan2(b, a) in code or check signs of a and b manually.
• Adding moduli when you should add vectors. |z + w| = |z| + |w| only when z and w point the same direction. In general, the triangle inequality leaves slack.
• Confusing e^(iθ) · e^(iφ) with e^(iθ + iφ). They're equal — exponential of sum equals product of exponentials. The error is treating them as different and reaching for the multiplication formula instead.