A complex number has the form a + bi, where a and b are real numbers and i is defined by i² = −1 — a quantity with no counterpart among the ordinary real numbers. Once dismissed as "imaginary" and fictitious, complex numbers turned out to be essential to electrical engineering, quantum mechanics, fluid dynamics, and pure mathematics, including the only fully satisfying setting in which every polynomial equation has a solution.
What is the square root of −1? With ordinary numbers, there's no answer — squaring any real number (positive or negative) always gives a non-negative result. Mathematicians eventually decided: let's simply invent a new number that solves this. They called it i, defined precisely by i² = −1.
The name "imaginary number" was originally coined dismissively by René Descartes in 1637, who doubted these numbers had genuine mathematical legitimacy. The name unfortunately stuck, even though complex numbers are every bit as mathematically legitimate, rigorously well-defined, and practically useful as the "real" numbers — the name reflects historical skepticism, not any actual mathematical deficiency.
Addition: (a+bi)+(c+di) = (a+c)+(b+d)i. Multiplication: (a+bi)(c+di) = (ac−bd)+(ad+bc)i, using i²=−1 to simplify. Complex numbers form a field ℂ (see the Field entry) — every non-zero complex number has a multiplicative inverse.
Every complex number a+bi can be plotted as a point (a,b) on a 2D plane — the real part on the horizontal axis, the imaginary part on the vertical axis. This geometric representation (developed by Wessel, 1799, and popularised by Argand, 1806, hence sometimes called the "Argand plane") makes complex arithmetic visually intuitive: addition corresponds to vector addition, and multiplication corresponds to a combination of rotation and scaling.
Any complex number can also be written in polar form, z = r(cos θ + i sin θ) = re^(iθ), where r is the distance from the origin (the "modulus") and θ is the angle from the positive real axis (the "argument"). This connects directly to Euler's Identity (see that entry): multiplying two complex numbers in polar form multiplies their moduli and adds their arguments — turning complex multiplication into simple rotation-and-scaling.
The conjugate of a+bi is a−bi, written z̄. Multiplying a complex number by its conjugate always gives a non-negative real number: (a+bi)(a−bi) = a²+b² = |z|², which is exactly the squared modulus (distance from the origin). This trick is essential for dividing complex numbers and for many other calculations.
As covered in the Polynomial entry, every non-constant polynomial with complex coefficients has a root in ℂ (Gauss, 1799). This means ℂ is algebraically closed — no further number system extension is needed to solve polynomial equations. This is one of the deepest reasons complex numbers are considered the "right" and complete setting for algebra, not merely a convenient extension.
Surprisingly, complex numbers first arose not from trying to solve x²=−1 (which mathematicians were content to simply declare impossible for centuries), but from Gerolamo Cardano's and Rafael Bombelli's work on cubic equations in the 1500s. Cardano's formula for solving cubics sometimes required taking the square root of a negative number as an intermediate step, even when the final answer was a perfectly ordinary real number — Bombelli (1572) showed that formally manipulating these "impossible" square roots according to consistent rules, then simplifying, correctly yielded the right real answer, providing early practical evidence that these strange quantities were mathematically meaningful and useful, whatever their philosophical status.
Full mathematical acceptance came gradually. Gauss's rigorous proof of the Fundamental Theorem of Algebra (1799) and his systematic use of the complex plane (which he referred to, more neutrally than "imaginary," as representing numbers of a "lateral" or "complex" nature) were pivotal in establishing complex numbers as legitimate, indispensable mathematical objects rather than mere calculational curiosities or tricks.
Functions of a complex variable that are complex-differentiable ("holomorphic") turn out to have an extraordinarily rigid structure compared to ordinary real-valued calculus: a holomorphic function is automatically infinitely differentiable, and is completely determined everywhere by its values on any small open region (the "identity theorem" and "analytic continuation"). Cauchy's Integral Formula and Cauchy's Integral Theorem, developed substantially by Augustin-Louis Cauchy in the early-to-mid 1800s, are foundational results in this theory, with applications extending into number theory (as seen in the Riemann Hypothesis entry) and mathematical physics.
Unlike most classical physics, where complex numbers are often merely a convenient calculational tool that could in principle be avoided, quantum mechanics uses complex numbers in an essential, seemingly unavoidable way — the wavefunction ψ is fundamentally complex-valued, and quantities like probability amplitudes rely fundamentally on complex phase relationships between different possible quantum states. A 2021 series of experimental physics papers (Renou et al., published in Nature) provided empirical evidence, based on specific quantum information protocols, that complex numbers (not just an equivalent real-number reformulation) are genuinely required to correctly describe certain observed quantum phenomena — an active and still-developing area of foundational physics research.
William Rowan Hamilton (1843) extended complex numbers further to quaternions — four-dimensional numbers of the form a+bi+cj+dk, useful for representing 3D rotations (widely used in computer graphics and robotics) but sacrificing commutativity (ij≠ji in general). This illustrates a broader pattern in the historical extension of number systems: each successive extension (ℕ→ℤ→ℚ→ℝ→ℂ→quaternions→octonions) gains new capabilities while typically sacrificing some previously-held algebraic property.