Complex number

A complex number is an element of a number system that extends the real numbers by introducing an imaginary unit \(i\) satisfying \(i^2 = -1\). A complex number can be written uniquely in the form \(a + bi\), where \(a\) and \(b\) are real numbers; here \(a\) is called the real part and \(b\) the imaginary part. Complex numbers form an algebraically closed field, meaning that every non-constant polynomial equation with complex coefficients has a solution in the complex numbers. They have fundamental applications in mathematics, physics, engineering, and many other sciences, providing a natural framework for describing oscillations, waves, quantum phenomena, and electrical circuits, among others.

Definition and basic properties

Formally, the set of complex numbers \(\mathbb{C}\) is defined as the set of all ordered pairs \((a, b)\) of real numbers, equipped with two operations: addition \((a, b) + (c, d) = (a + c, b + d)\) and multiplication \((a, b) \cdot (c, d) = (ac - bd, ad + bc)\). The pair \((0, 0)\) is the additive identity and \((1, 0)\) is the multiplicative identity. The imaginary unit \(i\) corresponds to the pair \((0, 1)\), and any complex number \((a, b)\) can be expressed as \(a + bi\) where \(a, b \in \mathbb{R}\). The real part of \(z = a + bi\) is \(\operatorname{Re}(z) = a\) and the imaginary part is \(\operatorname{Im}(z) = b\). A complex number with \(b = 0\) is purely real; one with \(a = 0\) and \(b \neq 0\) is purely imaginary.

Equality of complex numbers means \(a + bi = c + di\) if and only if \(a = c\) and \(b = d\). The set \(\mathbb{C}\) is a field under the defined addition and multiplication, and contains the real numbers as a subfield via the identification \(a \leftrightarrow a + 0i\). Notably, \(\mathbb{C}\) is not an ordered field: no total order can be defined that respects the arithmetic operations.

History

The concept of complex numbers emerged from the study of polynomial equations. In the 16th century, Italian mathematicians such as Gerolamo Cardano encountered square roots of negative numbers when solving cubic equations. Cardano formally introduced them in his Ars Magna (1545), though he called them “fictitious.” Rafael Bombelli later established the rules for operating with them. The term “imaginary” was coined by René Descartes in 1637 as a derogatory term, but the utility of these numbers grew. Leonhard Euler in the 18th century introduced the notation \(i\) for the imaginary unit and discovered the fundamental identity \(e^{i\theta} = \cos\theta + i\sin\theta\). The geometric representation as points in a plane was developed by Caspar Wessel, Jean-Robert Argand, and Carl Friedrich Gauss, leading to the now-standard interpretation. The rigorous foundation as ordered pairs of reals was given by William Rowan Hamilton in the 19th century.

Geometric representation: the complex plane

A complex number \(z = a + bi\) can be represented as the point \((a, b)\) in a Cartesian coordinate plane, where the horizontal axis is called the real axis and the vertical axis the imaginary axis. This is the complex plane or Argand diagram. The modulus (or absolute value) of \(z\) is the distance from the origin: \(|z| = \sqrt{a^2 + b^2}\). The argument of \(z\), denoted \(\arg(z)\), is the angle \(\theta\) between the positive real axis and the line from the origin to the point, measured in radians. The principal value of the argument usually lies in \((-\pi, \pi]\). A complex number with \(|z| = 1\) lies on the unit circle. The geometric view makes operations like addition correspond to vector addition, and multiplication to a combination of scaling and rotation.

Polar form and Euler's formula

Using the modulus \(r = |z|\) and argument \(\theta = \arg(z)\), a complex number can be written in polar form:

\[
z = r(\cos\theta + i\sin\theta).
\]

Euler's formula, \(e^{i\theta} = \cos\theta + i\sin\theta\), provides a compact exponential representation:

\[
z = r e^{i\theta}.
\]

This representation greatly simplifies multiplication: if \(z_1 = r_1 e^{i\theta_1}\) and \(z_2 = r_2 e^{i\theta_2}\), then \(z_1 z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)}\). Division, powers, and roots are also elegantly expressed. De Moivre's theorem, \((\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)\), follows directly. The polar form reveals the periodic nature of the argument: \(e^{i\theta} = e^{i(\theta + 2\pi k)}\) for any integer \(k\).

Operations and their properties

Addition and subtraction are performed component-wise: \((a + bi) \pm (c + di) = (a \pm c) + (b \pm d)i\). Multiplication uses \(i^2 = -1\): \((a + bi)(c + di) = (ac - bd) + (ad + bc)i\). The complex conjugate of \(z = a + bi\) is \(\overline{z} = a - bi\). It satisfies \(z \overline{z} = |z|^2\), which is a real number. Division of \(z_1\) by \(z_2 \neq 0\) is defined as \(z_1 / z_2 = (z_1 \overline{z_2}) / |z_2|^2\). The conjugate respects addition and multiplication: \(\overline{z + w} = \overline{z} + \overline{w}\) and \(\overline{zw} = \overline{z} \, \overline{w}\). The modulus is multiplicative: \(|zw| = |z| |w|\), and satisfies the triangle inequality: \(|z + w| \le |z| + |w|\). The real and imaginary parts can be recovered by \(\operatorname{Re}(z) = (z + \overline{z})/2\) and \(\operatorname{Im}(z) = (z - \overline{z})/(2i)\).

The field of complex numbers

\(\mathbb{C}\) with the operations defined above is a field. It is algebraically closed, a property stated by the fundamental theorem of algebra: every non-constant polynomial \(P(z)\) with complex coefficients has at least one complex root. Indeed, a polynomial of degree \(n\) has exactly \(n\) roots when counted with multiplicity. The complex numbers are the algebraic closure of the real numbers. There is no larger field that contains \(\mathbb{C}\) and is both commutative and a finite-dimensional extension of \(\mathbb{R}\); any further structure (like quaternions) sacrifices commutativity.

Complex functions and complex analysis

A complex function \(f: D \subseteq \mathbb{C} \to \mathbb{C}\) maps complex numbers to complex numbers. Complex analysis studies functions that are holomorphic (complex differentiable) in an open set. A function is holomorphic if the limit defining the derivative exists: \(f'(z) = \lim_{h \to 0} \frac{f(z+h) - f(z)}{h}\). Holomorphic functions are extremely rigid: they are infinitely differentiable, analytic (locally representable by power series), and satisfy the Cauchy–Riemann equations. Important examples include the exponential function \(e^z = e^x (\cos y + i \sin y)\), trigonometric functions \(\sin z, \cos z\), and the complex logarithm \(\log z = \ln|z| + i \arg(z)\). Complex integration along curves leads to Cauchy's integral theorem and the residue theorem, which are powerful tools for evaluating real integrals.

Exponential and trigonometric functions

The complex exponential extends the real exponential: for \(z = x + iy\), \(e^z = e^x e^{iy} = e^x (\cos y + i \sin y)\). It is periodic with period \(2\pi i\). The sine and cosine can be expressed as:

\[
\sin z = \frac{e^{iz} - e^{-iz}}{2i}, \quad \cos z = \frac{e^{iz} + e^{-iz}}{2}.
\]

These definitions yield hyperbolic functions and identities entirely analogous to their real counterparts. The complex logarithm is multivalued: \(\log z = \ln |z| + i(\theta + 2\pi k)\); a principal branch is usually chosen, typically with \(\theta \in (-\pi, \pi]\). Complex powers \(z^\alpha\) for \(\alpha \in \mathbb{C}\) are defined via the exponential and logarithm: \(z^\alpha = e^{\alpha \log z}\).

Roots of unity and cyclotomy

An \(n\)-th root of unity is a complex number \(\zeta\) satisfying \(\zeta^n = 1\). These are given by \(\zeta_k = e^{2\pi i k / n}\) for \(k = 0, 1, \dots, n-1\). They lie at the vertices of a regular \(n\)-gon inscribed in the unit circle. The primitive \(n\)-th roots are those for which \(\gcd(k,n)=1\); they generate the cyclic group of all \(n\)-th roots. Roots of unity are fundamental in discrete Fourier transforms, number theory (cyclotomic fields), and the factorization of polynomials.

Applications

Complex numbers are ubiquitous in science and engineering.

In electrical engineering: Alternating current circuits are analyzed using complex impedance \(Z = R + iX\), where resistance and reactance combine. Phasor representations treat sinusoidal signals as complex exponentials, greatly simplifying calculations.

In signal processing: The Fourier transform uses complex exponentials to decompose signals into frequency components. The discrete and fast Fourier transforms are implemented with complex arithmetic.

In quantum mechanics: The state of a system is described by a complex-valued wavefunction; the Schrödinger equation is fundamentally complex. Observables correspond to Hermitian operators, and probability amplitudes are complex.

In control theory and dynamic systems: Poles and zeros in the complex plane determine stability and response characteristics of linear systems.

In geometry and graphics: The complex plane provides a natural setting for 2D conformal mappings, rotations, and fractals (e.g., the Mandelbrot set, generated by iterating \(z_{n+1} = z_n^2 + c\) in \(\mathbb{C}\)).

In number theory: Complex numbers appear in the Riemann zeta function and analytic number theory, and elliptic curves over \(\mathbb{C}\) are tori.

Generalizations

Complex numbers can be extended in several ways. Quaternions \(\mathbb{H}\) are a 4‑dimensional non‑commutative algebra over \(\mathbb{R}\) with basis \(\{1, i, j, k\}\) and relations \(i^2=j^2=k^2=ijk=-1\). Octonions form an 8‑dimensional non‑associative algebra. More abstract algebraic constructions view \(\mathbb{C}\) as an algebraic closure of \(\mathbb{R}\) or as a real algebra of dimension 2. In analysis one studies several complex variables, leading to the theory of complex manifolds. In physics, complexification of Lie algebras and the use of twistor theory rely heavily on complex structures.

See also

• Imaginary unit
• Complex plane
• Euler's formula
• Complex analysis
• Quaternion
• Mandelbrot set
• Phasor
• Complex conjugate