Complex number
This article needs additional citations for verification. (July 2022) |
In
Complex numbers allow solutions to all
Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule along with the
The complex numbers also form a
The complex numbers form a rich structure that is simultaneously an
Definition and basic operations
A complex number is an expression
For a complex number a + bi, the real number a is called its real part , and the real number b (not the complex number bi) is its imaginary part.[4][5] The real part of a complex number z is denoted Re(z), , or ; the imaginary part is Im(z), , or : for example,, .
A complex number z can be identified with the ordered pair of real numbers , which may be interpreted as coordinates of a point in a Euclidean plane with standard coordinates, which is then called the 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 a can be regarded as a complex number a + 0i, whose imaginary part is 0. A purely imaginary number bi is a complex number 0 + bi, whose real part is zero. As with polynomials, it is common to write a + 0i = a, 0 + bi = bi, and a + (−b)i = a − bi; for example, 3 + (−4)i = 3 − 4i.
The set of all complex numbers is denoted by (blackboard bold) or C (upright bold).
In some disciplines such as electromagnetism and electrical engineering, j is used instead of i, as i frequently represents electric current,[8][9] and complex numbers are written as a + bj or a + jb.
Addition and subtraction
Two complex numbers and are added by separately adding their real and imaginary parts. That is to say:
The addition can be geometrically visualized as follows: the sum of two complex numbers a and b, interpreted as points in the complex plane, is the point obtained by building a parallelogram from the three vertices O, and the points of the arrows labeled a and b (provided that they are not on a line). Equivalently, calling these points A, B, respectively and the fourth point of the parallelogram X the triangles OAB and XBA are congruent.
Multiplication
The product of two complex numbers is computed as follows:
For example, . In particular, this includes as a special case the fundamental formula
This formula distinguishes the complex number i from any real number, since the square of any (negative or positive) real number x always satisfies .
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.[10]
Complex conjugate, absolute value and argument
The complex conjugate of the complex number z = x + yi is defined as .[11] It is also denoted by some authors by . Geometrically, z is the "reflection" of z about the real axis. Conjugating twice gives the original complex number: 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 z = x + yi , the product
is a non-negative real number. This allows to define the absolute value (or modulus or magnitude) of z to be the square root [12]
Using the conjugate, the reciprocal of a nonzero complex number can be computed to be
The argument of z (sometimes called the "phase" φ)[7] is the angle of the radius Oz with the positive real axis, and is written as arg z, expressed in radians in this article. The angle is defined only up to adding integer multiples of , since a rotation by (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 , which is referred to as the principal value.[13] The argument can be computed from the rectangular form x + yi by means of the
Polar form
For any complex number z, with absolute value and argument , the equation
holds. This identity is referred to as the polar form of z. It is sometimes abbreviated as . In
If two complex numbers are given in polar form, i.e., z1 = r1(cos φ1 + i sin φ1) and z2 = r2(cos φ2 + i sin φ2), the product and division can be computed as
Powers and roots
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:
The n nth roots of a complex number z are given by
In general there is no natural way of distinguishing one particular complex nth 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 nth root is a n-valued function of z.
Fundamental theorem of algebra
The fundamental theorem of algebra, of Carl Friedrich Gauss and Jean le Rond d'Alembert, states that for any complex numbers (called coefficients) a0, ..., an, the equation
Because of this fact, 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
The solution in
Work on the problem of general polynomials ultimately led to the
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
The earliest fleeting reference to
The impetus to study complex numbers as a topic in itself first arose in the 16th century when
The term "imaginary" for these quantities was coined by René Descartes in 1637, who was at pains to stress their unreal nature:[26]
... sometimes only imaginary, that is one can imagine as many as I said in each equation, but sometimes there exists no quantity that matches that which we imagine.
[... quelquefois seulement imaginaires c'est-à-dire que l'on peut toujours en imaginer autant que j'ai dit en chaque équation, mais qu'il n'y a quelquefois aucune quantité qui corresponde à celle qu'on imagine.]
A further source of confusion was that the equation seemed to be capriciously inconsistent with the algebraic identity , which is valid for non-negative real numbers a and b, and which was also used in complex number calculations with one of a, b positive and the other negative. The incorrect use of this identity in the case when both a and b are negative, and the related identity , even bedeviled Leonhard Euler. This difficulty eventually led to the convention of using the special symbol i in place of to guard against this mistake.[citation needed] 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:
In 1748, Euler went further and obtained Euler's formula of complex analysis:[27]
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,[28] although it had been anticipated as early as 1685 in Wallis's A Treatise of Algebra.[29]
Wessel's memoir appeared in the Proceedings of the
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, 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,[34][35] Mourey,[36] Warren,[37][38][39] Français and his brother, Bellavitis.[40][41]
The English mathematician
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 cos φ + i sin φ the direction factor, and the modulus;[d][43] Cauchy (1821) called cos φ + i sin φ the reduced form (l'expression réduite)[44] and apparently introduced the term argument; Gauss used i for ,[e] introduced the term complex number for a + bi,[f] and called a2 + b2 the norm.[g] The expression direction coefficient, often used for cos φ + i sin φ, is due to Hankel (1867),[48] 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 is via polynomials, i.e., expressions of the form
This function is
between the quotient ring and . Some authors take this as the definition of .[49]
Accepting that is algebraically closed, because it is an algebraic extension of in this approach, is therefore the algebraic closure of
Matrix representation of complex numbers
Complex numbers a + bi can also be represented by 2 × 2 matrices that have the form
A simple computation shows that the map
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 (x, y) corresponds to the multiplication of x + iy by a + ib. In particular, if the determinant is 1, there is a real number t such that the matrix has the form
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,
Convergence
The notions of
Complex exponential
Like in real analysis, this notion of convergence is used to construct a number of
Complex logarithm
For any positive real number t, there is a unique real number x such that . This leads to the definition of the natural logarithm as the inverse of the exponential function. The situation is different for complex numbers, since
by the functional equation and Euler's identity. For example, eiπ = e3iπ = −1 , so both iπ and 3iπ are possible values for the complex logarithm of −1.
In general, given any non-zero complex number w, any number z solving the equation
is called a complex logarithm of w, denoted . It can be shown that these numbers satisfy
If is not a non-positive real number (a positive or a non-real number), the resulting principal value of the complex logarithm is obtained with −π < φ < π. It is an analytic function outside the negative real numbers, but it cannot be prolongated to a function that is continuous at any negative real number , where the principal value is ln z = ln(−z) + iπ.[h]
Complex exponentiation zω is defined as
Complex numbers, unlike real numbers, do not in general satisfy the unmodified power and logarithm identities, particularly when naïvely treated as single-valued functions; see failure of power and logarithm identities. For example, they do not satisfy
Complex sine and cosine
The series defining the real trigonometric functions
Holomorphic functions
A function → is called holomorphic or complex differentiable at a point if the limit
exists (in which case it is denoted by ). This mimics the definition for real differentiable functions, except that all quantities are complex numbers. Loosely speaking, the freedom of approaching in different directions imposes a much stronger condition than being (real) differentiable. For example, the function
is differentiable as a function , but is not complex differentiable. A real differentiable function is complex differentiable if and only if it satisfies the Cauchy–Riemann equations, which are sometimes abbreviated as
Complex analysis shows some features not apparent in real analysis. For example, the
Applications
Complex numbers have applications in many scientific areas, including signal processing, control theory, electromagnetism, fluid dynamics, quantum mechanics, cartography, and vibration analysis. Some of these applications are described below.
Complex conjugation is also employed in inversive geometry, a branch of geometry studying reflections more general than ones about a line. In the network analysis of electrical circuits, the complex conjugate is used in finding the equivalent impedance when the maximum power transfer theorem is looked for.
Geometry
Shapes
Three non-collinear points in the plane determine the shape of the triangle . Locating the points in the complex plane, this shape of a triangle may be expressed by complex arithmetic as
Fractal geometry
The Mandelbrot set is a popular example of a fractal formed on the complex plane. It is defined by plotting every location where iterating the sequence does not
Triangles
Every triangle has a unique Steiner inellipse – an ellipse inside the triangle and tangent to the midpoints of the three sides of the triangle. The foci of a triangle's Steiner inellipse can be found as follows, according to Marden's theorem:[51][52] Denote the triangle's vertices in the complex plane as a = xA + yAi, b = xB + yBi, and c = xC + yCi. Write the cubic equation , 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 . 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 , the algebraic closure of , which also contains all algebraic numbers, 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
Another example is the Gaussian integers; that is, numbers of the form x + iy, where x and y are integers, which can be used to classify sums of squares.
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 ζ(s) is related to the distribution of prime numbers.
Improper integrals
In applied fields, complex numbers are often used to compute certain real-valued
Dynamic equations
In
Linear algebra
Since is algebraically closed, any non-empty complex
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
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
Signal analysis
Complex numbers are used in
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
and
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
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 j, to avoid confusion with I, which is generally in use to denote electric current, or, more particularly, i, which is generally in use to denote instantaneous electric current.
Because the
To obtain the measurable quantity, the real part is taken:
The complex-valued signal V(t) is called the analytic representation of the real-valued, measurable signal v(t). [53]
Fluid dynamics
In
Quantum mechanics
The complex number field is intrinsic to the
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.
Algebraic characterization
The field has the following three properties:
- First, it has characteristic 0. This means that 1 + 1 + ⋯ + 1 ≠ 0 for any number of summands (all of which equal one).
- Second, its transcendence degreeover , theprime fieldof is the cardinality of the continuum.
- Third, it is algebraically closed(see above).
It can be shown that any field having these properties is
Characterization as a topological field
The preceding characterization of describes only the algebraic aspects of That is to say, the properties of
, are not dealt with. The following description of 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. contains a subset P (namely the set of positive real numbers) of nonzero elements satisfying the following three conditions:- P is closed under addition, multiplication and taking inverses.
- If x and y are distinct elements of P, then either x − y or y − x is in P.
- If S is any nonempty subset of P, then S + P = x + P for some x in
Moreover, has a nontrivial involutive automorphism x ↦ x* (namely the complex conjugation), such that x x* is in P for any nonzero x in
Any field F with these properties can be endowed with a topology by taking the sets B(x, p) = { y | p − (y − x)(y − x)* ∈ P } as a base, where x ranges over the field and p ranges over P. With this topology F is isomorphic as a topological field to
The only
Other number systems
rational numbers | real numbers | complex numbers | quaternions | octionions | sedenions | |
---|---|---|---|---|---|---|
complete | No | Yes | Yes | Yes | Yes | Yes |
dimension as an -vector space | [does not apply] | 1 | 2 | 4 | 8 | 16 |
ordered | Yes | Yes | No | No | No | No |
multiplication commutative () | Yes | Yes | Yes | No | No | No |
multiplication associative () | Yes | Yes | Yes | Yes | No | No |
normed division algebra (over )
|
[does not apply] | Yes | Yes | Yes | Yes | No |
The process of extending the field of reals to is an instance of the Cayley–Dickson construction. Applying this construction iteratively to then yields the quaternions, the octonions and the sedenions.[56] This construction turns out to diminish the structural properties of the involved number systems.
Unlike the reals, is not an ordered field, that is to say, it is not possible to define a relation z1 < z2 that is compatible with the addition and multiplication. In fact, in any ordered field, the square of any element is necessarily positive, so i2 = −1 precludes the existence of an ordering on [57] Passing from to the quaternions loses commutativity, while the octions (additionally to not being commutative) fail to be associative. The reals, complex numbers, quaternions and octonions are all
The Cayley–Dickson construction is closely related to the regular representation of thought of as an -
Hypercomplex numbers also generalize and For example, this notion contains the split-complex numbers, which are elements of the ring (as opposed to for complex numbers). In this ring, the equation a2 = 1 has four solutions.
The field is the completion of the field of
The fields and their finite field extensions, including are called local fields.
See also
- Circular motion using complex numbers
- Analytic continuation
- Complex-base system
- Complex geometry
- Geometry of numbers
- Dual-complex number
- Eisenstein integer
- Geometric algebra (which includes the complex plane as the 2-dimensional spinor subspace )
- Unit complex number
Notes
- ^ Solomentsev 2001: "The plane whose points are identified with the elements of is called the complex plane ... The complete geometric interpretation of complex numbers and operations on them appeared first in the work of C. Wessel (1799). The geometric representation of complex numbers, sometimes called the 'Argand diagram', came into use after the publication in 1806 and 1814 of papers by J.R. Argand, who rediscovered, largely independently, the findings of Wessel".
- ^ In the literature the imaginary unit often precedes the radical sign, even when preceded itself by an integer.[23]
- ^ It has been proved that imaginary numbers necessarily appear in the cubic formula when the equation has three real, different roots by Pierre Laurent Wantzel in 1843, Vincenzo Mollame in 1890, Otto Hölder in 1891, and Adolf Kneser in 1892. Paolo Ruffini also provided an incomplete proof in 1799.——S. Confalonieri (2015)[25]
- ^ Argand 1814, p. 204 defines the modulus of a complex number but he doesn't name it:
"Dans ce qui suit, les accens, indifféremment placés, seront employés pour indiquer la grandeur absolue des quantités qu'ils affectent; ainsi, si , et étant réels, on devra entendre que ou ."
[In what follows, accent marks, wherever they're placed, will be used to indicate the absolute size of the quantities to which they're assigned; thus if , and being real, one should understand that or .]
Argand 1814, p. 208 defines and names the module and the direction factor of a complex number: "... pourrait être appelé le module de , et représenterait la grandeur absolue de la ligne , tandis que l'autre facteur, dont le module est l'unité, en représenterait la direction."
[... could be called the module of and would represent the absolute size of the line (Argand represented complex numbers as vectors.) whereas the other factor [namely, ], whose module is unity [1], would represent its direction.] - ^ Gauss writes:[45]"Quemadmodum scilicet arithmetica sublimior in quaestionibus hactenus pertractatis inter solos numeros integros reales versatur, ita theoremata circa residua biquadratica tunc tantum in summa simplicitate ac genuina venustate resplendent, quando campus arithmeticae ad quantitates imaginarias extenditur, ita ut absque restrictione ipsius obiectum constituant numeri formae a + bi, denotantibus i, pro more quantitatem imaginariam , atque a, b indefinite omnes numeros reales integros inter - et +." [Of course just as the higher arithmetic has been investigated so far in problems only among real integer numbers, so theorems regarding biquadratic residues then shine in greatest simplicity and genuine beauty, when the field of arithmetic is extended to imaginary quantities, so that, without restrictions on it, numbers of the form a + bi — i denoting by convention the imaginary quantity , and the variables a, b [denoting] all real integer numbers between and — constitute an object.]
- ^ Gauss:[46]"Tales numeros vocabimus numeros integros complexos, ita quidem, ut reales complexis non opponantur, sed tamquam species sub his contineri censeantur." [We will call such numbers [namely, numbers of the form a + bi ] "complex integer numbers", so that real [numbers] are regarded not as the opposite of complex [numbers] but [as] a type [of number that] is, so to speak, contained within them.]
- ^ Gauss:[47] "Productum numeri complexi per numerum ipsi conjunctum utriusque normam vocamus. Pro norma itaque numeri realis, ipsius quadratum habendum est." [We call a "norm" the product of a complex number [for example, a + ib ] with its conjugate [a - ib ]. Therefore the square of a real number should be regarded as its norm.]
- ^ However for another inverse function of the complex exponential function (and not the above defined principal value), the branch cut could be taken at any other ray thru the origin.
References
- ^ For an extensive account of the history of "imaginary" numbers, from initial skepticism to ultimate acceptance, see Bourbaki, Nicolas (1998). "Foundations of Mathematics § Logic: Set theory". Elements of the History of Mathematics. Springer. pp. 18–24.
- ^ "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.", Penrose 2005, pp.72–73.
- ISBN 9780470470770.
- ISBN 978-0-07-161569-3.
- ^ Aufmann, Barker & Nation 2007, p. 66, Chapter P
- ISBN 978-0-486-65812-4.
- ^ a b Weisstein, Eric W. "Complex Number". mathworld.wolfram.com. Retrieved 12 August 2020.
- S2CID 51647814. Retrieved 24 June 2023. p. 789:
The use of i (or Greek ı) for the imaginary symbol is nearly universal in mathematical work, which is a very strong reason for retaining it in the applications of mathematics in electrical engineering. Aside, however, from the matter of established conventions and facility of reference to mathematical literature, the substitution of the symbol j is objectionable because of the vector terminology with which it has become associated in engineering literature, and also because of the confusion resulting from the divided practice of engineering writers, some using j for +i and others using j for −i.
- ISBN 978-0-07-912147-9. p. 2:
In electrical engineering, the letter j is used instead of i.
- ^ Apostol 1981, pp. 15–16.
- ^ Apostol 1981, pp. 15–16
- ^ Apostol 1981, p. 18.
- ^ Other authors, including Ebbinghaus et al. 1991, §6.1, chose the argument to be in the interval .
- ISBN 978-81-203-2641-5.
- ^
Nilsson, James William; Riedel, Susan A. (2008). "Chapter 9". Electric circuits (8th ed.). Prentice Hall. p. 338. ISBN 978-0-13-198925-2.
- ^ Bourbaki 1998, §VIII.1
- ^ Kline, Morris. A history of mathematical thought, volume 1. p. 253.
- OCLC 1080410598.
- ^ O'Connor and Robertson (2016), "Girolamo Cardano."
- ^ Nahin, Paul J. An Imaginary Tale: The Story of √−1. Princeton: Princeton University Press, 1998.
- ISBN 978-0-321-16193-2.
- ^ Hamilton, Wm. (1844). "On a new species of imaginary quantities connected with a theory of quaternions". Proceedings of the Royal Irish Academy. 2: 424–434.
- ISBN 978-0-691-12798-9. Archivedfrom the original on 12 October 2012. Retrieved 20 April 2011.
- ^ ISBN 978-3658092757.
- ISBN 978-0-486-60068-0. Retrieved 20 April 2011.
- ^ Euler, Leonard (1748). Introductio in Analysin Infinitorum [Introduction to the Analysis of the Infinite] (in Latin). Vol. 1. Lucerne, Switzerland: Marc Michel Bosquet & Co. p. 104.
- ^ Wessel, Caspar (1799). "Om Directionens analytiske Betegning, et Forsog, anvendt fornemmelig til plane og sphæriske Polygoners Oplosning" [On the analytic representation of direction, an effort applied in particular to the determination of plane and spherical polygons]. Nye Samling af det Kongelige Danske Videnskabernes Selskabs Skrifter [New Collection of the Writings of the Royal Danish Science Society] (in Danish). 5: 469–518.
- ^ Wallis, John (1685). A Treatise of Algebra, Both Historical and Practical ... London, England: printed by John Playford, for Richard Davis. pp. 264–273.
- ^ Argand (1806). Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques [Essay on a way to represent complex quantities by geometric constructions] (in French). Paris, France: Madame Veuve Blanc.
- ^ 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)
- ISBN 9780198505358. Retrieved 18 March 2020.
- ^ Gauss 1831.
- ^ "Adrien Quentin Buée (1745–1845): MacTutor".
- S2CID 110394048.
- ^ Mourey, C.V. (1861). La vraies théore des quantités négatives et des quantités prétendues imaginaires [The true theory of negative quantities and of alleged imaginary quantities] (in French). Paris, France: Mallet-Bachelier. 1861 reprint of 1828 original.
- ^ Warren, John (1828). A Treatise on the Geometrical Representation of the Square Roots of Negative Quantities. Cambridge, England: Cambridge University Press.
- S2CID 186211638.
- S2CID 125699726.
- ^ Français, J.F. (1813). "Nouveaux principes de géométrie de position, et interprétation géométrique des symboles imaginaires" [New principles of the geometry of position, and geometric interpretation of complex [number] symbols]. Annales des mathématiques pures et appliquées (in French). 4: 61–71.
- ISBN 978-3-7643-7186-9.
- ISBN 978-0-19-921986-5.
- ^ Jeff Miller (21 September 1999). "MODULUS". Earliest Known Uses of Some of the Words of Mathematics (M). Archived from the original on 3 October 1999.
{{cite web}}
: CS1 maint: unfit URL (link) - ^ Cauchy, Augustin-Louis (1821). Cours d'analyse de l'École royale polytechnique (in French). Vol. 1. Paris, France: L'Imprimerie Royale. p. 183.
- ^ Gauss 1831, p. 96
- ^ Gauss 1831, p. 96
- ^ Gauss 1831, p. 98
- ^ Hankel, Hermann (1867). Vorlesungen über die complexen Zahlen und ihre Functionen [Lectures About the Complex Numbers and Their Functions] (in German). Vol. 1. Leipzig, [Germany]: Leopold Voss. p. 71. 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".)
- ^ Bourbaki 1998, §VIII.1
- S2CID 121095307.
- from the original on 8 March 2012. Retrieved 1 January 2012.
- Journal of Online Mathematics and Its Applications. Archivedfrom the original on 8 February 2012. Retrieved 1 January 2012.
- ISBN 978-0-471-92712-9.
- MR 1477154.
- ^ Bourbaki 1998, §VIII.4.
- MR2014924
- ^ Apostol 1981, p. 25.
- ISBN 978-0-07-000657-7.
- Andreescu, Titu; Andrica, Dorin (2014), Complex Numbers from A to ... Z (Second ed.), New York: Springer, ISBN 978-0-8176-8414-3
- Apostol, Tom(1981). Mathematical analysis. Addison-Wesley.
- Aufmann, Richard N.; Barker, Vernon C.; Nation, Richard D. (2007). College Algebra and Trigonometry (6 ed.). Cengage Learning. ISBN 978-0-618-82515-8.
- Conway, John B. (1986). Functions of One Complex Variable I. Springer. ISBN 978-0-387-90328-6.
- ISBN 978-0-309-09657-7.
- Joshi, Kapil D. (1989). Foundations of Discrete Mathematics. New York: ISBN 978-0-470-21152-6.
- Needham, Tristan (1997). Visual Complex Analysis. Clarendon Press. ISBN 978-0-19-853447-1.
- ISBN 978-0-486-65812-4.
- ISBN 978-0-679-45443-4.
- Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. (2007). "Section 5.5 Complex Arithmetic". Numerical Recipes: The art of scientific computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8. Archived from the originalon 13 March 2020. Retrieved 9 August 2011.
- Solomentsev, E.D. (2001) [1994], "Complex number", Encyclopedia of Mathematics, EMS Press
Historical references
- Argand (1814). "Reflexions sur la nouvelle théorie des imaginaires, suives d'une application à la demonstration d'un theorème d'analise" [Reflections on the new theory of complex numbers, followed by an application to the proof of a theorem of analysis]. Annales de mathématiques pures et appliquées (in French). 5: 197–209.
- Bourbaki, Nicolas (1998). "Foundations of mathematics § logic: set theory". Elements of the history of mathematics. Springer.
- Burton, David M. (1995). The History of Mathematics (3rd ed.). New York: ISBN 978-0-07-009465-9.
- Gauss, C. F. (1831). "Theoria residuorum biquadraticorum. Commentatio secunda" [Theory of biquadratic residues. Second memoir.]. Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores (in Latin). 7: 89–148.
- Katz, Victor J. (2004). A History of Mathematics, Brief Version. ISBN 978-0-321-16193-2.
- Nahin, Paul J. (1998). An Imaginary Tale: The Story of . Princeton University Press. ISBN 978-0-691-02795-1. — A gentle introduction to the history of complex numbers and the beginnings of complex analysis.
- Ebbinghaus, H. D.; Hermes, H.; Hirzebruch, F.; Koecher, M.; Mainzer, K.; Neukirch, J.; Prestel, A.; Remmert, R. (1991). Numbers (hardcover ed.). Springer. ISBN 978-0-387-97497-2. — An advanced perspective on the historical development of the concept of number.