Complex number explained

i ^{2} = -1 ; every complex number can be expressed in the form a + bi , where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number is called the , and is called the . The set of complex numbers is denoted by either of the symbols C or . Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world.^{[1] [2]}

Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equation

(x +1 ) ^{2} = -9 has no real solution, because the square of a real number cannot be negative, but has the two nonreal complex solutions -1+3 i and -1-3 i .

Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule

i ^{2} =-1 along with the associative, commutative, and distributive laws. Every nonzero complex number has a multiplicative inverse. This makes the complex numbers a field with the real numbers as a subfield.

The complex numbers also form a real vector space of dimension two, with

\{ 1 ,i\} as a standard basis. This standard basis makes the complex numbers a Cartesian plane, called the complex plane. This allows a geometric interpretation of the complex numbers and their operations, and conversely some geometric objects and operations can be expressed in terms of complex numbers. For example, the real numbers form the real line, which is pictured as the horizontal axis of the complex plane, while real multiples of i are the vertical axis. A complex number can also be defined by its geometric polar coordinates: the radius is called the absolute value of the complex number, while the angle from the positive real axis is called the argument of the complex number. The complex numbers of absolute value one form the unit circle. Adding a fixed complex number to all complex numbers defines a translation in the complex plane, and multiplying by a fixed complex number is a similarity centered at the origin (dilating by the absolute value, and rotating by the argument). The operation of complex conjugation is the reflection symmetry with respect to the real axis.

The complex numbers form a rich structure that is simultaneously an algebraically closed field, a commutative algebra over the reals, and a Euclidean vector space of dimension two.

Definition and basic operations

A complex number is an expression of the form, where and are real numbers, and is an abstract symbol, the so-called imaginary unit, whose meaning will be explained further below. For example, is a complex number.^[3]

For a complex number, the real number is called its real part, and the real number (not the complex number) is its imaginary part.^[4] The real part of a complex number is denoted,

l{Re}(z) , or ak{R}(z) ; the imaginary part is, l{Im}(z) , or ak{I}(z) : for example,$\backslash operatorname(2\; +\; 3i)\; =\; 2$, \operatorname{Im}( 2 + 3 i) = 3 .

A complex number can be identified with the ordered pair of real numbers

(\Re (z),\Im (z)) , which may be interpreted as coordinates of a point in a Euclidean plane with standard coordinates, which is then called the complex plane or Argand diagram,^[5] .^[6] The horizontal axis is generally used to display the real part, with increasing values to the right, and the imaginary part marks the vertical axis, with increasing values upwards.

A real number can be regarded as a complex number, whose imaginary part is 0. A purely imaginary number is a complex number, whose real part is zero. As with polynomials, it is common to write,, and ; for example, .

The set of all complex numbers is denoted by

\Complex (blackboard bold) or (upright bold).

In some disciplines such as electromagnetism and electrical engineering, is used instead of, as frequently represents electric current, and complex numbers are written as or .

Addition and subtraction

Two complex numbers

a = x + yi and b = u + vi are added by separately adding their real and imaginary parts. That is to say:

$$a\; +\; b\; =(x+yi)\; +\; (u+vi)\; =\; (x+u)\; +\; (y+v)i.$$Similarly, subtraction can be performed as$$a\; -\; b\; =(x+yi)\; -\; (u+vi)\; =\; (x-u)\; +\; (y-v)i.$$

The addition can be geometrically visualized as follows: the sum of two complex numbers and, interpreted as points in the complex plane, is the point obtained by building a parallelogram from the three vertices, and the points of the arrows labeled and (provided that they are not on a line). Equivalently, calling these points,, respectively and the fourth point of the parallelogram the triangles and are congruent.

Multiplication

The product of two complex numbers is computed as follows:

(a + bi) ⋅ (c + di) = ac - bd + (ad + bc)i. For example, ( 3+2 i)( 4- i) = 3 ⋅ 4 - ( 2 ⋅ ( -1 )) + ( 3 ⋅ ( -1 ) + 2 ⋅ 4 )i = 14 +5 i. In particular, this includes as a special case the fundamental formulai ^{2} = i ⋅ i = -1 . This formula distinguishes the complex number i from any real number, since the square of any (negative or positive) real number x always satisfies x ^{2} \ge 0 .

With this definition of multiplication and addition, familiar rules for the arithmetic of rational or real numbers continue to hold for complex numbers. More precisely, the distributive property, the commutative properties (of addition and multiplication) hold. Therefore, the complex numbers form an algebraic structure known as a field, the same way as the rational or real numbers do.

Complex conjugate, absolute value and argument

The complex conjugate of the complex number is defined as

\overline z = x - yi. It is also denoted by some authors by z ^{*} . Geometrically, is the "reflection" of about the real axis. Conjugating twice gives the original complex number: \overline{\overline{z}} = z. A complex number is real if and only if it equals its own conjugate. The unary operation of taking the complex conjugate of a complex number cannot be expressed by applying only their basic operations addition, subtraction, multiplication and division.

For any complex number, the product

z ⋅ \overline z = (x + iy)(x - iy) = x ^{2} + y ^{2} is a non-negative real number. This allows to define the absolute value (or modulus or magnitude) of z to be the square root $$|z|=\backslash sqrt.$$By Pythagoras' theorem, |z| is the distance from the origin to the point representing the complex number z in the complex plane. In particular, the circle of radius one around the origin consists precisely of the numbers z such that |z| = 1 . If z = x = x + 0 i is a real number, then |z| = |x| : its absolute value as a complex number and as a real number are equal.

Using the conjugate, the reciprocal of a nonzero complex number

z = x + yi can be computed to be

$$\backslash frac=\; \backslash frac=\; \backslash frac$$

^2

= \frac= \frac - \fraci.More generally, the division of an arbitrary complex number w = u + vi by a non-zero complex number z = x + yi equals$$\backslash frac=\; \backslash frac$$

^2

= \frac= \frac + \fraci.This process is sometimes called "rationalization" of the denominator (although the denominator in the final expression might be an irrational real number), because it resembles the method to remove roots from simple expressions in a denominator.

The argument of (sometimes called the "phase") is the angle of the radius with the positive real axis, and is written as, expressed in radians in this article. The angle is defined only up to adding integer multiples of

2 \pi , since a rotation by 2 \pi (or 360°) around the origin leaves all points in the complex plane unchanged. One possible choice to uniquely specify the argument is to require it to be within the interval ( - \pi,\pi] , which is referred to as the principal value.^[7]

Fractal geometry

The Mandelbrot set is a popular example of a fractal formed on the complex plane. It is defined by plotting every location

c where iterating the sequence f ^{2+ c} _{c(z) = z} does not diverge when iterated infinitely. Similarly, Julia sets have the same rules, except where c remains constant.

Triangles

(x - a)(x - b)(x - c) =0 , take its derivative, and equate the (quadratic) derivative to zero. Marden's theorem says that the solutions of this equation are the complex numbers denoting the locations of the two foci of the Steiner inellipse.

Algebraic number theory

As mentioned above, any nonconstant polynomial equation (in complex coefficients) has a solution in

C . A fortiori, the same is true if the equation has rational coefficients. The roots of such equations are called algebraic numbers – they are a principal object of study in algebraic number theory. Compared to \overline{ Q }, the algebraic closure of Q , which also contains all algebraic numbers, C has the advantage of being easily understandable in geometric terms. In this way, algebraic methods can be used to study geometric questions and vice versa. With algebraic methods, more specifically applying the machinery of field theory to the number field containing roots of unity, it can be shown that it is not possible to construct a regular nonagon using only compass and straightedge – a purely geometric problem.

Another example is the Gaussian integers; that is, numbers of the form, where and are integers, which can be used to classify sums of squares.

Analytic number theory

See main article: Analytic number theory. Analytic number theory studies numbers, often integers or rationals, by taking advantage of the fact that they can be regarded as complex numbers, in which analytic methods can be used. This is done by encoding number-theoretic information in complex-valued functions. For example, the Riemann zeta function is related to the distribution of prime numbers.

Improper integrals

In applied fields, complex numbers are often used to compute certain real-valued improper integrals, by means of complex-valued functions. Several methods exist to do this; see methods of contour integration.

Dynamic equations

In differential equations, it is common to first find all complex roots of the characteristic equation of a linear differential equation or equation system and then attempt to solve the system in terms of base functions of the form . Likewise, in difference equations, the complex roots of the characteristic equation of the difference equation system are used, to attempt to solve the system in terms of base functions of the form .

Linear algebra

Since

\C is algebraically closed, any non-empty complex square matrix has at least one (complex) eigenvalue. By comparison, real matrices do not always have real eigenvalues, for example rotation matrices (for rotations of the plane for angles other than 0° or 180°) leave no direction fixed, and therefore do not have any real eigenvalue. The existence of (complex) eigenvalues, and the ensuing existence of eigendecomposition is a useful tool for computing matrix powers and matrix exponentials.

Complex numbers often generalize concepts originally conceived in the real numbers. For example, the conjugate transpose generalizes the transpose, hermitian matrices generalize symmetric matrices, and unitary matrices generalize orthogonal matrices.

In applied mathematics

Control theory

In control theory, systems are often transformed from the time domain to the complex frequency domain using the Laplace transform. The system's zeros and poles are then analyzed in the complex plane. The root locus, Nyquist plot, and Nichols plot techniques all make use of the complex plane.

In the root locus method, it is important whether zeros and poles are in the left or right half planes, that is, have real part greater than or less than zero. If a linear, time-invariant (LTI) system has poles that are

• in the right half plane, it will be unstable,
• all in the left half plane, it will be stable,
• on the imaginary axis, it will have marginal stability.

If a system has zeros in the right half plane, it is a nonminimum phase system.

Signal analysis

Complex numbers are used in signal analysis and other fields for a convenient description for periodically varying signals. For given real functions representing actual physical quantities, often in terms of sines and cosines, corresponding complex functions are considered of which the real parts are the original quantities. For a sine wave of a given frequency, the absolute value of the corresponding is the amplitude and the argument is the phase.

If Fourier analysis is employed to write a given real-valued signal as a sum of periodic functions, these periodic functions are often written as complex-valued functions of the form

$$x(t)\; =\; \backslash operatorname\; \backslash$$

and

$$X(t)\; =\; A\; e^\; =\; a\; e^\; e^\; =\; a\; e^$$

where ω represents the angular frequency and the complex number A encodes the phase and amplitude as explained above.

This use is also extended into digital signal processing and digital image processing, which use digital versions of Fourier analysis (and wavelet analysis) to transmit, compress, restore, and otherwise process digital audio signals, still images, and video signals.

Another example, relevant to the two side bands of amplitude modulation of AM radio, is:

In physics

Electromagnetism and electrical engineering

See main article: Alternating current. In electrical engineering, the Fourier transform is used to analyze varying voltages and currents. The treatment of resistors, capacitors, and inductors can then be unified by introducing imaginary, frequency-dependent resistances for the latter two and combining all three in a single complex number called the impedance. This approach is called phasor calculus.

In electrical engineering, the imaginary unit is denoted by, to avoid confusion with, which is generally in use to denote electric current, or, more particularly,, which is generally in use to denote instantaneous electric current.

Because the voltage in an AC circuit is oscillating, it can be represented as

$$V(t)\; =\; V\_0\; e^\; =\; V\_0\; \backslash left\; (\backslash cos\backslash omega\; t\; +\; j\; \backslash sin\backslash omega\; t\; \backslash right),$$

To obtain the measurable quantity, the real part is taken:

$$v(t)\; =\; \backslash operatorname(V)\; =\; \backslash operatorname\backslash left\; [V\_0\; e^\{j\; \backslash omega\; t\}\; \backslash right\; ]\; =\; V\_0\; \backslash cos\; \backslash omega\; t.$$

The complex-valued signal is called the analytic representation of the real-valued, measurable signal .^[38]

Fluid dynamics

In fluid dynamics, complex functions are used to describe potential flow in two dimensions.

Quantum mechanics

The complex number field is intrinsic to the mathematical formulations of quantum mechanics, where complex Hilbert spaces provide the context for one such formulation that is convenient and perhaps most standard. The original foundation formulas of quantum mechanics – the Schrödinger equation and Heisenberg's matrix mechanics – make use of complex numbers.

Relativity

In special and general relativity, some formulas for the metric on spacetime become simpler if one takes the time component of the spacetime continuum to be imaginary. (This approach is no longer standard in classical relativity, but is used in an essential way in quantum field theory.) Complex numbers are essential to spinors, which are a generalization of the tensors used in relativity.

Characterizations, generalizations and related notions

Algebraic characterization

The field

\Complex has the following three properties:

• First, it has characteristic 0. This means that for any number of summands (all of which equal one).
• Second, its transcendence degree over

\Q , the prime field of \Complex, is the cardinality of the continuum.

• Third, it is algebraically closed (see above).

It can be shown that any field having these properties is isomorphic (as a field) to

\Complex. For example, the algebraic closure of the field \Q _{p} of the -adic number also satisfies these three properties, so these two fields are isomorphic (as fields, but not as topological fields).^[39] Also, \Complex is isomorphic to the field of complex Puiseux series. However, specifying an isomorphism requires the axiom of choice. Another consequence of this algebraic characterization is that \Complex contains many proper subfields that are isomorphic to \Complex .

Characterization as a topological field

The preceding characterization of

\Complex describes only the algebraic aspects of \Complex. That is to say, the properties of nearness and continuity, which matter in areas such as analysis and topology, are not dealt with. The following description of \Complex as a topological field (that is, a field that is equipped with a topology, which allows the notion of convergence) does take into account the topological properties. \Complex contains a subset (namely the set of positive real numbers) of nonzero elements satisfying the following three conditions:

• is closed under addition, multiplication and taking inverses.
• If and are distinct elements of, then either or is in .
• If is any nonempty subset of, then for some in

\Complex. Moreover, \Complex has a nontrivial involutive automorphism (namely the complex conjugation), such that is in for any nonzero in \Complex.

Any field with these properties can be endowed with a topology by taking the sets as a base, where ranges over the field and ranges over . With this topology is isomorphic as a topological field to

\Complex.

The only connected locally compact topological fields are

\R and \Complex. This gives another characterization of \Complex as a topological field, because \Complex can be distinguished from \R because the nonzero complex numbers are connected, while the nonzero real numbers are not.

Other number systems

See main article: Cayley–Dickson construction, Quaternion and Octonion.

Number systems

	rational numbers \Q	real numbers \R	complex numbers \C	quaternions H	octonions O	sedenions S
complete
dimension as an \R -vector space	[does not apply]	1	2	4	8	16
multiplication commutative
multiplication associative
	[does not apply]

The process of extending the field

R of reals to C is an instance of the Cayley–Dickson construction. Applying this construction iteratively to \C then yields the quaternions, the octonions and the sedenions.^[40] This construction turns out to diminish the structural properties of the involved number systems.

Unlike the reals,

\Complex is not an ordered field, that is to say, it is not possible to define a relation that is compatible with the addition and multiplication. In fact, in any ordered field, the square of any element is necessarily positive, so precludes the existence of an ordering on \Complex. Passing from \C to the quaternions H loses commutativity, while the octonions (additionally to not being commutative) fail to be associative. The reals, complex numbers, quaternions and octonions are all normed division algebras over R . By Hurwitz's theorem they are the only ones; the sedenions, the next step in the Cayley–Dickson construction, fail to have this structure.

The Cayley–Dickson construction is closely related to the regular representation of

C, thought of as an R -algebra (an R -vector space with a multiplication), with respect to the basis . This means the following: the R -linear mapfor some fixed complex number can be represented by a matrix (once a basis has been chosen). With respect to the basis, this matrix isthat is, the one mentioned in the section on matrix representation of complex numbers above. While this is a linear representation of C in the 2 × 2 real matrices, it is not the only one. Any matrixhas the property that its square is the negative of the identity matrix: . Then$$\backslash$$is also isomorphic to the field C, and gives an alternative complex structure on R ^{2.} This is generalized by the notion of a linear complex structure.

Hypercomplex numbers also generalize

R, C, H, and O . For example, this notion contains the split-complex numbers, which are elements of the ring R[x]/(x ^{2-1)} (as opposed to R[x]/(x ^{2+1)} for complex numbers). In this ring, the equation has four solutions.

The field

R is the completion of Q, the field of rational numbers, with respect to the usual absolute value metric. Other choices of metrics on Q lead to the fields Q _{p} of -adic numbers (for any prime number), which are thereby analogous to R . There are no other nontrivial ways of completing Q than R and Q _{p,} by Ostrowski's theorem. The algebraic closures \overline { Q _{p}} of Q _{p} still carry a norm, but (unlike C ) are not complete with respect to it. The completion C _{p} of \overline { Q _{p}} turns out to be algebraically closed. By analogy, the field is called -adic complex numbers.

The fields

R, Q _{p,} and their finite field extensions, including C, are called local fields.

See also

• Analytic continuation
• Circular motion using complex numbers
• Complex-base system
• Complex coordinate space
• Complex geometry
• Geometry of numbers
• Dual-complex number
• Eisenstein integer
• Geometric algebra (which includes the complex plane as the 2-dimensional spinor subspace

l{G} ^{+} _{2} )

• Unit complex number

References

• Book: Ahlfors, Lars . Lars Ahlfors . 1979 . Complex analysis . 3rd . McGraw-Hill . registration . 978-0-07-000657-7.
  • Book: Apostol, Tom . Tom Apostol . 1981 . Mathematical analysis . Addison-Wesley.
• Book: Aufmann . Richard N. . College Algebra and Trigonometry . Barker . Vernon C. . Nation . Richard D. . Cengage Learning . 2007 . 978-0-618-82515-8 . 6 .
• Book: Conway, John B. . none . Functions of One Complex Variable I . 1986 . Springer . 978-0-387-90328-6.
• Book: Derbyshire, John . John Derbyshire . 2006 . Unknown Quantity: A real and imaginary history of algebra . Joseph Henry Press . 978-0-309-09657-7 .
• Book: none . Joshi . Kapil D. . Foundations of Discrete Mathematics . . New York . 978-0-470-21152-6 . 1989.
• Book: Needham, Tristan . 1997 . Visual Complex Analysis . Clarendon Press . 978-0-19-853447-1.
• Book: Pedoe, Dan . none . Daniel Pedoe . Geometry: A comprehensive course . Dover . 1988 . 978-0-486-65812-4.
• Book: Penrose, Roger . Roger Penrose . 2005 . The Road to Reality: A complete guide to the laws of the universe . Alfred A. Knopf . 978-0-679-45443-4 .
• Book: Press . W.H. . Teukolsky . S.A. . Vetterling . W.T. . Flannery . B.P. . 2007 . Numerical Recipes: The art of scientific computing . 3rd . Cambridge University Press . New York . 978-0-521-88068-8 . Section 5.5 Complex Arithmetic . http://apps.nrbook.com/empanel/index.html?pg=225 . 9 August 2011 . 13 March 2020 . https://web.archive.org/web/20200313111530/http://apps.nrbook.com/empanel/index.html?pg=225 . dead .

Historical references

• Argand . 1814 . Reflexions sur la nouvelle théorie des imaginaires, suives d'une application à la demonstration d'un theorème d'analise . Annales de mathématiques pures et appliquées . 5 . 197–209 . Reflections on the new theory of complex numbers, followed by an application to the proof of a theorem of analysis . fr.
• Book: Bourbaki, Nicolas . none . Nicolas Bourbaki . Elements of the history of mathematics . Foundations of mathematics § logic: set theory . Springer . 1998.
• Book: none . Burton . David M. . The History of Mathematics . . New York . 3rd . 978-0-07-009465-9 . 1995.
• Gauss . C. F. . 1831 . Theoria residuorum biquadraticorum. Commentatio secunda. . Theory of biquadratic residues. Second memoir. . Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores . 7 . 89–148 . la . Carl Friedrich Gauss.
• Book: none . Katz . Victor J. . A History of Mathematics, Brief Version . . 978-0-321-16193-2 . 2004.
• Book: Nahin, Paul J. . none . An Imaginary Tale: The Story of \scriptstyle\sqrt{ -1 } . Princeton University Press . 978-0-691-02795-1 . 1998. — A gentle introduction to the history of complex numbers and the beginnings of complex analysis.
• Book: H. D. . Ebbinghaus . H. . Hermes . F. . Hirzebruch . M. . Koecher . K. . Mainzer . J. . Neukirch . A. . Prestel . R. . Remmert . Numbers . Springer . 978-0-387-97497-2 . hardcover . 1991. — An advanced perspective on the historical development of the concept of number.

Notes and References

1. For an extensive account of the history of "imaginary" numbers, from initial skepticism to ultimate acceptance, see Book: Bourbaki, Nicolas . Nicolas Bourbaki . 1998 . Elements of the History of Mathematics . Foundations of Mathematics § Logic: Set theory . 18–24 . Springer.
2. "Complex numbers, as much as reals, and perhaps even more, find a unity with nature that is truly remarkable. It is as though Nature herself is as impressed by the scope and consistency of the complex-number system as we are ourselves, and has entrusted to these numbers the precise operations of her world at its minutest scales.", .
3. Book: Axler, Sheldon . College algebra . limited . 262. Wiley. 2010. 9780470470770 .
4. Book: Spiegel . M.R. . Complex Variables . Lipschutz . S. . Schiller . J.J. . Spellman . D. . 14 April 2009 . McGraw Hill . 978-0-07-161569-3 . 2nd . Schaum's Outline Series.
5. Book: Pedoe, Dan . Daniel Pedoe . Geometry: A comprehensive course . Dover . 1988 . 978-0-486-65812-4.
6. Web site: Weisstein . Eric W. . Complex Number . 2020-08-12 . mathworld.wolfram.com.
7. Book: Complex Variables: Theory And Applications. 2nd. Chapter 1. H.S.. Kasana. PHI Learning Pvt. Ltd. 2005. 978-81-203-2641-5. 14. https://books.google.com/books?id=rFhiJqkrALIC&pg=PA14.
8. Book: Nilsson . James William . Electric circuits . Riedel . Susan A. . Prentice Hall . 2008 . 978-0-13-198925-2 . 8th . 338 . Chapter 9 . https://books.google.com/books?id=sxmM8RFL99wC&pg=PA338.
9. Book: Kline, Morris . A history of mathematical thought, volume 1. 253.
10. Book: Jurij., Kovič. Tristan Needham, Visual Complex Analysis, Oxford University Press Inc., New York, 1998, 592 strani. 1080410598.
11. O'Connor and Robertson (2016), "Girolamo Cardano."
12. Nahin, Paul J. An Imaginary Tale: The Story of √−1. Princeton: Princeton University Press, 1998.
13. Book: Katz . Victor J. . A History of Mathematics, Brief Version . 9.1.4 . . 978-0-321-16193-2 . 2004.
14. Hamilton . Wm. . On a new species of imaginary quantities connected with a theory of quaternions . Proceedings of the Royal Irish Academy . 1844 . 2 . 424–434 .
15. Book: Nahin, Paul J. . An Imaginary Tale: The Story of √−1 . 2007 . . 978-0-691-12798-9 . 20 April 2011 . https://web.archive.org/web/20121012090553/http://mathforum.org/kb/thread.jspa?forumID=149&threadID=383188&messageID=1181284 . 12 October 2012 . live .
16. Book: Descartes, René . La Géométrie The Geometry of René Descartes with a facsimile of the first edition . René Descartes . 1954 . 1637 . . 978-0-486-60068-0 . 20 April 2011 .
17. Book: Euler . Leonard . Introductio in Analysin Infinitorum . Introduction to the Analysis of the Infinite . 1748 . Marc Michel Bosquet & Co. . Lucerne, Switzerland . 1 . 104 . la.
18. Wessel . Caspar . Om Directionens analytiske Betegning, et Forsog, anvendt fornemmelig til plane og sphæriske Polygoners Oplosning . Nye Samling af det Kongelige Danske Videnskabernes Selskabs Skrifter [New Collection of the Writings of the Royal Danish Science Society] . 1799 . 5 . 469–518 . On the analytic representation of direction, an effort applied in particular to the determination of plane and spherical polygons . da.
19. Book: Wallis, John . 1685 . A Treatise of Algebra, Both Historical and Practical ... . London, England . printed by John Playford, for Richard Davis . 264–273 .
20. Book: Argand . Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques . Essay on a way to represent complex quantities by geometric constructions . 1806 . Madame Veuve Blanc . Paris, France . fr.
21. Gauss, Carl Friedrich (1799) "Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse." [New proof of the theorem that any rational integral algebraic function of a single variable can be resolved into real factors of the first or second degree.] Ph.D. thesis, University of Helmstedt, (Germany). (in Latin)
22. Book: Ewald, William B. . 1996 . From Kant to Hilbert: A Source Book in the Foundations of Mathematics . 1 . 313 . Oxford University Press . 9780198505358. 18 March 2020.
23. Web site: Adrien Quentin Buée (1745–1845): MacTutor.
24. Buée . Mémoire sur les quantités imaginaires . Philosophical Transactions of the Royal Society of London . 1806 . 96 . 23–88 . 10.1098/rstl.1806.0003 . 110394048 . Memoir on imaginary quantities . fr.
25. Book: Mourey . C.V. . La vraies théore des quantités négatives et des quantités prétendues imaginaires . The true theory of negative quantities and of alleged imaginary quantities . 1861 . Mallet-Bachelier . Paris, France . fr. 1861 reprint of 1828 original.
26. Book: Warren . John . A Treatise on the Geometrical Representation of the Square Roots of Negative Quantities . 1828 . Cambridge University Press . Cambridge, England .
27. Warren . John . Consideration of the objections raised against the geometrical representation of the square roots of negative quantities . Philosophical Transactions of the Royal Society of London . 1829 . 119 . 241–254 . 186211638 . 10.1098/rstl.1829.0022 . free .
28. Warren . John . On the geometrical representation of the powers of quantities, whose indices involve the square roots of negative numbers . Philosophical Transactions of the Royal Society of London . 1829 . 119 . 339–359 . 125699726 . 10.1098/rstl.1829.0031 . free .
29. Français . J.F. . Nouveaux principes de géométrie de position, et interprétation géométrique des symboles imaginaires . Annales des mathématiques pures et appliquées . 1813 . 4 . 61–71 . New principles of the geometry of position, and geometric interpretation of complex [number] symbols . fr.
30. Book: Two Cultures . Kim Williams . Caparrini . Sandro . On the Common Origin of Some of the Works on the Geometrical Interpretation of Complex Numbers . 2000 . Birkhäuser . 978-3-7643-7186-9 . 139 . https://books.google.com/books?id=voFsJ1EhCnYC&pg=PA139.
31. Book: An Introduction to the Theory of Numbers . Hardy . G.H. . Wright . E.M. . 2000 . 1938 . . 978-0-19-921986-5 . 189 (fourth edition).
32. Web site: Jeff Miller . Sep 21, 1999 . MODULUS . https://web.archive.org/web/19991003034827/http://members.aol.com/jeff570/m.html . Earliest Known Uses of Some of the Words of Mathematics (M) . 1999-10-03 . usurped.
33. Book: Cauchy, Augustin-Louis . 1821 . Cours d'analyse de l'École royale polytechnique . Paris, France . L'Imprimerie Royale . 1 . 183 . fr .
34. Book: Hankel, Hermann . 1867 . Vorlesungen über die complexen Zahlen und ihre Functionen . Lectures About the Complex Numbers and Their Functions . Leipzig, [Germany] . Leopold Voss . 1 . 71 . de . From p. 71: "Wir werden den Factor (cos φ + i sin φ) haüfig den Richtungscoefficienten nennen." (We will often call the factor (cos φ + i sin φ) the "coefficient of direction".)
35. Other authors, including, chose the argument to be in the interval [ 0 , 2 \pi) .The argument can be computed from the rectangular form by means of the arctan (inverse tangent) function.^[7]

    Polar form

    See main article: Polar coordinate system.

    For any complex number z, with absolute value

    r = |z| and argument \varphi , the equation z = r(\cos\varphi + i\sin\varphi) holds. This identity is referred to as the polar form of z. It is sometimes abbreviated as $z\; =\; r\; \backslash operatorname\backslash mathrm\; \backslash varphi$. In electronics, one represents a phasor with amplitude and phase in angle notation:^[8] $$z\; =\; r\; \backslash angle\; \backslash varphi\; .$$

    If two complex numbers are given in polar form, i.e., and, the product and division can be computed as $$z\_1\; z\_2\; =\; r\_1\; r\_2\; (\backslash cos(\backslash varphi\_1\; +\; \backslash varphi\_2)\; +\; i\; \backslash sin(\backslash varphi\_1\; +\; \backslash varphi\_2)).$$$$\backslash frac\; =\; \backslash frac\; \backslash left(\backslash cos(\backslash varphi\_1\; -\; \backslash varphi\_2)\; +\; i\; \backslash sin(\backslash varphi\_1\; -\; \backslash varphi\_2)\backslash right),\; \backslash textz\_2\; \backslash ne\; 0.$$(These are a consequence of the trigonometric identities for the sine and cosine function.)In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product. The picture at the right illustrates the multiplication of$$(2+i)(3+i)=5+5i.$$Because the real and imaginary part of are equal, the argument of that number is 45 degrees, or (in radian). On the other hand, it is also the sum of the angles at the origin of the red and blue triangles are arctan(1/3) and arctan(1/2), respectively. Thus, the formula$$\backslash frac\; =\; \backslash arctan\backslash left(\backslash frac\backslash right)\; +\; \backslash arctan\backslash left(\backslash frac\backslash right)$$holds. As the arctan function can be approximated highly efficiently, formulas like this – known as Machin-like formulas – are used for high-precision approximations of .

    Powers and roots

    See also: Square root#Square roots of negative and complex numbers and l1=Square roots of negative and complex numbers. The n-th power of a complex number can be computed using de Moivre's formula, which is obtained by repeatedly applying the above formula for the product:$$z^=\backslash underbrace\_\; =\; (r(\backslash cos\; \backslash varphi\; +\; i\backslash sin\; \backslash varphi))^n\; =\; r^n\; \backslash ,\; (\backslash cos\; n\backslash varphi\; +\; i\; \backslash sin\; n\; \backslash varphi).$$For example, the first few powers of the imaginary unit i are

    i, i ^{2} = -1 , i ^{3} = - i, i ^{4} = 1 , i ^{5} = i, ... .

    The th roots of a complex number are given by$$z^\; =\; \backslash sqrt[n]r\; \backslash left(\backslash cos\; \backslash left(\backslash frac\backslash right)\; +\; i\; \backslash sin\; \backslash left(\backslash frac\backslash right)\backslash right)$$for . (Here

    \sqrt[n]r is the usual (positive) th root of the positive real number .) Because sine and cosine are periodic, other integer values of do not give other values. For any z \ne 0 , there are, in particular n distinct complex n-th roots. For example, there are 4 fourth roots of 1, namelyz _{1} = 1 , z _{2} = i, z _{3} = -1 , z _{4} = - i. In general there is no natural way of distinguishing one particular complex th root of a complex number. (This is in contrast to the roots of a positive real number x, which has a unique positive real n-th root, which is therefore commonly referred to as the n-th root of x.) One refers to this situation by saying that the th root is a -valued function of .

    Fundamental theorem of algebra

    \Q (the polynomial does not have a rational root, because is not a rational number) nor the real numbers \R (the polynomial does not have a real root, because the square of is positive for any real number).

    Because of this fact,

    \Complex is called an algebraically closed field. It is a cornerstone of various applications of complex numbers, as is detailed further below. There are various proofs of this theorem, by either analytic methods such as Liouville's theorem, or topological ones such as the winding number, or a proof combining Galois theory and the fact that any real polynomial of odd degree has at least one real root.

    History

    See also: Negative number#History. The solution in radicals (without trigonometric functions) of a general cubic equation, when all three of its roots are real numbers, contains the square roots of negative numbers, a situation that cannot be rectified by factoring aided by the rational root test, if the cubic is irreducible; this is the so-called casus irreducibilis ("irreducible case"). This conundrum led Italian mathematician Gerolamo Cardano to conceive of complex numbers in around 1545 in his Ars Magna,^[9] though his understanding was rudimentary; moreover, he later described complex numbers as being "as subtle as they are useless".^[10] Cardano did use imaginary numbers, but described using them as "mental torture."^[11] This was prior to the use of the graphical complex plane. Cardano and other Italian mathematicians, notably Scipione del Ferro, in the 1500s created an algorithm for solving cubic equations which generally had one real solution and two solutions containing an imaginary number. Because they ignored the answers with the imaginary numbers, Cardano found them useless.^[12]

    Work on the problem of general polynomials ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher. Complex numbers thus form an algebraically closed field, where any polynomial equation has a root.

    Many mathematicians contributed to the development of complex numbers. The rules for addition, subtraction, multiplication, and root extraction of complex numbers were developed by the Italian mathematician Rafael Bombelli.^[13] A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton, who extended this abstraction to the theory of quaternions.^[14]

    The earliest fleeting reference to square roots of negative numbers can perhaps be said to occur in the work of the Greek mathematician Hero of Alexandria in the 1st century AD, where in his Stereometrica he considered, apparently in error, the volume of an impossible frustum of a pyramid to arrive at the term

    \sqrt{ 81 - 144 } in his calculations, which today would simplify to \sqrt{ -63 } = 3 i\sqrt{ 7 } . Negative quantities were not conceived of in Hellenistic mathematics and Hero merely replaced it by its positive \sqrt{ 144 - 81 } = 3 \sqrt{ 7 }. ^[15]

    The impetus to study complex numbers as a topic in itself first arose in the 16th century when algebraic solutions for the roots of cubic and quartic polynomials were discovered by Italian mathematicians (Niccolò Fontana Tartaglia and Gerolamo Cardano). It was soon realized (but proved much later) that these formulas, even if one were interested only in real solutions, sometimes required the manipulation of square roots of negative numbers. In fact, it was proved later that the use of complex numbers is unavoidable when all three roots are real and distinct. However, the general formula can still be used in this case, with some care to deal with the ambiguity resulting from the existence of three cubic roots for nonzero complex numbers. Rafael Bombelli was the first to address explicitly these seemingly paradoxical solutions of cubic equations and developed the rules for complex arithmetic, trying to resolve these issues.

    The term "imaginary" for these quantities was coined by René Descartes in 1637, who was at pains to stress their unreal nature:^[16] A further source of confusion was that the equation

    \sqrt{ -1 } ^{2} = \sqrt{ -1 }\sqrt{ -1 } = -1 seemed to be capriciously inconsistent with the algebraic identity \sqrt{a}\sqrt{b} = \sqrt{ab} , which is valid for non-negative real numbers and, and which was also used in complex number calculations with one of, positive and the other negative. The incorrect use of this identity in the case when both and are negative, and the related identity $\backslash frac\; =\; \backslash sqrt$, even bedeviled Leonhard Euler. This difficulty eventually led to the convention of using the special symbol in place of \sqrt{ -1 } to guard against this mistake. Even so, Euler considered it natural to introduce students to complex numbers much earlier than we do today. In his elementary algebra text book, Elements of Algebra, he introduces these numbers almost at once and then uses them in a natural way throughout.

    In the 18th century complex numbers gained wider use, as it was noticed that formal manipulation of complex expressions could be used to simplify calculations involving trigonometric functions. For instance, in 1730 Abraham de Moivre noted that the identities relating trigonometric functions of an integer multiple of an angle to powers of trigonometric functions of that angle could be re-expressed by the following de Moivre's formula:

    $$(\backslash cos\; \backslash theta\; +\; i\backslash sin\; \backslash theta)^\; =\; \backslash cos\; n\; \backslash theta\; +\; i\backslash sin\; n\; \backslash theta.$$

    In 1748, Euler went further and obtained Euler's formula of complex analysis:^[17]

    $$e\; ^\; =\; \backslash cos\; \backslash theta\; +\; i\backslash sin\; \backslash theta$$

    by formally manipulating complex power series and observed that this formula could be used to reduce any trigonometric identity to much simpler exponential identities.

    The idea of a complex number as a point in the complex plane (above) was first described by Danish–Norwegian mathematician Caspar Wessel in 1799,^[18] although it had been anticipated as early as 1685 in Wallis's A Treatise of Algebra.^[19]

    Wessel's memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed. In 1806 Jean-Robert Argand independently issued a pamphlet on complex numbers and provided a rigorous proof of the fundamental theorem of algebra.^[20] Carl Friedrich Gauss had earlier published an essentially topological proof of the theorem in 1797 but expressed his doubts at the time about "the true metaphysics of the square root of -1".^[21] It was not until 1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane,^[22] largely establishing modern notation and terminology:

    If one formerly contemplated this subject from a false point of view and therefore found a mysterious darkness, this is in large part attributable to clumsy terminology. Had one not called +1, −1,

    \sqrt{ -1 } positive, negative, or imaginary (or even impossible) units, but instead, say, direct, inverse, or lateral units, then there could scarcely have been talk of such darkness.

    In the beginning of the 19th century, other mathematicians discovered independently the geometrical representation of the complex numbers: Buée,^{[23] [24]} Mourey,^[25] Warren,^{[26] [27] [28]} Français and his brother, Bellavitis.^{[29] [30]}

    The English mathematician G.H. Hardy remarked that Gauss was the first mathematician to use complex numbers in "a really confident and scientific way" although mathematicians such as Norwegian Niels Henrik Abel and Carl Gustav Jacob Jacobi were necessarily using them routinely before Gauss published his 1831 treatise.^[31]

    Augustin-Louis Cauchy and Bernhard Riemann together brought the fundamental ideas of complex analysis to a high state of completion, commencing around 1825 in Cauchy's case.

    The common terms used in the theory are chiefly due to the founders. Argand called the direction factor, and

    r = \sqrt{a ^{2} + b ^{2}} the modulus;^[32] Cauchy (1821) called the reduced form (l'expression réduite)^[33] and apparently introduced the term argument; Gauss used for \sqrt{ -1 } , introduced the term complex number for, and called the norm. The expression direction coefficient, often used for, is due to Hankel (1867),^[34] and absolute value, for modulus, is due to Weierstrass.

    Later classical writers on the general theory include Richard Dedekind, Otto Hölder, Felix Klein, Henri Poincaré, Hermann Schwarz, Karl Weierstrass and many others. Important work (including a systematization) in complex multivariate calculus has been started at beginning of the 20th century. Important results have been achieved by Wilhelm Wirtinger in 1927.

    Abstract algebraic aspects

    While the above low-level definitions, including the addition and multiplication, accurately describes the complex numbers, there are other, equivalent approaches that reveal the abstract algebraic structure of the complex numbers more immediately.

    Construction as a quotient field

    One approach to

    \C is via polynomials, i.e., expressions of the form$$p(X)\; =\; a\_nX^n+\backslash dotsb+a\_1X+a\_0,$$where the coefficients are real numbers. The set of all such polynomials is denoted by \R[X] . Since sums and products of polynomials are again polynomials, this set \R[X] forms a commutative ring, called the polynomial ring (over the reals). To every such polynomial p, one may assign the complex number p(i) = a _{n} i ^{n} + ... b + a _{1} i + a _{0} , i.e., the value obtained by setting X = i . This defines a function\R[X] / (X ^{2} + 1 ) \stackrel \cong \to \C between the quotient ring and \C . Some authors take this as the definition of \C .

    Accepting that

    \Complex is algebraically closed, because it is an algebraic extension of R in this approach, \Complex is therefore the algebraic closure of \R.

    Matrix representation of complex numbers

    Complex numbers can also be represented by matrices that have the formHere the entries and are real numbers. As the sum and product of two such matrices is again of this form, these matrices form a subring of the ring of matrices.

    A simple computation shows that the mapis a ring isomorphism from the field of complex numbers to the ring of these matrices, proving that these matrices form a field. This isomorphism associates the square of the absolute value of a complex number with the determinant of the corresponding matrix, and the conjugate of a complex number with the transpose of the matrix.

    The geometric description of the multiplication of complex numbers can also be expressed in terms of rotation matrices by using this correspondence between complex numbers and such matrices. The action of the matrix on a vector corresponds to the multiplication of by . In particular, if the determinant is, there is a real number such that the matrix has the form

    In this case, the action of the matrix on vectors and the multiplication by the complex number

    \cos t + i\sin t are both the rotation of the angle .

    Complex analysis

    See main article: Complex analysis.

    The study of functions of a complex variable is known as complex analysis and has enormous practical use in applied mathematics as well as in other branches of mathematics. Often, the most natural proofs for statements in real analysis or even number theory employ techniques from complex analysis (see prime number theorem for an example).

    Unlike real functions, which are commonly represented as two-dimensional graphs, complex functions have four-dimensional graphs and may usefully be illustrated by color-coding a three-dimensional graph to suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane.

    Convergence

    The notions of convergent series and continuous functions in (real) analysis have natural analogs in complex analysis. A sequence of complex numbers is said to converge if and only if its real and imaginary parts do. This is equivalent to the (ε, δ)-definition of limits, where the absolute value of real numbers is replaced by the one of complex numbers. From a more abstract point of view,

    C , endowed with the metric$$\backslash operatorname(z\_1,\; z\_2)\; =\; |z\_1\; -\; z\_2|$$is a complete metric space, which notably includes the triangle inequality$$|z\_1\; +\; z\_2|\; \backslash le\; |z\_1|\; +\; |z\_2|$$for any two complex numbers and .

    Complex exponential

    Like in real analysis, this notion of convergence is used to construct a number of elementary functions: the exponential function, also written, is defined as the infinite series, which can be shown to converge for any z:$$\backslash exp\; z:=\; 1+z+\backslash frac+\backslash frac+\backslash cdots\; =\; \backslash sum\_^\; \backslash frac.$$For example,

    \exp ( 1 ) is Euler's constant e ≈ 2 . 718 . Euler's formula states:$$\backslash exp(i\backslash varphi)\; =\; \backslash cos\; \backslash varphi\; +\; i\backslash sin\; \backslash varphi$$for any real number . This formula is a quick consequence of general basic facts about convergent power series and the definitions of the involved functions as power series. As a special case, this includes Euler's identity$$\backslash exp(i\; \backslash pi)\; =\; -1.$$

    Complex logarithm

    For any positive real number t, there is a unique real number x such that

    \exp(x) = t . This leads to the definition of the natural logarithm as the inverse ln \colon \R ^{+} \to \R ; x \mapsto ln x of the exponential function. The situation is different for complex numbers, since\exp(z +2 \pi i) = \exp z \exp ( 2 \pi i) = \exp z by the functional equation and Euler's identity. For example,, so both and are possible values for the complex logarithm of .

    In general, given any non-zero complex number w, any number z solving the equation

    \exp z = w is called a complex logarithm of, denoted log w . It can be shown that these numbers satisfy$$z\; =\; \backslash log\; w\; =\; \backslash ln|w|\; +\; i\backslash arg\; w,$$where arg is the argument defined above, and ln the (real) natural logarithm. As arg is a multivalued function, unique only up to a multiple of, log is also multivalued. The principal value of log is often taken by restricting the imaginary part to the interval . This leads to the complex logarithm being a bijective function taking values in the strip \R ^{+} + i \left( - \pi, \pi\right] (that is denoted S _{0} in the above illustration)$$\backslash ln\; \backslash colon\; \backslash ;\; \backslash Complex^\backslash times\; \backslash ;\; \backslash to\; \backslash ;\; \backslash ;\; \backslash ;\; \backslash R^+\; +\; \backslash ;\; i\; \backslash ,\; \backslash left(-\backslash pi,\; \backslash pi\backslash right$$
36. Kalman. Dan. An Elementary Proof of Marden's Theorem . . 115 . 4 . 330–38 . 2008a . 10.1080/00029890.2008.11920532 . 13222698 . 0002-9890 . 1 January 2012 . https://web.archive.org/web/20120308104622/http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=3338&pf=1 . 8 March 2012. live.
37. Kalman . Dan . The Most Marvelous Theorem in Mathematics . . 2008b . 1 January 2012. https://web.archive.org/web/20120208014954/http://mathdl.maa.org/mathDL/4/?pa=content&sa=viewDocument&nodeId=1663 . 8 February 2012 . live.
38. Book: Grant . I.S. . Electromagnetism . 2008. 2 . Manchester Physics Series . 978-0-471-92712-9 . Phillips . W.R..
39. Book: Marker, David . Marker . D. . Messmer . M. . Pillay . A. . Introduction to the Model Theory of Fields . https://projecteuclid.org/euclid.lnl/1235423155 . 978-3-540-60741-0 . 1477154 . 1–37 . Springer-Verlag . Berlin . Lecture Notes in Logic . Model theory of fields . 5 . 1996.
40. Book: McCrimmon, Kevin . 2004 . A Taste of Jordan Algebras . 64 . Universitext . Springer . 0-387-95447-3.