Complex number

• 1. Introduction

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  René Descartes called them imaginary. In 1545, the Italian mathematician Gerolamo Cardano described them as being "as subtle as they are useless," and confessed that working with them felt like "mental torture." The thing tormenting him was a complex number: an element of a number system that extends the real numbers by adding a single new element, denoted i, the imaginary unit. No real number satisfies the equation that defines i, because the square of any real number, positive or negative, is never negative. So mathematicians invented a number that does. Every complex number can be written using two real parts, one called the real part and one called the imaginary part. Despite the dismissive name imaginary, these numbers have a mathematical existence as firm as that of the real numbers. They are fundamental tools in the scientific description of the natural world. How did a quantity born of frustration over cubic equations become essential to quantum mechanics, electrical engineering, and the geometry of the plane? Why did even Leonhard Euler stumble over them? And how did a number nobody could picture eventually become a point you could draw?

• 2. A Field That Extends The Reals

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  Addition, subtraction, and multiplication of complex numbers all follow the associative, commutative, and distributive laws familiar from ordinary arithmetic. Two complex numbers are added by separately adding their real and imaginary parts, and subtraction works the same way. Multiplication takes advantage of the rule that defines i, which produces a special case known as the fundamental formula. This single rule is what distinguishes i from any real number. Every nonzero complex number has a multiplicative inverse, which means you can divide by any complex number except zero. These properties together make the complex numbers a field, the same algebraic structure shared by the rational and the real numbers, with the real numbers sitting inside as a subfield. The complex conjugate plays a quiet but essential role here. It is the reflection of a number about the real axis, and conjugating twice returns the original number. A complex number is real if and only if it equals its own conjugate. Multiplying a number by its conjugate yields a non-negative real number, and the square root of that product defines the absolute value, also called the modulus or magnitude. The conjugate is what makes division possible, a process sometimes called rationalization of the denominator. Curiously, taking the conjugate cannot be expressed using only addition, subtraction, multiplication, and division, which marks it as an operation of a different kind.

• 3. The Plane You Can Draw

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  A complex number can be identified with an ordered pair of real numbers, plotted as a point in a Euclidean plane called the complex plane or Argand diagram. The horizontal axis displays the real part, increasing to the right, while the imaginary part marks the vertical axis, increasing upward. The real numbers form the horizontal real line, and the unit complex numbers, those of absolute value one, form the unit circle around the origin. By Pythagoras' theorem, the absolute value is simply the distance from the origin to the point. This geometric picture turns arithmetic into motion. Adding a fixed complex number to every other one produces a translation across the plane. Multiplying by a fixed complex number is a similarity centered at the origin, dilating by the absolute value and rotating by the argument. The argument, sometimes called the phase, is the angle the radius makes with the positive real axis, measured in radians. It is defined only up to multiples of a full turn, since a rotation of 360 degrees leaves every point unchanged, so mathematicians often restrict it to a principal value. The polar form captures all of this at once, writing a number in terms of its absolute value and its argument. When two numbers in polar form are multiplied, their absolute values multiply while their arguments add. This neat behavior of angles is what links complex numbers to trigonometry, and it gives rise to formulas precise enough to compute the digits of pi.

• 4. Powers, Roots, And Closure

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  De Moivre's formula gives the n-th power of a complex number by repeatedly applying the rule that arguments add under multiplication. Running the formula in reverse produces roots, and here the complex numbers reveal a striking generosity. Any nonzero complex number has exactly n distinct complex n-th roots, evenly spaced because sine and cosine repeat. There are, for instance, four fourth roots of 1. Unlike a positive real number, which has a single natural positive n-th root, a complex number offers no natural way to single out one root over the others, so the n-th root is described as an n-valued function. This abundance of solutions reaches its grandest statement in the fundamental theorem of algebra, credited to Carl Friedrich Gauss and Jean le Rond d'Alembert. It asserts that every non-constant polynomial equation with real or complex coefficients has at least one complex solution. The rational numbers fail this test, and so do the real numbers, where a polynomial whose square is always positive has no real root. Because the complex numbers never fail it, they are called an algebraically closed field. The theorem has many proofs, drawing on analytic tools like Liouville's theorem, topological ones like the winding number, and arguments blending Galois theory with the fact that any real polynomial of odd degree has a real root. This closure is the cornerstone beneath nearly every application that follows.

• 5. From Mental Torture To Lateral Units

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  The square roots of negative numbers may have first flickered into view with Hero of Alexandria in the 1st century AD, who, in his Stereometrica, considered the volume of an impossible frustum of a pyramid and, apparently in error, simply replaced a negative value with its positive counterpart. Negative quantities themselves were not conceived of in Hellenistic mathematics. The real impetus came in the 16th century, when Italian mathematicians, among them Niccolò Fontana Tartaglia, Scipione del Ferro, and Cardano, discovered algebraic solutions for the roots of cubic and quartic polynomials. Their formulas sometimes demanded square roots of negative numbers even when the answers sought were entirely real, a snag known as casus irreducibilis. Cardano set down his ideas around 1545 in his Ars Magna, though his grasp remained rudimentary, and he ignored the imaginary answers as useless. Rafael Bombelli was the first to confront these paradoxical solutions head-on, developing the rules for addition, subtraction, multiplication, and root extraction of complex numbers. The dismissive label stuck when René Descartes coined the term imaginary in 1637, stressing their unreal nature and writing that sometimes "there exists no quantity that matches that which we imagine." Even Euler was bedeviled by a false identity that misbehaved when two quantities were both negative, a difficulty that led to the convention of a special symbol to guard against the mistake. Yet Euler thought it natural to introduce students to these numbers early, doing exactly that in his textbook Elements of Algebra. The full rehabilitation took longer. William Rowan Hamilton built a more abstract formalism and carried it onward into the theory of quaternions.

• 6. The Point In The Plane

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  In 1799, the Danish-Norwegian mathematician Caspar Wessel first described a complex number as a point in the plane, an idea anticipated as early as 1685 in Wallis's A Treatise of Algebra. Wessel's memoir appeared in the Proceedings of the Copenhagen Academy and went largely unnoticed. In 1806, Jean-Robert Argand independently issued a pamphlet on the subject and offered a rigorous proof of the fundamental theorem of algebra. Gauss had published an essentially topological proof of that theorem back in 1797, yet he confessed doubts about "the true metaphysics of the square root of minus one." Not until 1831 did he overcome them and publish his treatise treating complex numbers as points in the plane, fixing much of the modern notation and terminology. Gauss blamed earlier confusion on clumsy language, suggesting that had the units been called direct, inverse, or lateral rather than positive, negative, or imaginary, "there could scarcely have been talk of such darkness." Other mathematicians, including Buée, Mourey, Warren, and Bellavitis, discovered the geometric representation independently. The English mathematician G.H. Hardy remarked that Gauss was the first to use complex numbers in "a really confident and scientific way," even as Niels Henrik Abel and Carl Gustav Jacob Jacobi were already using them routinely. The vocabulary of the field still bears the fingerprints of its founders. Argand named the modulus, Cauchy introduced the term argument, Gauss coined the phrase complex number itself, and Weierstrass gave us absolute value.

• 7. The Reach Into Science

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  Complex numbers turn up across signal processing, control theory, electromagnetism, fluid dynamics, quantum mechanics, cartography, and vibration analysis. In electrical engineering, the imaginary unit is written j rather than i, since i already denotes electric current. There, the treatment of resistors, capacitors, and inductors is unified by combining them into a single complex number called the impedance, an approach known as phasor calculus. Signal analysis leans on the same idea: a sine wave of a given frequency is encoded so that the absolute value gives the amplitude and the argument gives the phase, a technique that extends into digital audio, image, and video processing through Fourier and wavelet analysis. The reach goes deeper still into physics. The complex number field is intrinsic to the mathematical formulations of quantum mechanics, where complex Hilbert spaces provide the standard setting, and both the Schrödinger equation and Heisenberg's matrix mechanics depend on it. In special and general relativity, certain metric formulas on spacetime simplify if the time component is taken to be imaginary, and complex numbers are essential to spinors. In control theory, systems move from the time domain to the complex frequency domain through the Laplace transform, and the placement of zeros and poles in the left or right half plane decides whether a system is stable, unstable, or marginally stable. The mathematics turns inward too. Marden's theorem locates the foci of a triangle's Steiner inellipse using the roots of a cubic, and the Mandelbrot set, that famous fractal, is defined by plotting every point on the complex plane where an iterated sequence refuses to diverge.

• 8. Where Complex Numbers Sit

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  Beyond the picture of a point in the plane, complex numbers can be defined entirely through abstract algebra, as the quadratic extension field of the real numbers in which a particular polynomial splits. They can even be built as a quotient of the polynomial ring over the reals, by an ideal generated by an irreducible polynomial, or represented concretely as a special family of two-by-two real matrices whose sums and products stay within the family. In that matrix picture, the determinant corresponds to the square of the absolute value, the transpose corresponds to the conjugate, and the polar form appears as scaled rotation matrices. The complex field can also be pinned down by three properties: it has characteristic zero, its transcendence degree over the rationals is the cardinality of the continuum, and it is algebraically closed. Any field with these properties is isomorphic to it, including the algebraic closure of the p-adic numbers, though specifying that isomorphism requires the axiom of choice. The construction that builds the complex numbers from the reals, the Cayley-Dickson construction, can be applied again and again to yield the quaternions, the octonions, the sedenions, and the trigintaduonions. Each step costs something. The complex numbers are not an ordered field, because in any ordered field the square of every element must be positive. Passing to the quaternions loses commutativity, and the octonions lose associativity as well. By Hurwitz's theorem, the reals, complex numbers, quaternions, and octonions are the only normed division algebras, and the sedenions, the next rung, fall short. The complex field is the completion of the rationals under the usual absolute value, and the completion of the p-adic numbers turns out to be algebraically closed too, earning the name p-adic complex numbers.