Complex Numbers

What a complex number actually is

A complex number is an ordered pair of real numbers, written z = a + bi. The symbol i stands for a unit perpendicular to the real number line, defined by the single rule i² = −1. Everything else — addition, subtraction, multiplication, division — follows mechanically by treating i like any other symbol and substituting −1 wherever i² appears.

The geometric picture, the Argand plane, plots z = a + bi at coordinates (a, b):

        imaginary axis
              │
            3 ┤        • z = 2 + 3i
            2 ┤      ╱
            1 ┤    ╱
   ─────────┼──╱─────────── real axis
           −1┤
           −2┤
              │

So a complex number is just a 2D vector with a special multiplication rule. That rule is what makes complex numbers powerful — and what distinguishes them from plain pairs (a, b).

The four operations, derived

Take two complex numbers z = a + bi and w = c + di. The arithmetic falls out of distributing and replacing i² = −1:

Addition:        z + w = (a + c) + (b + d)i
Subtraction:     z − w = (a − c) + (b − d)i
Multiplication:  z·w   = (a + bi)(c + di)
                       = ac + adi + bci + bdi²
                       = (ac − bd) + (ad + bc)i
Division:        z/w   = z·w̄ / (w·w̄)
                       = ((a + bi)(c − di)) / (c² + d²)
                       = ((ac + bd) + (bc − ad)i) / (c² + d²)

Division uses the conjugate trick: multiply top and bottom by w̄ = c − di so the denominator becomes c² + d² = |w|², a positive real number. The complex denominator vanishes; the rest is plain real division.

Modulus and argument — polar form

Every complex number has two natural descriptions:

• Cartesian: z = a + bi.
• Polar: z = r(cos θ + i sin θ) = r·e^(iθ), where r = |z| = √(a² + b²) and θ = arg(z) = atan2(b, a).

Polar form makes multiplication trivial: r₁e^(iθ₁) · r₂e^(iθ₂) = (r₁r₂) e^(i(θ₁+θ₂)). Multiply moduli, add arguments. Multiplication by a unit-modulus complex number is exactly rotation. This is the secret behind De Moivre's theorem, FFT roots of unity, and quantum phase.

Complex numbers vs real numbers

	Real numbers ℝ	Complex numbers ℂ
Dimension over ℝ	1	2
Solutions to x² + 1 = 0	None	±i
Algebraically closed	No	Yes (FTA)
Total order respecting +, ×	Yes	No
Geometric meaning of ×	Scaling	Rotation + scaling
Roots of unity	±1 only	n distinct n-th roots for every n
Natural for waves/oscillation	Awkward (sum of sin/cos)	Single e^(iωt) handles both

The fundamental theorem of algebra

Adding i to the reals doesn't just close x² + 1 = 0. It closes every polynomial. The fundamental theorem of algebra, proved rigorously by Gauss in his 1799 dissertation, states:

Every non-constant polynomial p(z) = a_n z^n + ... + a_0 with complex coefficients has at least one complex root. By induction, it has exactly n roots counted with multiplicity.

This is a far stronger property than the reals provide. x² + 1 is irreducible over ℝ but factors as (x − i)(x + i) over ℂ. Once you allow i, every polynomial factors into linear pieces. There is no further extension needed — ℂ is the algebraic closure of ℝ.

Where complex numbers earn their keep

Fourier analysis and signal processing

The Fourier transform writes a signal as a sum of complex exponentials e^(iωt) = cos(ωt) + i sin(ωt). One complex exponential carries both sine and cosine, with amplitude in the modulus and phase in the argument. JPEG, MP3, MRI, and 5G modulation all depend on this representation.

Electrical engineering — impedance

For a sinusoidal current at angular frequency ω, a resistor has impedance R, an inductor has iωL, and a capacitor has 1/(iωC). Complex impedance reduces every linear circuit to Ohm's law in ℂ — phase shifts and amplitude scaling are encoded in a single complex number.

Quantum mechanics — amplitudes

Quantum states are vectors with complex coefficients. Probabilities come from the squared modulus |⟨φ|ψ⟩|². Phase differences between complex amplitudes produce interference — the double-slit pattern is the modulus-squared of a sum of two complex exponentials.

Control theory — root locus

A linear feedback system is stable iff every root of its characteristic polynomial lies in the left half of the complex plane (Re(z) < 0). Engineers literally plot poles and zeros on the Argand plane to design controllers.

Geometry and graphics

2D rotation by angle θ is multiplication by e^(iθ). Graphics libraries that pre-date matrix math used complex numbers directly; modern engines lift the same idea to 3D via quaternions, which are 4D analogs of ℂ.

Number theory — roots of unity

The n-th roots of unity e^(2πik/n) form a cyclic group on the unit circle. They power the Fast Fourier Transform (O(n log n) instead of O(n²)), Gaussian-period proofs of quadratic reciprocity, and lattice-based cryptography.

Counterexamples and cautions

• √(ab) = √a·√b fails for negatives. Naively, √(−1)·√(−1) = √((−1)(−1)) = √1 = 1, but √(−1)·√(−1) = i·i = −1. The identity only holds when at least one of a, b is non-negative.
• Complex log is multivalued. log(z) = ln|z| + i·arg(z), but arg is determined only modulo 2π. Choosing a single value (the principal branch) requires cutting the plane along a ray, usually the negative real axis. Cross the cut and your log jumps by 2πi.
• arg vs Arg. Lowercase arg(z) is the multivalued set; capital Arg(z) ∈ (−π, π] is the principal branch. Mixing them silently gives wrong answers off the cut.
• You can't say "z₁ < z₂" between two complex numbers. No order is compatible with addition and multiplication. Compare moduli, real parts, or arguments — never the numbers themselves.
• Powers z^w are also multivalued. z^w = e^(w log z) inherits log's branch ambiguity. i^i, for example, equals e^(−π/2) on the principal branch but e^(−π/2 + 2πk) for any integer k.

Common mistakes

• Distributing (a + bi)(c + di) as ac + bdi. You must expand all four products and use i² = −1. Result is (ac − bd) + (ad + bc)i, not ac + bdi.
• Forgetting the conjugate when dividing. Writing (a + bi)/(c + di) and just dividing termwise is wrong. Multiply numerator and denominator by c − di first.
• Confusing arg and Arg. Asking "what's the argument of −1?" Both 0 + π and 0 − π are valid; the principal branch picks π.
• Thinking imaginary means fictional. The name is historical baggage. Complex numbers are as well-defined as integers; they just happen to live in 2D rather than 1D.
• Computing |z|² as |a|² + |b|². The modulus squared is a² + b² for z = a + bi — no absolute value bars on the components. Confusing the two is mostly harmless until a or b is itself complex (in higher constructions).
• Mistaking e^(iπ) = 1. It's −1, not 1. The full rotation 2π brings you back to 1; π takes you halfway around.