A complex numberz can be visually represented as a pair of numbers(a, b) forming aposition vector (blue) or a point (red) on a diagram called an Argand diagram, representing the complex plane.Re is the real axis,Im is the imaginary axis, andi is the "imaginary unit", that satisfiesi2 = −1.
In mathematics, acomplex number is an element of anumber system that extends thereal numbers with a specific element denotedi, called theimaginary unit and satisfying the equation; because no real number satisfies the above equation,i was called animaginary number byRené Descartes. Every complex number can be expressed in the form, wherea andb are real numbers,a is called thereal part, andb is called theimaginary part. The set of complex numbers is denoted by either of the symbols orC. 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 allpolynomial equations, even those that have no solutions in real numbers. More precisely, thefundamental 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 equationhas no real solution, because the square of a real number cannot be negative, but has the two nonreal complex solutions and.
Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule along with theassociative,commutative, anddistributive laws. Every nonzero complex number has amultiplicative inverse. This makes the complex numbers afield with the real numbers as a subfield. Because of these properties,, and which form is written depends upon convention and style considerations.
The complex numbers also form areal vector space ofdimension two, with as astandard basis. This standard basis makes the complex numbers aCartesian 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 thereal line, which is pictured as the horizontal axis of the complex plane, while real multiples of are the vertical axis. A complex number can also be defined by its geometricpolar coordinates: the radius is called theabsolute 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 theunit circle. Adding a fixed complex number to all complex numbers defines atranslation in the complex plane, and multiplying by a fixed complex number is asimilarity centered at the origin (dilating by the absolute value, and rotating by the argument). The operation ofcomplex conjugation is thereflection symmetry with respect to the real axis.
Various complex numbers depicted in the complex plane.
A complex number is an expression of the forma +bi, wherea andb are real numbers, andi is an abstract symbol, the so-called imaginary unit, whose meaning will be explained further below. For example,2 + 3i is a complex number.[3]
For a complex numbera +bi, the real numbera is called itsreal part, and the real numberb (not the complex numberbi) is itsimaginary part.[4][5] The real part of a complex numberz is denotedRe(z),, or; the imaginary part isIm(z),, or: for example,,.
A complex numberz can be identified with theordered pair of real numbers, which may be interpreted as coordinates of a point in a Euclidean plane with standard coordinates, which is then called thecomplex plane orArgand diagram.[6][7][a] 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 numbera can be regarded as a complex numbera + 0i, whose imaginary part is 0. A purely imaginary numberbi is a complex number0 +bi, whose real part is zero. It is common to writea + 0i =a,0 +bi =bi, anda + (−b)i =a −bi; for example,3 + (−4)i = 3 − 4i.
In some disciplines such as electromagnetism and electrical engineering,j is used instead ofi, asi frequently represents electric current,[8][9] and complex numbers are written asa +bj ora +jb.
The addition can be geometrically visualized as follows: the sum of two complex numbersa andb, interpreted as points in the complex plane, is the point obtained by building aparallelogram from the three verticesO, and the points of the arrows labeleda andb (provided that they are not on a line). Equivalently, calling these pointsA,B, respectively and the fourth point of the parallelogramX thetrianglesOAB andXBA arecongruent.
Multiplication of complex numbers2−i and3+4i visualized with vectors
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 numberi from any real number, since the square of any (negative or positive) real number is always a non-negative real number.
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, thedistributive property, thecommutative properties (of addition and multiplication) hold. Therefore, the complex numbers form an algebraic structure known as afield, the same way as the rational or real numbers do.[10]
Complex conjugate, absolute value, argument and division
Geometric representation ofz and its conjugatez in the complex plane.
Thecomplex conjugate of the complex numberz =x +yi is defined as[11] It is also denoted by some authors by. Geometrically,z is the"reflection" ofz 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. Theunary operation of taking the complex conjugate of a complex number cannot be expressed by applying only the basic operations of addition, subtraction, multiplication and division.
Argumentφ and modulusr locate a point in the complex plane.
For any complex numberz =x +yi , the product
is anon-negative real number. This allows to define theabsolute value (ormodulus ormagnitude) ofz to be the square root[12]ByPythagoras' theorem, is the distance from the origin to the point representing the complex numberz in the complex plane. In particular, thecircle of radius one around the origin consists precisely of the numbersz such that. If is a real number, then: its absolute value as a complex number and as a real number are equal.
Using the conjugate, thereciprocal of a nonzero complex number can be computed to be
More generally, the division of an arbitrary complex number by a non-zero complex number equalsThis process is sometimes called "rationalization" of the denominator (although the denominator in the final expression may be an irrational real number), because it resembles the method to remove roots from simple expressions in a denominator.[13][14]
Theargument ofz (sometimes called the "phase"φ)[7] is the angle of theradiusOz with the positive real axis, and is written asargz, expressed inradians 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 theprincipal value.[15]The argument can be computed from the rectangular formx + yi by means of thearctan (inverse tangent) function.[16]
"Polar form" redirects here. For the higher-dimensional analogue, seePolar decomposition.
Multiplication of2 +i (blue triangle) and3 +i (red triangle). The red triangle is rotated to match the vertex of the blue one (the adding of both angles in the termsφ1+φ2 in the equation) and stretched by the length of thehypotenuse of the blue triangle (the multiplication of both radiuses, as per termr1r2 in the equation).
For any complex numberz, with absolute value and argument, the equation
holds. This identity is referred to as the polar form ofz. It is sometimes abbreviated as. In electronics, one represents aphasor with amplituder and phaseφ inangle notation:[17]
If two complex numbers are given in polar form, i.e.,z1 =r1(cos φ1 +i sin φ1) andz2 =r2(cos φ2 +i sin φ2), the product and division can be computed as(These are a consequence of thetrigonometric identities for the sine and cosine function.)In other words, the absolute values aremultiplied and the arguments areadded to yield the polar form of the product. The picture at the right illustrates the multiplication ofBecause the real and imaginary part of5 + 5i are equal, the argument of that number is 45 degrees, orπ/4 (inradian). On the other hand, it is also the sum of the angles at the origin of the red and blue triangles arearctan(1/3) and arctan(1/2), respectively. Thus, the formulaholds. As thearctan function can be approximated highly efficiently, formulas like this – known asMachin-like formulas – are used for high-precision approximations ofπ:[18]
Then-th power of a complex number can be computed usingde Moivre's formula, which is obtained by repeatedly applying the above formula for the product:For example, the first few powers of the imaginary uniti are.
Geometric representation of the 2nd to 6th roots of a complex numberz, in polar formreiφ wherer = |z | andφ = argz. Ifz is real,φ = 0 orπ. Principal roots are shown in black.
Thennth roots of a complex numberz are given byfor0 ≤k ≤n − 1. (Here is the usual (positive)nth root of the positive real numberr.) Because sine and cosine are periodic, other integer values ofk do not give other values. For any, there are, in particularn distinct complexn-th roots. For example, there are 4 fourth roots of 1, namely
In general there isno natural way of distinguishing one particular complexnth root of a complex number. (This is in contrast to the roots of a positive real numberx, which has a unique positive realn-th root, which is therefore commonly referred to asthen-th root ofx.) One refers to this situation by saying that thenth root is an-valued function ofz.
Thefundamental theorem of algebra, ofCarl Friedrich Gauss andJean le Rond d'Alembert, states that for any complex numbers (calledcoefficients)a0, ..., an, the equationhas at least one complex solutionz, provided that at least one of the higher coefficientsa1, ..., an is nonzero.[19] This property does not hold for thefield of rational numbers (the polynomialx2 − 2 does not have a rational root, because√2 is not a rational number) nor the real numbers (the polynomialx2 + 4 does not have a real root, because the square ofx is positive for any real numberx).
Because of this fact, is called analgebraically 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 asLiouville's theorem, ortopological ones such as thewinding number, or a proof combiningGalois theory and the fact that any real polynomial ofodd degree has at least one real root.
The solution inradicals (withouttrigonometric functions) of a generalcubic equation, when all three of its roots are real numbers, contains the square roots ofnegative numbers, a situation that cannot be rectified by factoring aided by therational root test, if the cubic isirreducible; this is the so-calledcasus irreducibilis ("irreducible case"). This conundrum led Italian mathematicianGerolamo Cardano to conceive of complex numbers in around 1545 in hisArs Magna,[20] though his understanding was rudimentary; moreover, he later described complex numbers as being "as subtle as they are useless".[21] Cardano did use imaginary numbers, but described using them as "mental torture".[22] This was prior to the use of the graphical complex plane. Cardano and other Italian mathematicians, notablyScipione 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.[23]
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 everypolynomial equation of degree one or higher. Complex numbers thus form analgebraically closed field, where any polynomial equation has aroot.
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 mathematicianRafael Bombelli.[24] A more abstract formalism for the complex numbers was further developed by the Irish mathematicianWilliam Rowan Hamilton, who extended this abstraction to the theory ofquaternions.[25]
The earliest fleeting reference tosquare roots ofnegative numbers can perhaps be said to occur in the work of the Greek mathematicianHero of Alexandria in the 1st centuryAD, where in hisStereometrica he considered, apparently in error, the volume of an impossiblefrustum of apyramid to arrive at the term in his calculations, which today would simplify to.[b] Negative quantities were not conceived of inHellenistic mathematics and Hero merely replaced the negative value by its positive[27]
The impetus to study complex numbers as a topic in itself first arose in the 16th century whenalgebraic solutions for the roots ofcubic andquarticpolynomials were discovered by Italian mathematicians (Niccolò Fontana Tartaglia andGerolamo Cardano). It was soon realized (but proved much later)[28] 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 numbersis unavoidable when all three roots are real and distinct.[c] 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 byRené Descartes in 1637, who was at pains to stress their unreal nature:[29]
... 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 numbersa andb, and which was also used in complex number calculations with one ofa,b positive and the other negative. The incorrect use of this identity in the case when botha andb are negative, and the related identity, even bedeviledLeonhard Euler. This difficulty eventually led to the convention of using the special symboli in place of to guard against this mistake.[30][31] 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 1730Abraham 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 followingde Moivre's formula:
Euler's formula relates the complex exponential function of an imaginary argument, which can be thought of as describinguniform circular motion in the complex plane, to the cosine and sine functions, geometrically its projections onto the real and imaginary axes, respectively.
by formally manipulating complexpower series and observed that this formula could be used to reduce any trigonometric identity to much simpler exponential identities.
Wessel's memoir appeared in the Proceedings of theCopenhagen Academy but went largely unnoticed. In 1806Jean-Robert Argand independently issued a pamphlet on complex numbers and provided a rigorous proof of thefundamental theorem of algebra.[35]Carl Friedrich Gauss had earlier published an essentiallytopological proof of the theorem in 1797 but expressed his doubts at the time about "the true metaphysics of the square root of −1".[36] It was not until 1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane,[37] largely establishing modern notation and terminology:[38]
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.
The English mathematicianG.H. Hardy remarked that Gauss was the first mathematician to use complex numbers in "a really confident and scientific way" although mathematicians such as NorwegianNiels Henrik Abel andCarl Gustav Jacob Jacobi were necessarily using them routinely before Gauss published his 1831 treatise.[47]
The common terms used in the theory are chiefly due to the founders. Argand calledcosφ +i sinφ thedirection factor, and themodulus;[d][48] Cauchy (1821) calledcosφ +i sinφ thereduced form (l'expression réduite)[49] and apparently introduced the termargument; Gauss usedi for,[e] introduced the termcomplex number fora +bi,[f] and calleda2 +b2 thenorm.[g] The expressiondirection coefficient, often used forcosφ +i sinφ, is due to Hankel (1867),[53] andabsolute value, formodulus, is due to Weierstrass.
While the above low-level definitions, including the addition and multiplication, accurately describe the complex numbers, there are other, equivalent approaches that reveal the abstract algebraic structure of the complex numbers more immediately.
One approach to is viapolynomials, i.e., expressions of the formwhere thecoefficientsa0, ..., an are real numbers. The set of all such polynomials is denoted by. Since sums and products of polynomials are again polynomials, this set forms acommutative ring, called thepolynomial ring (over the reals). To every such polynomialp, one may assign the complex number, i.e., the value obtained by setting. This defines a function
This function issurjective since every complex number can be obtained in such a way: the evaluation of alinear polynomial at is. However, the evaluation of polynomial ati is 0, since This polynomial isirreducible, i.e., cannot be written as a product of two linear polynomials. Basic facts ofabstract algebra then imply that thekernel of the above map is anideal generated by this polynomial, and that the quotient by this ideal is a field, and that there is anisomorphism
between the quotient ring and. Some authors take this as the definition of.[54] This definition expresses as aquadratic algebra.
Complex numbersa +bi can also be represented by2 × 2matrices that have the formHere the entriesa andb are real numbers. As the sum and product of two such matrices is again of this form, these matrices form asubring of the ring of2 × 2 matrices.
A simple computation shows that the mapis aring 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 thedeterminant of the corresponding matrix, and the conjugate of a complex number with thetranspose of the matrix.
Thepolar form representation of complex numbers explicitly gives these matrices as scaledrotation matrices.In particular, the case ofr = 1, which is, gives (unscaled) rotation matrices.
The study of functions of a complex variable is known ascomplex analysis and has enormous practical use inapplied mathematics as well as in other branches of mathematics. Often, the most natural proofs for statements inreal analysis or evennumber theory employ techniques from complex analysis (seeprime number theorem for an example).
Adomain coloring graph of the function(z2 − 1)(z − 2 −i)2/z2 + 2 + 2i. Darker spots mark moduli near zero, brighter spots are farther away from the origin. The color encodes the argument. The function has zeros for±1, (2 +i) andpoles at
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 athree-dimensional graph to suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane.
Illustration of the behavior of the sequence for three different values ofz (all having the same argument): for the sequence converges to 0 (inner spiral), while it diverges for (outer spiral).
The notions ofconvergent series andcontinuous functions in (real) analysis have natural analogs in complex analysis. A sequence of complex numbers is said toconverge 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,, endowed with themetricis a completemetric space, which notably includes thetriangle inequalityfor any two complex numbersz1 andz2.
Illustration of the complex exponential function mapping the complex plane,w = exp (z). The left plane shows a square mesh with mesh size 1, with the three complex numbers 0, 1, andi highlighted. The two rectangles (in magenta and green) are mapped to circular segments, while the lines parallel to thex-axis are mapped to rays emanating from, but not containing the origin. Lines parallel to they-axis are mapped to circles.
Like in real analysis, this notion of convergence is used to construct a number ofelementary functions: theexponential functionexpz, also writtenez, is defined as theinfinite series, which can be shown toconverge for anyz:For example, isEuler's number.Euler's formula states:for any real numberφ. 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 includesEuler's identity
The exponential function maps complex numbersz differing by a multiple of to the same complex numberw.
For any positive real numbert, there is a unique real numberx such that. This leads to the definition of thenatural logarithm as theinverse 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 bothiπ and3iπ are possible values for the complex logarithm of−1.
In general, given any non-zero complex numberw, any numberz solving the equation
is called acomplex logarithm ofw, denoted. It can be shown that these numbers satisfywhere is theargument definedabove, and the (real)natural logarithm. As arg is amultivalued function, unique only up to a multiple of2π, log is also multivalued. Theprincipal value of log is often taken by restricting the imaginary part to theinterval(−π,π]. This leads to the complex logarithm being abijective function taking values in the strip (that is denoted in the above illustration)
If is not a non-positive real number (a positive or a non-real number), the resultingprincipal value of the complex logarithm is obtained with−π <φ <π. It is ananalytic 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 islnz = ln(−z) +iπ.[h]
Complexexponentiationzω is defined asand is multi-valued, except whenω is an integer. Forω = 1 /n, for some natural numbern, this recovers the non-uniqueness ofnth roots mentioned above. Ifz > 0 is real (andω an arbitrary complex number), one has a preferred choice of, the real logarithm, which can be used to define a preferred exponential function.
Complex numbers, unlike real numbers, do not in general satisfy the unmodified power and logarithm identities, particularly when naïvely treated as single-valued functions; seefailure of power and logarithm identities. For example, they do not satisfyBoth sides of the equation are multivalued by the definition of complex exponentiation given here, and the values on the left are a subset of those on the right.
The series defining the real trigonometric functionssin andcos, as well as thehyperbolic functionssinh andcosh, also carry over to complex arguments without change. For the other trigonometric and hyperbolic functions, such astan, things are slightly more complicated, as the defining series do not converge for all complex values. Therefore, one must define them either in terms of sine, cosine and exponential, or, equivalently, by using the method ofanalytic continuation.
The value of a trigonometric or hyperbolic function of a complex number can be expressed in terms of those functions evaluated on real numbers, via angle-addition formulas. Forz =x +iy,
Where these expressions are not well defined, because a trigonometric or hyperbolic function evaluates to infinity or there is division by zero, they are nonetheless correct aslimits.
Color wheel graph of the functionsin(1/z) that is holomorphic except atz = 0, which is an essential singularity of this function. White parts inside refer to numbers having large absolute values.
A function → is calledholomorphic orcomplex 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 isnot complex differentiable.A real differentiable function is complex differentiableif and only if it satisfies theCauchy–Riemann equations, which are sometimes abbreviated as
Complex analysis shows some features not apparent in real analysis. For example, theidentity theorem asserts that two holomorphic functionsf andg agree if they agree on an arbitrarily smallopen subset of.Meromorphic functions, functions that can locally be written asf(z)/(z −z0)n with a holomorphic functionf, still share some of the features of holomorphic functions. Other functions haveessential singularities, such assin(1/z) atz = 0.
Threenon-collinear points in the plane determine theshape of the triangle. Locating the points in the complex plane, this shape of a triangle may be expressed by complex arithmetic asThe shape of a triangle will remain the same, when the complex plane is transformed by translation or dilation (by anaffine transformation), corresponding to the intuitive notion of shape, and describingsimilarity. Thus each triangle is in asimilarity class of triangles with the same shape.[55]
The Mandelbrot set with the real and imaginary axes labeled.
TheMandelbrot 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 notdiverge wheniterated infinitely. Similarly,Julia sets have the same rules, except where remains constant.
Every triangle has a uniqueSteiner inellipse – anellipse inside the triangle and tangent to the midpoints of the three sides of the triangle. Thefoci of a triangle's Steiner inellipse can be found as follows, according toMarden's theorem:[56][57] Denote the triangle's vertices in the complex plane asa =xA +yAi,b =xB +yBi, andc =xC +yCi. Write thecubic 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.
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 calledalgebraic numbers – they are a principal object of study inalgebraic 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 offield theory to thenumber field containingroots of unity, it can be shown that it is not possible to construct a regularnonagonusing only compass and straightedge – a purely geometric problem.
Another example is theGaussian integers; that is, numbers of the formx +iy, wherex andy are integers, which can be used to classifysums of squares.
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, theRiemann zeta functionζ(s) is related to the distribution ofprime numbers.
In applied fields, complex numbers are often used to compute certain real-valuedimproper integrals, by means of complex-valued functions. Several methods exist to do this; seemethods of contour integration.
Indifferential equations, it is common to first find all complex rootsr of thecharacteristic equation of alinear differential equation or equation system and then attempt to solve the system in terms of base functions of the formf(t) =ert. Likewise, indifference equations, the complex rootsr of the characteristic equation of the difference equation system are used, to attempt to solve the system in terms of base functions of the formf(t) =rt.
Since is algebraically closed, any non-empty complexsquare matrix has at least one (complex)eigenvalue. By comparison, real matrices do not always have real eigenvalues, for examplerotation matrices (for rotations of the plane for angles other than 0° or 180°) leave no direction fixed, and therefore do not have anyreal eigenvalue. The existence of (complex) eigenvalues, and the ensuing existence ofeigendecomposition is a useful tool for computing matrix powers andmatrix exponentials.
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
Complex numbers are used insignal 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 asine wave of a givenfrequency, the absolute value|z| of the correspondingz is theamplitude and theargumentargz is thephase.
IfFourier 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 theangular frequency and the complex numberA encodes the phase and amplitude as explained above.
In electrical engineering, the imaginary unit is denoted byj, to avoid confusion withI, 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 voltage in an AC circuit is oscillating, it can be represented as
To obtain the measurable quantity, the real part is taken:
The complex-valued signalV(t) is called theanalytic representation of the real-valued, measurable signalv(t).[58]
It can be shown that any field having these properties isisomorphic (as a field) to For example, thealgebraic closure of the field of thep-adic number also satisfies these three properties, so these two fields are isomorphic (as fields, but not as topological fields).[59] Also, is isomorphic to the field of complexPuiseux series. However, specifying an isomorphism requires theaxiom of choice. Another consequence of this algebraic characterization is that contains many proper subfields that are isomorphic to.
The preceding characterization of describes only the algebraic aspects of That is to say, the properties ofnearness andcontinuity, which matter in areas such asanalysis andtopology, are not dealt with. The following description of as atopological field (that is, a field that is equipped with atopology, which allows the notion of convergence) does take into account the topological properties. contains a subsetP (namely the set of positive real numbers) of nonzero elements satisfying the following three conditions:
P is closed under addition, multiplication and taking inverses.
Ifx andy are distinct elements ofP, then eitherx −y ory −x is inP.
IfS is any nonempty subset ofP, thenS +P =x +P for somex in
Moreover, has a nontrivialinvolutiveautomorphismx ↦x* (namely the complex conjugation), such thatx x* is inP for any nonzerox in
Any fieldF with these properties can be endowed with a topology by taking the setsB(x, p) = { y |p − (y −x)(y −x)* ∈P } as abase, wherex ranges over the field andp ranges overP. With this topologyF is isomorphic as atopological field to
The onlyconnectedlocally compacttopological fields are and This gives another characterization of as a topological field, because can be distinguished from because the nonzero complex numbers areconnected, while the nonzero real numbers are not.[60]
The process of extending the field of reals to is an instance of theCayley–Dickson construction. Applying this construction iteratively to then yields thequaternions, theoctonions,[61] thesedenions, and thetrigintaduonions. This construction turns out to diminish the structural properties of the involved number systems.
Unlike the reals, is not anordered field, that is to say, it is not possible to define a relationz1 <z2 that is compatible with the addition and multiplication. In fact, in any ordered field, the square of any element is necessarily positive, soi2 = −1 precludes the existence of anordering on[62] Passing from to the quaternions loses commutativity, while the octonions (additionally to not being commutative) fail to be associative. The reals, complex numbers, quaternions and octonions are allnormed division algebras over. ByHurwitz's theorem they are the only ones; thesedenions, the next step in the Cayley–Dickson construction, fail to have this structure.
The Cayley–Dickson construction is closely related to theregular representation of thought of as an-algebra (an-vector space with a multiplication), with respect to the basis(1, i). This means the following: the-linear mapfor some fixed complex numberw can be represented by a2 × 2 matrix (once a basis has been chosen). With respect to the basis(1, i), this matrix isthat is, the one mentioned in the section on matrix representation of complex numbers above. While this is alinear representation of 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:J2 = −I. Thenis also isomorphic to the field and gives an alternative complex structure on This is generalized by the notion of alinear complex structure.
Hypercomplex numbers also generalize and For example, this notion contains thesplit-complex numbers, which are elements of the ring (as opposed to for complex numbers). In this ring, the equationa2 = 1 has four solutions.
The field is the completion of the field ofrational numbers, with respect to the usualabsolute valuemetric. Other choices ofmetrics on lead to the fields ofp-adic numbers (for anyprime numberp), which are thereby analogous to. There are no other nontrivial ways of completing than and byOstrowski's theorem. The algebraic closures of still carry a norm, but (unlike) are not complete with respect to it. The completion of turns out to be algebraically closed. By analogy, the field is calledp-adic complex numbers.
The fields and their finite field extensions, including are calledlocal fields.
^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.[26]
^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)[28]
^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 themodule and thedirection factor of a complex number:"... pourrait être appelé lemodule de, et représenterait lagrandeur absolue de la ligne, tandis que l'autre facteur, dont le module est l'unité, en représenterait la direction." [... could be called themodule of and would represent theabsolute 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:[50]"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 quantitatesimaginarias extenditur, ita ut absque restrictione ipsius obiectum constituant numeri formaea + bi, denotantibusi, pro more quantitatem imaginariam, atquea, 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 toimaginary quantities, so that, without restrictions on it, numbers of the forma + bi —i denoting by convention the imaginary quantity, and the variablesa, b [denoting] all real integer numbers between and — constitute an object.]
^Gauss:[51]"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 forma + 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:[52]"Productum numeri complexi per numerum ipsi conjunctum utriusquenormam 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 otherray thru the origin.
^For an extensive account of the history of "imaginary" numbers, from initial skepticism to ultimate acceptance, seeBourbaki, 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.
^abWeisstein, Eric W."Complex Number".mathworld.wolfram.com. Retrieved12 August 2020.
^Campbell, George Ashley (April 1911)."Cisoidal oscillations"(PDF).Proceedings of the American Institute of Electrical Engineers.XXX (1–6).American Institute of Electrical Engineers: 789–824 [Fig. 13 on p. 810].Bibcode:1911PAIEE..30d.789C.doi:10.1109/PAIEE.1911.6659711.S2CID51647814. Retrieved24 June 2023. p. 789:The use ofi (or Greekı) 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 symbolj 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 usingj for +i and others usingj for −i.
^Brown, James Ward; Churchill, Ruel V. (1996).Complex variables and applications (6 ed.). New York, USA:McGraw-Hill. p. 2.ISBN978-0-07-912147-9. p. 2:In electrical engineering, the letterj is used instead ofi.
^abConfalonieri, Sara (2015).The Unattainable Attempt to Avoid the Casus Irreducibilis for Cubic Equations: Gerolamo Cardano's De Regula Aliza. Springer. pp. 15–16 (note 26).ISBN978-3658092757.
^Euler, Leonhard (1748).Introductio in Analysin Infinitorum [Introduction to the Analysis of the Infinite] (in Latin). Vol. 1. Lucerne, Switzerland: Marc Michel Bosquet & Co. p. 104.
^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 + isin φ) haüfig denRichtungscoefficienten nennen." (We will often call the factor (cos φ + i sin φ) the "coefficient of direction".)
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.ISBN978-0-521-88068-8. Archived fromthe original on 13 March 2020. Retrieved9 August 2011.
Nahin, Paul J. (1998).An Imaginary Tale: The Story of. Princeton University Press.ISBN978-0-691-02795-1. — 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.ISBN978-0-387-97497-2. — An advanced perspective on the historical development of the concept of number.
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