# Complex number
A complex number is a number of the form \( z = x + iy \), where \( x \) and \( y \) are real numbers and \( i \) is a root of the polynomial \( X^2 + 1 \), so that \( i^2 = -1 \). Algebraically, the set \( \mathbb{C} \) of complex numbers is the extension of the field of real numbers \( \mathbb{R} \) obtained by adjoining such a root \( i \)[1](https://encyclopediaofmath.org/wiki/Complex_number). The form \( x + iy \) is called the algebraic or Cartesian form; \( x \) is the real part and \( y \) the imaginary part[1](https://encyclopediaofmath.org/wiki/Complex_number). Complex numbers complete the real numbers in a precise sense: \( \mathbb{C} \) is algebraically closed, and it is the algebraic closure of \( \mathbb{R} \)[1](https://encyclopediaofmath.org/wiki/Complex_number).
| Key fact | Detail |
|---|---|
| Definition | \( z = x + iy \) with \( x, y \in \mathbb{R} \) and \( i^2 = -1 \); equivalently the extension of \( \mathbb{R} \) by a root of \( X^2+1 \)[1](https://encyclopediaofmath.org/wiki/Complex_number) |
| Arithmetic | \( z_1 z_2 = x_1 x_2 - y_1 y_2 + i(x_1 y_2 + x_2 y_1) \); division uses the conjugate \( \overline{z} = x - iy \)[2](https://dlmf.nist.gov/1.9) |
| Modulus and phase | \( \left|z\right| = r = (x^2+y^2)^{1/2} \), \( \mathrm{ph}\,z = \theta + 2n\pi \)[2](https://dlmf.nist.gov/1.9) |
| Polar form | \( z = r(\cos\theta + i\sin\theta) = r e^{i\theta} \), with \( e^{i\theta} = \cos\theta + i\sin\theta \)[2](https://dlmf.nist.gov/1.9) |
| Multiplication rule | Moduli multiply and phases add: \( \left|z_1 z_2\right| = \left|z_1\right|\left|z_2\right| \), \( \mathrm{ph}(z_1 z_2) = \mathrm{ph}\,z_1 + \mathrm{ph}\,z_2 \)[2](https://dlmf.nist.gov/1.9) |
| Ordering | \( \mathbb{C} \) cannot be made into an ordered field, because a field admits an order if and only if \( -1 \) cannot be written as a sum of squares[3](https://encyclopediaofmath.org/wiki/Ordered_field) |
| Terminology | Gauss introduced the symbol \( i \) for \( \sqrt{-1} \), the term "complex number" for \( a+bi \), and called \( a^2+b^2 \) the "norm"[4](https://etc.usf.edu/lit2go/103/history-of-modern-mathematics/1729/article-4-complex-numbers/) |
## Arithmetic
Addition and subtraction act componentwise: for \( z_1 = x_1 + iy_1 \) and \( z_2 = x_2 + iy_2 \), \( z_1 \pm z_2 = x_1 \pm x_2 + i(y_1 \pm y_2) \)[2](https://dlmf.nist.gov/1.9). Multiplication follows from expanding the product and replacing \( i^2 \) with \( -1 \), giving \( z_1 z_2 = x_1 x_2 - y_1 y_2 + i(x_1 y_2 + x_2 y_1) \)[2](https://dlmf.nist.gov/1.9). Division is performed with the complex conjugate \( \overline{z} = x - iy \): since \( z\overline{z} = \left|z\right|^2 \) is real, \( z_1/z_2 = z_1\overline{z}_2/\left|z_2\right|^2 \) for \( z_2 \ne 0 \)[2](https://dlmf.nist.gov/1.9).
The conjugate satisfies \( \left|\overline{z}\right| = \left|z\right| \) and \( \mathrm{ph}\,\overline{z} = -\mathrm{ph}\,z \), and it converts quotients and products into manageable real quantities[2](https://dlmf.nist.gov/1.9). Powers follow De Moivre's theorem, \( \cos n\theta + i\sin n\theta = (\cos\theta + i\sin\theta)^n \) for \( n \in \mathbb{Z} \)[2](https://dlmf.nist.gov/1.9).
## Geometry and the polar form
A complex number \( x + iy \) can be plotted as the point \( (x, y) \) of the complex plane, with the horizontal axis carrying the real component and the vertical axis the imaginary component[5](https://openstax.org/books/precalculus/pages/3-1-complex-numbers). In this picture the modulus \( r = (x^2+y^2)^{1/2} \) is the distance from the origin and the phase \( \theta \) is the angle from the positive real axis[2](https://dlmf.nist.gov/1.9).
The polar form \( z = r e^{i\theta} \) rests on [Euler's formula](https://www.edgepedia.org/eulers-formula) \( e^{i\theta} = \cos\theta + i\sin\theta \)[2](https://dlmf.nist.gov/1.9). Under this representation, addition behaves like vector addition while multiplication acquires a geometric meaning: moduli multiply and phases add, so multiplying by \( e^{i\theta} \) rotates a point by the angle \( \theta \) and scaling the modulus stretches it[2](https://dlmf.nist.gov/1.9). The triangle inequality \( \left|\left|z_1\right|-\left|z_2\right|\right| \le \left|z_1+z_2\right| \le \left|z_1\right|+\left|z_2\right| \) expresses the same geometry as the corresponding statement for vectors[2](https://dlmf.nist.gov/1.9).
## Why the complex numbers cannot be ordered
The real numbers carry their usual order, and the order interacts with addition and multiplication: sums and products of positive elements are positive. The complex numbers admit no such order. A field can be made into an ordered field if and only if \( -1 \) cannot be written as a sum of squares; in \( \mathbb{C} \), \( -1 = i^2 \) is itself a square, so the condition fails[3](https://encyclopediaofmath.org/wiki/Ordered_field). Equivalently, in any ordered field every square is nonnegative, while the square of the imaginary unit is \( -1 \)[6](https://androma.org/theorems/10622). This obstruction holds for any total order on \( \mathbb{C} \), whether or not it restricts to the usual order on the embedded real line[6](https://androma.org/theorems/10622).
A field in which \( -1 \) is not a finite sum of squares is called formally real, and every ordered field is formally real; \( \mathbb{C} \) fails this test at the first step[3](https://encyclopediaofmath.org/wiki/Ordered_field). The contrast with \( \mathbb{R} \) is structural: \( \mathbb{R} \) is a complete ordered field, and adjoining \( i \) destroys orderability while gaining algebraic closure[3](https://encyclopediaofmath.org/wiki/Ordered_field).
## History
The recorded first encounter with complex numbers is attributed to [Gerolamo Cardano](https://www.edgepedia.org/gerolamo-cardano) (1501–1576), who used them in chapter 37 of his *Ars Magna*, published in 1545, as a tool for finding real solutions of cubic equations[7](https://mast.queensu.ca/~math211/M211OH/M211OH33.pdf). Rafael Bombelli (1526–1572) took the next step in his *Algebra* (1572), formulating the rules for calculating with these quantities[7](https://mast.queensu.ca/~math211/M211OH/M211OH33.pdf). After Descartes, leading mathematicians including Bernoulli, Moivre, and Euler made free use of complex numbers[7](https://mast.queensu.ca/~math211/M211OH/M211OH33.pdf).
**Geometric interpretation.** The idea of representing complex numbers graphically appeared as early as 1685 in Wallis's *De Algebra tractatus*. In the eighteenth century the surveyor Caspar Wessel made decided advances; his memoir appeared in the Proceedings of the Copenhagen Academy for 1799 and is described as exceedingly clear and complete even in comparison with modern works[4](https://etc.usf.edu/lit2go/103/history-of-modern-mathematics/1729/article-4-complex-numbers/). In 1804 the Abbé Buée independently arrived at the idea that \( \pm\sqrt{-1} \) should represent a unit line perpendicular to the real axis, publishing in 1806, the year Argand also issued a pamphlet on the same subject. The scientific foundation for the graphic representation is now generally referred to Argand's essay[4](https://etc.usf.edu/lit2go/103/history-of-modern-mathematics/1729/article-4-complex-numbers/).
**Terminology and acceptance.** Argand called \( \cos\varphi + i\sin\varphi \) the "direction factor" and \( r = \sqrt{a^2+b^2} \) the "modulus"; Cauchy in 1828 called \( \cos\varphi + i\sin\varphi \) the "reduced form"; Gauss used \( i \) for \( \sqrt{-1} \), introduced the term "complex number" for \( a+bi \), and called \( a^2+b^2 \) the "norm"[4](https://etc.usf.edu/lit2go/103/history-of-modern-mathematics/1729/article-4-complex-numbers/). Acceptance was slow: in 1831 Gauss found the theory quite unknown, and in 1832 he published his chief memoir on the subject, bringing it prominently before the mathematical world[4](https://etc.usf.edu/lit2go/103/history-of-modern-mathematics/1729/article-4-complex-numbers/). In that same year the logician [Augustus De Morgan](https://www.edgepedia.org/augustus-de-morgan) wrote in *On the Study and Difficulties of Mathematics* that \( \sqrt{-a} \) had been shown "to be void of meaning, or rather self-contradictory and absurd", while conceding that a part of algebra established by means of such symbols was of great utility[8](https://complexanalysis.org/web/sec_origin.html).
## Applications
Complex numbers are essential in the analysis and solution of mathematical problems across science and engineering[9](https://math.nist.gov/javanumerics/reports/jgfnwg-01.html). In electrical engineering they describe sinusoidal signals and impedances: a sinusoid \( A\cos(\omega t + \varphi) \) corresponds to the phasor \( Ae^{j\varphi} \), whose real part recovers the signal, and whose modulus \( A \) and phase \( \varphi \) give the amplitude and time shift[10](https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/d3be4ea78b036a6362230fb41780cf54_MIT6_013S09_notes.pdf). A capacitor's frequency-dependent impedance is \( Z_c = 1/(j\omega C) \), and [Ohm's law](https://www.edgepedia.org/ohms-law) applies with complex impedance replacing resistance[11](https://courses.media.mit.edu/2022spring/mas836/problem_sets/ps2.pdf).
In software, complex arithmetic is usually provided as a library type rather than a primitive. Java has no primitive complex type; the NIST Jampack library represents a complex number as a class \( \mathsf{Z} \) holding a real part and an imaginary part, each a double-precision float[12](https://math.nist.gov/pub/Jampack/Jampack/Doc/03_Complex.html). The Java Grande Forum's Numerics Working Group argued that complex operations should be as convenient and efficient as operations on `float` and `double`, noting that efficient complex arithmetic is only a few times slower than its real counterpart and that object overhead and reference semantics make a naive class implementation unacceptably inefficient[9](https://math.nist.gov/javanumerics/reports/jgfnwg-01.html).
## References
1. [Complex number, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Complex_number)
2. [DLMF §1.9 Calculus of a Complex Variable, NIST Digital Library of Mathematical Functions](https://dlmf.nist.gov/1.9)
3. [Ordered field, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Ordered_field)
4. [David Eugene Smith, Article 4: Complex Numbers, History of Modern Mathematics, Lit2Go ETC](https://etc.usf.edu/lit2go/103/history-of-modern-mathematics/1729/article-4-complex-numbers/)
5. [3.1 Complex Numbers, OpenStax Precalculus](https://openstax.org/books/precalculus/pages/3-1-complex-numbers)
6. [Complex Numbers Are Not an Ordered Field, Androma Theorem #10622](https://androma.org/theorems/10622)
7. [Complex Numbers, MATH 211 course notes, Queen's University](https://mast.queensu.ca/~math211/M211OH/M211OH33.pdf)
8. [The Origin of Complex Numbers, complexanalysis.org](https://complexanalysis.org/web/sec_origin.html)
9. [Improving Java for Numerical Computation, Java Grande Forum Numerics Working Group](https://math.nist.gov/javanumerics/reports/jgfnwg-01.html)
10. [6.013 Electromagnetics and Applications, Course Notes, MIT OpenCourseWare](https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/d3be4ea78b036a6362230fb41780cf54_MIT6_013S09_notes.pdf)
11. [MAS.836 Sensor Systems Problem Set 2, MIT Media Lab](https://courses.media.mit.edu/2022spring/mas836/problem_sets/ps2.pdf)
12. [The Jampack Owners Manual: Complex numbers, NIST](https://math.nist.gov/pub/Jampack/Jampack/Doc/03_Complex.html)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Arithmetic and number systems › Number systems › Real and complex number constructions › Complex numbers*
*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*
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