A complex number is a number that can be expressed in the form a + bi,
where a and b are real numbers, and i is a solution of the equation x2
= −1, which is called an imaginary number because there is no real
number that satisfies this equation. For the complex number a + bi, a
is called the real part, and b is called the imaginary part. Despite
the historical nomenclature "imaginary", complex numbers are regarded
in the mathematical sciences as just as "real" as the real numbers,
and are fundamental in many aspects of the scientific description of
the natural world.[1][2]
The complex number system can be defined as the algebraic extension of
the ordinary real numbers by an imaginary number i.[3] This means that
complex numbers can be added, subtracted, and multiplied, as
polynomials in the variable i, with the rule i2 = −1 imposed.
Furthermore, complex numbers can also be divided by nonzero complex
numbers. Overall, the complex number system is a field.
Most importantly the complex numbers give rise to the fundamental
theorem of algebra: every non-constant polynomial equation with
complex coefficients has a complex solution. This property is true of
the complex numbers, but not the reals. The 16th century Italian
mathematician
Contents 1 Overview 1.1 Definition 1.2 Cartesian form and definition via ordered pairs 1.3 Complex plane 1.4 History in brief 1.5 Notation 2 Equality and order relations 3 Elementary operations 3.1 Conjugate
3.2
4 Polar form 4.1
5 Exponentiation 5.1 Euler's formula
5.2 Natural logarithm
5.3
6 Properties 6.1 Field structure 6.2 Solutions of polynomial equations 6.3 Algebraic characterization 6.4 Characterization as a topological field 7 Formal construction 7.1 Construction as ordered pairs 7.2 Construction as a quotient field 7.3 Matrix representation of complex numbers 8 Complex analysis 8.1 Complex exponential and related functions 8.2 Holomorphic functions 9 Applications 9.1 Control theory
9.2 Improper integrals
9.3 Fluid dynamics
9.4 Dynamic equations
9.5
9.9.1 Fractals 9.9.2 Triangles 9.10
10 History 11 Generalizations and related notions 12 See also 13 Notes 14 References 14.1 Mathematical references 14.2 Historical references 15 Further reading 16 External links Overview[edit] Complex numbers allow solutions to certain equations that have no solutions in real numbers. For example, the equation ( x + 1 ) 2 = − 9 displaystyle (x+1)^ 2 =-9, has no real solution, since the square of a real number cannot be negative. Complex numbers provide a solution to this problem. The idea is to extend the real numbers with an indeterminate i (sometimes called the imaginary unit) that is taken to satisfy the relation i2 = −1, so that solutions to equations like the preceding one can be found. In this case the solutions are −1 + 3i and −1 − 3i, as can be verified using the fact that i2 = −1: ( ( − 1 + 3 i ) + 1 ) 2 = ( 3 i ) 2 = ( 3 2 ) ( i 2 ) = 9 ( − 1 ) = − 9 , displaystyle ((-1+3i)+1)^ 2 =(3i)^ 2 =(3^ 2 )(i^ 2 )=9(-1)=-9, ( ( − 1 − 3 i ) + 1 ) 2 = ( − 3 i ) 2 = ( − 3 ) 2 ( i 2 ) = 9 ( − 1 ) = − 9. displaystyle ((-1-3i)+1)^ 2 =(-3i)^ 2 =(-3)^ 2 (i^ 2 )=9(-1)=-9. According to the fundamental theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. Definition[edit] An illustration of the complex plane. The real part of a complex number z = x + iy is x, and its imaginary part is y. A complex number is a number of the form a + bi, where a and b are real numbers and i is an indeterminate satisfying i2 = −1. For example, 2 + 3i is a complex number.[5] A complex number may therefore be defined as a polynomial in the single indeterminate i, with the relation i2 + 1 = 0 imposed. From this definition, complex numbers can be added or multiplied, using the addition and multiplication for polynomials. Formally, the set of complex numbers is the quotient ring of the polynomial ring in the indeterminate i, by the ideal generated by the polynomial i2 + 1 (see below).[6] The set of all complex numbers is denoted by C displaystyle mathbf C (upright bold) or C displaystyle mathbb C (blackboard bold). The real number a is called the real part of the complex number a + bi; the real number b is called the imaginary part of a + bi. By this convention, the imaginary part does not include a factor of i: hence b, not bi, is the imaginary part.[7][8] The real part of a complex number z is denoted by Re(z) or ℜ(z); the imaginary part of a complex number z is denoted by Im(z) or ℑ(z). For example, Re ( 2 + 3 i ) = 2 Im ( 2 + 3 i ) = 3. displaystyle begin aligned operatorname Re (2+3i)&=2\operatorname Im (2+3i)&=3.end aligned 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. It is common to write a for a + 0i and
bi for 0 + bi. Moreover, when the imaginary part is negative, it is
common to write a − bi with b > 0 instead of a + (−b)i, for
example 3 − 4i instead of 3 + (−4)i.
Cartesian form and definition via ordered pairs[edit]
A complex number can thus be identified with an ordered pair
(Re(z),Im(z)) in the Cartesian plane, an identification sometimes
known as the Cartesian form of z. In fact, a complex number can be
defined as an ordered pair (a,b), but then rules for addition and
multiplication must also be included as part of the definition (see
below).[9]
Figure 1: A complex number z, plotted as a point (red) and position vector (blue) on an Argand diagram; a+bi is its rectangular expression. A complex number can be viewed as a point or position vector in a
two-dimensional
Main section: History The solution in radicals (without trigonometric functions) of a
general cubic equation contains the square roots of negative numbers
when all three roots are real numbers, a situation that cannot be
rectified by factoring aided by the rational root test if the cubic is
irreducible (the so-called casus irreducibilis). This conundrum led
Italian mathematician
z 1 displaystyle z_ 1 and z 2 displaystyle z_ 2 are equal if and only if Re ( z 1 ) = Re ( z 2 ) displaystyle operatorname Re (z_ 1 )=operatorname Re (z_ 2 ) and Im ( z 1 ) = Im ( z 2 ) displaystyle operatorname Im (z_ 1 )=operatorname Im (z_ 2 ) . If the complex numbers are written in polar form, they are equal if and only if they have the same argument and the same magnitude. Because complex numbers are naturally thought of as existing on a two-dimensional plane, there is no natural linear ordering on the set of complex numbers. Furthermore, there is no linear ordering on the complex numbers that is compatible with addition and multiplication – the complex numbers cannot have the structure of an ordered field. This is because any square in an ordered field is at least 0, but i2 = −1. Elementary operations[edit] Conjugate[edit] See also: Complex conjugate Geometric representation of z and its conjugate z ¯ displaystyle bar z in the complex plane The complex conjugate of the complex number z = x + yi is defined to be x − yi. It is denoted by either z ¯ displaystyle overline z or z*.[15] Geometrically, z ¯ displaystyle bar z is the "reflection" of z about the real axis. Conjugating twice gives the original complex number: z ¯ ¯ = z displaystyle bar bar z =z . The real and imaginary parts of a complex number z can be extracted using the conjugate: Re ( z ) = z + z ¯ 2 , displaystyle operatorname Re (z)= dfrac z+ overline z 2 ,, Im ( z ) = z − z ¯ 2 i . displaystyle operatorname Im (z)= dfrac z- overline z 2i ., Moreover, a complex number is real if and only if it equals its own conjugate. Conjugation distributes over the standard arithmetic operations: z + w ¯ = z ¯ + w ¯ , displaystyle overline z+w = overline z + overline w ,, z − w ¯ = z ¯ − w ¯ , displaystyle overline z-w = overline z - overline w ,, z ⋅ w ¯ = z ¯ ⋅ w ¯ , displaystyle overline zcdot w = overline z cdot overline w ,, z / w ¯ = z ¯ / w ¯ . displaystyle overline z/w = overline z / overline w .,
Complex numbers are added by separately adding the real and imaginary parts of the summands. That is to say: ( a + b i ) + ( c + d i ) = ( a + c ) + ( b + d ) i . displaystyle (a+bi)+(c+di)=(a+c)+(b+d)i. Similarly, subtraction is defined by ( a + b i ) − ( c + d i ) = ( a − c ) + ( b − d ) i . displaystyle (a+bi)-(c+di)=(a-c)+(b-d)i. Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers A and B, interpreted as points of the complex plane, is the point X obtained by building a parallelogram, three of whose vertices are O, A and B. Equivalently, X is the point such that the triangles with vertices O, A, B, and X, B, A, are congruent. Multiplication and division[edit] The multiplication of two complex numbers is defined by the following formula: ( a + b i ) ( c + d i ) = ( a c − b d ) + ( b c + a d ) i . displaystyle (a+bi)(c+di)=(ac-bd)+(bc+ad)i. In particular, the square of i is −1: i 2 = i × i = − 1. displaystyle i^ 2 =itimes i=-1. The preceding definition of multiplication of general complex numbers follows naturally from this fundamental property of i. Indeed, if i is treated as a number so that di means d times i, the above multiplication rule is identical to the usual rule for multiplying two sums of two terms. ( a + b i ) ( c + d i ) = a c + b c i + a d i + b i d i displaystyle (a+bi)(c+di)=ac+bci+adi+bidi (distributive property) = a c + b i d i + b c i + a d i displaystyle =ac+bidi+bci+adi (commutative property of addition—the order of the summands can be changed) = a c + b d i 2 + ( b c + a d ) i displaystyle =ac+bdi^ 2 +(bc+ad)i (commutative and distributive properties) = ( a c − b d ) + ( b c + a d ) i displaystyle =(ac-bd)+(bc+ad)i (fundamental property of i). The division of two complex numbers is defined in terms of complex multiplication, which is described above, and real division. When at least one of c and d is non-zero, we have a + b i c + d i = ( a c + b d c 2 + d 2 ) + ( b c − a d c 2 + d 2 ) i . displaystyle frac a+bi c+di =left( ac+bd over c^ 2 +d^ 2 right)+left( bc-ad over c^ 2 +d^ 2 right)i. Division can be defined in this way because of the following observation: a + b i c + d i = ( a + b i ) ⋅ ( c − d i ) ( c + d i ) ⋅ ( c − d i ) = ( a c + b d c 2 + d 2 ) + ( b c − a d c 2 + d 2 ) i . displaystyle frac a+bi c+di = frac left(a+biright)cdot left(c-diright) left(c+diright)cdot left(c-diright) =left( ac+bd over c^ 2 +d^ 2 right)+left( bc-ad over c^ 2 +d^ 2 right)i. As shown earlier, c − di is the complex conjugate of the denominator c + di. At least one of the real part c and the imaginary part d of the denominator must be nonzero for division to be defined. This is called "rationalization" of the denominator (although the denominator in the final expression might be an irrational real number). Reciprocal[edit] The reciprocal of a nonzero complex number z = x + yi is given by 1 z = z ¯ z z ¯ = z ¯ x 2 + y 2 = x x 2 + y 2 − y x 2 + y 2 i . displaystyle frac 1 z = frac bar z z bar z = frac bar z x^ 2 +y^ 2 = frac x x^ 2 +y^ 2 - frac y x^ 2 +y^ 2 i. This formula can be used to compute the multiplicative inverse of a complex number if it is given in rectangular coordinates. Inversive geometry, a branch of geometry studying reflections more general than ones about a line, can also be expressed in terms of complex numbers. In the network analysis of electrical circuits, the complex conjugate is used in finding the equivalent impedance when the maximum power transfer theorem is used. Square root[edit] See also: Square roots of negative and complex numbers The square roots of a + bi (with b ≠ 0) are ± ( γ + δ i ) displaystyle pm (gamma +delta i) , where γ = a + a 2 + b 2 2 displaystyle gamma = sqrt frac a+ sqrt a^ 2 +b^ 2 2 and δ = sgn ( b ) − a + a 2 + b 2 2 , displaystyle delta =operatorname sgn (b) sqrt frac -a+ sqrt a^ 2 +b^ 2 2 , where sgn is the signum function. This can be seen by squaring ± ( γ + δ i ) displaystyle pm (gamma +delta i) to obtain a + bi.[16][17] Here a 2 + b 2 displaystyle sqrt a^ 2 +b^ 2 is called the modulus of a + bi, and the square root sign indicates the square root with non-negative real part, called the principal square root; also a 2 + b 2 = z z ¯ displaystyle sqrt a^ 2 +b^ 2 = sqrt z bar z , where z = a + b i displaystyle z=a+bi .[18] Polar form[edit] Main article: Polar coordinate system "Polar form" redirects here. For the higher-dimensional analogue, see Polar decomposition. Figure 2: The argument φ and modulus r locate a point on an Argand diagram; r ( cos φ + i sin φ ) displaystyle r(cos varphi +isin varphi ) or r e i φ displaystyle re^ ivarphi are polar expressions of the point.
r =
z
= x 2 + y 2 . displaystyle textstyle r=z= sqrt x^ 2 +y^ 2 ., If z is a real number (that is, if y = 0), then r = x . That is, the absolute value of a real number equals its absolute value as a complex number. By Pythagoras' theorem, the absolute value of complex number is the distance to the origin of the point representing the complex number in the complex plane. The square of the absolute value is
z
2 = z z ¯ = x 2 + y 2 . displaystyle textstyle z^ 2 =z bar z =x^ 2 +y^ 2 ., where z ¯ displaystyle bar z is the complex conjugate of z displaystyle z . The argument of z (in many applications referred to as the "phase") is the angle of the radius OP with the positive real axis, and is written as arg ( z ) displaystyle arg(z) . As with the modulus, the argument can be found from the rectangular form x + y i displaystyle x+yi :[20] φ = arg ( z ) = arctan ( y x ) if x > 0 arctan ( y x ) + π if x < 0 and y ≥ 0 arctan ( y x ) − π if x < 0 and y < 0 π 2 if x = 0 and y > 0 − π 2 if x = 0 and y < 0 indeterminate if x = 0 and y = 0. displaystyle varphi =arg(z)= begin cases arctan left( dfrac y x right)& text if x>0\arctan left( dfrac y x right)+pi & text if x<0 text and ygeq 0\arctan left( dfrac y x right)-pi & text if x<0 text and y<0\ dfrac pi 2 & text if x=0 text and y>0\- dfrac pi 2 & text if x=0 text and y<0\ text indeterminate & text if x=0 text and y=0.end cases Visualisation of the square to sixth roots of a complex number z, in polar form reiφ where φ = arg z and r = z – if z is real, φ = 0 or π. Principal roots are in black. Normally, as given above, the principal value in the interval (−π,π] is chosen. Values in the range [0,2π) are obtained by adding 2π if the value is negative. The value of φ is expressed in radians in this article. It can increase by any integer multiple of 2π and still give the same angle. Hence, the arg function is sometimes considered as multivalued. The polar angle for the complex number 0 is indeterminate, but arbitrary choice of the angle 0 is common. The value of φ equals the result of atan2: φ = atan2 ( Im ( z ) , Re ( z ) ) . displaystyle varphi =operatorname atan2 left(operatorname Im (z),operatorname Re (z)right). Together, r and φ give another way of representing complex numbers, the polar form, as the combination of modulus and argument fully specify the position of a point on the plane. Recovering the original rectangular co-ordinates from the polar form is done by the formula called trigonometric form z = r ( cos φ + i sin φ ) . displaystyle z=r(cos varphi +isin varphi )., Using
z = r e i φ . displaystyle z=re^ ivarphi ., Using the cis function, this is sometimes abbreviated to z = r cis φ . displaystyle z=roperatorname cis varphi ., In angle notation, often used in electronics to represent a phasor with amplitude r and phase φ, it is written as[21] z = r ∠ φ . displaystyle z=rangle varphi ., Multiplication and division in polar form[edit] Multiplication of 2 + i (blue triangle) and 3 + i (red triangle). The red triangle is rotated to match the vertex of the blue one and stretched by √5, the length of the hypotenuse of the blue triangle. Formulas for multiplication, division and exponentiation are simpler in polar form than the corresponding formulas in Cartesian coordinates. Given two complex numbers z1 = r1(cos φ1 + i sin φ1) and z2 = r2(cos φ2 + i sin φ2), because of the well-known trigonometric identities cos ( a ) cos ( b ) − sin ( a ) sin ( b ) = cos ( a + b ) displaystyle cos(a)cos(b)-sin(a)sin(b)=cos(a+b) cos ( a ) sin ( b ) + sin ( a ) cos ( b ) = sin ( a + b ) displaystyle cos(a)sin(b)+sin(a)cos(b)=sin(a+b) we may derive z 1 z 2 = r 1 r 2 ( cos ( φ 1 + φ 2 ) + i sin ( φ 1 + φ 2 ) ) . displaystyle z_ 1 z_ 2 =r_ 1 r_ 2 (cos(varphi _ 1 +varphi _ 2 )+isin(varphi _ 1 +varphi _ 2 ))., In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product. For example, multiplying by i corresponds to a quarter-turn counter-clockwise, which gives back i2 = −1. The picture at the right illustrates the multiplication of ( 2 + i ) ( 3 + i ) = 5 + 5 i . displaystyle (2+i)(3+i)=5+5i., Since the real and imaginary part of 5 + 5i are equal, the argument of that number is 45 degrees, or π/4 (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 π 4 = arctan 1 2 + arctan 1 3 displaystyle frac pi 4 =arctan frac 1 2 +arctan frac 1 3 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 π. Similarly, division is given by z 1 z 2 = r 1 r 2 ( cos ( φ 1 − φ 2 ) + i sin ( φ 1 − φ 2 ) ) . displaystyle frac z_ 1 z_ 2 = frac r_ 1 r_ 2 left(cos(varphi _ 1 -varphi _ 2 )+isin(varphi _ 1 -varphi _ 2 )right).
e i x = cos x + i sin x displaystyle e^ ix =cos x+isin x , where e is the base of the natural logarithm. This can be proved through induction by observing that i 0 = 1 , i 1 = i , i 2 = − 1 , i 3 = − i , i 4 = 1 , i 5 = i , i 6 = − 1 , i 7 = − i , displaystyle begin aligned i^ 0 & =1,quad &i^ 1 & =i,quad &i^ 2 & =-1,quad &i^ 3 & =-i,\i^ 4 &= 1,quad &i^ 5 &= i,quad &i^ 6 & =-1,quad &i^ 7 & =-i,end aligned and so on, and by considering the
e i x = 1 + i x + ( i x ) 2 2 ! + ( i x ) 3 3 ! + ( i x ) 4 4 ! + ( i x ) 5 5 ! + ( i x ) 6 6 ! + ( i x ) 7 7 ! + ( i x ) 8 8 ! + ⋯ = 1 + i x − x 2 2 ! − i x 3 3 ! + x 4 4 ! + i x 5 5 ! − x 6 6 ! − i x 7 7 ! + x 8 8 ! + ⋯ = ( 1 − x 2 2 ! + x 4 4 ! − x 6 6 ! + x 8 8 ! − ⋯ ) + i ( x − x 3 3 ! + x 5 5 ! − x 7 7 ! + ⋯ ) = cos x + i sin x . displaystyle begin aligned e^ ix & =1+ix+ frac (ix)^ 2 2! + frac (ix)^ 3 3! + frac (ix)^ 4 4! + frac (ix)^ 5 5! + frac (ix)^ 6 6! + frac (ix)^ 7 7! + frac (ix)^ 8 8! +cdots \[8pt]& =1+ix- frac x^ 2 2! - frac ix^ 3 3! + frac x^ 4 4! + frac ix^ 5 5! - frac x^ 6 6! - frac ix^ 7 7! + frac x^ 8 8! +cdots \[8pt]& =left(1- frac x^ 2 2! + frac x^ 4 4! - frac x^ 6 6! + frac x^ 8 8! -cdots right)+ileft(x- frac x^ 3 3! + frac x^ 5 5! - frac x^ 7 7! +cdots right)\[8pt]& =cos x+isin x .end aligned The rearrangement of terms is justified because each series is
absolutely convergent.
Natural logarithm[edit]
It follows from
z = r ( cos φ + i sin φ ) displaystyle z=r(cos varphi +isin varphi ) where r is a non-negative real number, one possible value for the complex logarithm of z is ln ( z ) = ln ( r ) + φ i . displaystyle ln(z)=ln(r)+varphi i. Because cosine and sine are periodic functions, other possible values may be obtained. For example, e i π = e 3 i π = − 1 displaystyle e^ ipi =e^ 3ipi =-1 , so both i π displaystyle ipi and 3 i π displaystyle 3ipi are two possible values for the natural logarithm of − 1 displaystyle -1 . To deal with the existence of more the one possible value for a given input, the complex logarithm may be considered a multi-valued function, with ln ( z ) = ln ( r ) + ( φ + 2 π k ) i
k ∈ Z . displaystyle ln(z)=left ln(r)+(varphi +2pi k)i;;kin mathbb Z right . Alternatively, a branch cut can be used to define a single-valued
"branch" of the complex logarithm.
ln ( a b ) = b ln ( a ) displaystyle ln(a^ b )=bln(a) to define complex exponentiation, which is likewise multi-valued: ln ( z n ) = ln ( ( r ( cos φ + i sin φ ) ) n ) = n ln ( r ( cos φ + i sin φ ) ) = n ( ln ( r ) + ( φ + k 2 π ) i )
k ∈ Z = n ln ( r ) + n φ i + n k 2 π i
k ∈ Z . displaystyle begin aligned ln(z^ n )&=ln((r(cos varphi +isin varphi ))^ n )\&=nln(r(cos varphi +isin varphi ))\&= n(ln(r)+(varphi +k2pi )i) kin mathbb Z \&= nln(r)+nvarphi i+nk2pi i kin mathbb Z .end aligned When n is an integer, this simplifies to de Moivre's formula: z n = ( r ( cos φ + i sin φ ) ) n = r n ( cos n φ + i sin n φ ) . displaystyle z^ n =(r(cos varphi +isin varphi ))^ n =r^ n ,(cos nvarphi +isin nvarphi ). The nth roots of z are given by z n = r n ( cos ( φ + 2 k π n ) + i sin ( φ + 2 k π n ) ) displaystyle sqrt[ n ] z = sqrt[ n ] r left(cos left( frac varphi +2kpi n right)+isin left( frac varphi +2kpi n right)right) for any integer k satisfying 0 ≤ k ≤ n − 1. Here n√r is the usual (positive) nth root of the positive real number r. While the nth root of a positive real number r is chosen to be the positive real number c satisfying cn = r there is no natural way of distinguishing one particular complex nth root of a complex number. Therefore, the nth root of z is considered as a multivalued function (in z), as opposed to a usual function f, for which f(z) is a uniquely defined number. Formulas such as z n n = z displaystyle sqrt[ n ] z^ n =z (which holds for positive real numbers), do in general not hold for complex numbers. Properties[edit] Field structure[edit] The set C of complex numbers is a field.[22] Briefly, this means that the following facts hold: first, any two complex numbers can be added and multiplied to yield another complex number. Second, for any complex number z, its additive inverse −z is also a complex number; and third, every nonzero complex number has a reciprocal complex number. Moreover, these operations satisfy a number of laws, for example the law of commutativity of addition and multiplication for any two complex numbers z1 and z2: z 1 + z 2 = z 2 + z 1 , displaystyle z_ 1 +z_ 2 =z_ 2 +z_ 1 , z 1 z 2 = z 2 z 1 . displaystyle z_ 1 z_ 2 =z_ 2 z_ 1 . These two laws and the other requirements on a field can be proven by the formulas given above, using the fact that the real numbers themselves form a field. Unlike the reals, C 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 C.[23] When the underlying field for a mathematical topic or construct is the field of complex numbers, the topic's name is usually modified to reflect that fact. For example: complex analysis, complex matrix, complex polynomial, and complex Lie algebra. Solutions of polynomial equations[edit] Given any complex numbers (called coefficients) a0, …, an, the equation a n z n + ⋯ + a 1 z + a 0 = 0 displaystyle a_ n z^ n +dotsb +a_ 1 z+a_ 0 =0 has at least one complex solution z, provided that at least one of the
higher coefficients a1, …, an is nonzero.[24] This is the
statement of the fundamental theorem of algebra, of Carl Friedrich
Gauss and Jean le Rond d'Alembert. Because of this fact, C is called
an algebraically closed field. This property does not hold for the
field of rational numbers Q (the polynomial x2 − 2 does not have a
rational root, since √2 is not a rational number) nor the real
numbers R (the polynomial x2 + a does not have a real root for a >
0, since the square of x is positive for any real number x).
There are various proofs of this theorem, either by analytic methods
such as Liouville's theorem, or topological ones such as the winding
number, or a proof combining
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 C. Moreover, C has a nontrivial involutive automorphism x ↦ x* (namely the complex conjugation), such that x x* is in P for any nonzero x in C. 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 C. The only connected locally compact topological fields are R and C. This gives another characterization of C as a topological field, since C can be distinguished from R because the nonzero complex numbers are connected, while the nonzero real numbers are not.[26] Formal construction[edit] Construction as ordered pairs[edit] The set C of complex numbers can be defined as the set R2 of ordered pairs (a, b) of real numbers, in which the following rules for addition and multiplication are imposed:[27] ( a , b ) + ( c , d ) = ( a + c , b + d ) ( a , b ) ⋅ ( c , d ) = ( a c − b d , b c + a d ) . displaystyle begin aligned (a,b)+(c,d)&=(a+c,b+d)\(a,b)cdot (c,d)&=(ac-bd,bc+ad).end aligned It is then just a matter of notation to express (a, b) as a + bi. Construction as a quotient field[edit] Though this low-level construction does accurately describe the structure of the complex numbers, the following equivalent definition reveals the algebraic nature of C more immediately. This characterization relies on the notion of fields and polynomials. A field is a set endowed with addition, subtraction, multiplication and division operations that behave as is familiar from, say, rational numbers. For example, the distributive law ( x + y ) z = x z + y z displaystyle (x+y)z=xz+yz must hold for any three elements x, y and z of a field. The set R of real numbers does form a field. A polynomial p(X) with real coefficients is an expression of the form a n X n + ⋯ + a 1 X + a 0 displaystyle a_ n X^ n +dotsb +a_ 1 X+a_ 0 , where the a0, ..., an are real numbers. The usual addition and multiplication of polynomials endows the set R[X] of all such polynomials with a ring structure. This ring is called the polynomial ring over the real numbers. The set of complex numbers is defined as the quotient ring R[X]/(X 2 + 1).[28] This extension field contains two square roots of −1, namely (the cosets of) X and −X, respectively. (The cosets of) 1 and X form a basis of R[X]/(X 2 + 1) as a real vector space, which means that each element of the extension field can be uniquely written as a linear combination in these two elements. Equivalently, elements of the extension field can be written as ordered pairs (a, b) of real numbers. The quotient ring is a field, because the (X2 + 1) is a prime ideal in R[X], a principal ideal domain, and therefore is a maximal ideal. The formulas for addition and multiplication in the ring R[X], modulo the relation (X2 = 1 correspond to the formulas for addition and multiplication of complex numbers defined as ordered pairs. So the two definitions of the field C are isomorphic (as fields). Accepting that C is algebraically closed, since it is an algebraic extension of R in this approach, C is therefore the algebraic closure of R. Matrix representation of complex numbers[edit] Complex numbers a + bi can also be represented by 2 × 2 matrices that have the following form: ( a − b b a ) displaystyle begin pmatrix a&-b\b&;;aend pmatrix Here the entries a and b are real numbers. The sum and product of two such matrices is again of this form, and the sum and product of complex numbers corresponds to the sum and product of such matrices, the product being: ( a − b b a ) ( c − d d c ) = ( a c − b d − a d − b c b c + a d − b d + a c ) displaystyle begin pmatrix a&-b\b&;;aend pmatrix begin pmatrix c&-d\d&;;cend pmatrix = begin pmatrix ac-bd&-ad-bc\bc+ad&;;-bd+acend pmatrix 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. Moreover, the square of the absolute value of a complex number expressed as a matrix is equal to the determinant of that matrix:
z
2 =
a − b b a
= a 2 + b 2 . displaystyle z^ 2 = begin vmatrix a&-b\b&aend vmatrix =a^ 2 +b^ 2 . The conjugate z ¯ displaystyle overline z corresponds to the transpose of the matrix. Though this representation of complex numbers with matrices is the most common, many other representations arise from matrices other than ( 0 − 1 1 0 ) displaystyle bigl ( begin smallmatrix 0&-1\1&0end smallmatrix bigr ) that square to the negative of the identity matrix. See the article
on
Color wheel graph of sin(1/z). Black parts inside refer to numbers having large absolute values. 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. Complex exponential and related functions[edit] 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 d ( z 1 , z 2 ) =
z 1 − z 2
displaystyle operatorname d (z_ 1 ,z_ 2 )=z_ 1 -z_ 2 , is a complete metric space, which notably includes the triangle inequality
z 1 + z 2
≤
z 1
+
z 2
displaystyle z_ 1 +z_ 2 leq z_ 1 +z_ 2 for any two complex numbers z1 and z2. Like in real analysis, this notion of convergence is used to construct a number of elementary functions: the exponential function exp(z), also written ez, is defined as the infinite series exp ( z ) := 1 + z + z 2 2 ⋅ 1 + z 3 3 ⋅ 2 ⋅ 1 + ⋯ = ∑ n = 0 ∞ z n n ! . displaystyle exp(z):=1+z+ frac z^ 2 2cdot 1 + frac z^ 3 3cdot 2cdot 1 +cdots =sum _ n=0 ^ infty frac z^ n n! . The series defining the real trigonometric functions sine and cosine,
as well as the hyperbolic functions sinh and cosh, also carry over to
complex arguments without change. For the other trigonometric and
hyperbolic functions, such as tangent, 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 of
analytic continuation.
exp ( i φ ) = cos ( φ ) + i sin ( φ ) displaystyle exp(ivarphi )=cos(varphi )+isin(varphi ), for any real number φ, in particular exp ( i π ) = − 1 displaystyle exp(ipi )=-1, Unlike in the situation of real numbers, there is an infinitude of complex solutions z of the equation exp ( z ) = w displaystyle exp(z)=w, for any complex number w ≠ 0. It can be shown that any such solution z—called complex logarithm of w—satisfies log ( w ) = ln
w
+ i arg ( w ) , displaystyle log(w)=ln w+iarg(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 2π, log is also multivalued. The principal value of log is often taken by restricting the imaginary part to the interval (−π,π]. Complex exponentiation zω is defined as z ω = exp ( ω log z ) , displaystyle z^ omega =exp(omega log z), and is multi-valued, except when ω displaystyle omega is an integer. For ω = 1 / n, for some natural number n, this recovers the non-uniqueness of nth roots mentioned above. 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 a b c = ( a b ) c . displaystyle a^ bc =(a^ b )^ c . Both 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. Holomorphic functions[edit] A function f : C → C is called holomorphic if it satisfies the Cauchy–Riemann equations. For example, any R-linear map C → C can be written in the form f ( z ) = a z + b z ¯ displaystyle f(z)=az+b overline z with complex coefficients a and b. This map is holomorphic if and only if b = 0. The second summand b z ¯ displaystyle b overline z is real-differentiable, but does not satisfy the Cauchy–Riemann
equations.
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.
Improper integrals[edit]
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.
Fluid dynamics[edit]
In fluid dynamics, complex functions are used to describe potential
flow in two dimensions.
Dynamic equations[edit]
In differential equations, it is common to first find all complex
roots r 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 f(t) = ert. Likewise, in
difference equations, the complex roots r 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 f(t) = rt.
V ( t ) = V 0 e j ω t = V 0 ( cos ω t + j sin ω t ) , displaystyle V(t)=V_ 0 e^ jomega t =V_ 0 left(cos omega t+jsin omega tright), To obtain the measurable quantity, the real part is taken: v ( t ) = R e ( V ) = R e [ V 0 e j ω t ] = V 0 cos ω t . displaystyle v(t)=mathrm Re (V)=mathrm Re left[V_ 0 e^ jomega t right]=V_ 0 cos omega t. The complex-valued signal V ( t ) displaystyle V(t) is called the analytic representation of the real-valued, measurable signal v ( t ) displaystyle v(t) . [29]
Signal analysis[edit]
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 z of the
corresponding z is the amplitude and the argument arg(z) is the phase.
If
x ( t ) = Re X ( t ) displaystyle x(t)=operatorname Re X(t) , and X ( t ) = A e i ω t = a e i ϕ e i ω t = a e i ( ω t + ϕ ) displaystyle X(t)=Ae^ iomega t =ae^ iphi e^ iomega t =ae^ i(omega t+phi ) , 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 utilize 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: cos ( ( ω + α ) t ) + cos ( ( ω − α ) t ) = Re ( e i ( ω + α ) t + e i ( ω − α ) t ) = Re ( ( e i α t + e − i α t ) ⋅ e i ω t ) = Re ( 2 cos ( α t ) ⋅ e i ω t ) = 2 cos ( α t ) ⋅ Re ( e i ω t ) = 2 cos ( α t ) ⋅ cos ( ω t ) . displaystyle begin aligned cos((omega +alpha )t)+cos left((omega -alpha )tright)&=operatorname Re left(e^ i(omega +alpha )t +e^ i(omega -alpha )t right)\&=operatorname Re left((e^ ialpha t +e^ -ialpha t )cdot e^ iomega t right)\&=operatorname Re left(2cos(alpha t)cdot e^ iomega t right)\&=2cos(alpha t)cdot operatorname Re left(e^ iomega t right)\&=2cos(alpha t)cdot cos left(omega tright),.end aligned Quantum mechanics[edit]
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
( x − a ) ( x − b ) ( x − c ) = 0 displaystyle scriptstyle (x-a)(x-b)(x-c)=0 , take its derivative, and equate the (quadratic) derivative to zero.
Construction of a regular pentagon using straightedge and compass. 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 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 are 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[edit]
Main article: Analytic number theory
81 − 144 = 3 i 7 displaystyle scriptstyle sqrt 81-144 =3i sqrt 7 in his calculations, although negative quantities were not conceived
of in
144 − 81 = 3 7 displaystyle scriptstyle sqrt 144-81 =3 sqrt 7 ).[32] 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 (see Niccolò Fontana Tartaglia, Gerolamo Cardano). It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers. As an example, Tartaglia's formula for a cubic equation of the form x 3 = p x + q displaystyle scriptstyle x^ 3 =px+q [33] gives the solution to the equation x3 = x as 1 3 ( ( − 1 ) 1 / 3 + 1 ( − 1 ) 1 / 3 ) . displaystyle frac 1 sqrt 3 left(( sqrt -1 )^ 1/3 + frac 1 ( sqrt -1 )^ 1/3 right). At first glance this looks like nonsense. However formal calculations with complex numbers show that the equation z3 = i has solutions −i, 3 2 + 1 2 i displaystyle scriptstyle frac sqrt 3 2 + scriptstyle frac 1 2 i and − 3 2 + 1 2 i displaystyle scriptstyle frac - sqrt 3 2 + scriptstyle frac 1 2 i . Substituting these in turn for − 1 1 / 3 displaystyle scriptstyle sqrt -1 ^ 1/3 in Tartaglia's cubic formula and simplifying, one gets 0, 1 and −1
as the solutions of x3 − x = 0. Of course this particular equation
can be solved at sight but it does illustrate that when general
formulas are used to solve cubic equations with real roots then, as
later mathematicians showed rigorously, the use of complex numbers is
unavoidable.
[...] 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 − 1 2 = − 1 − 1 = − 1 displaystyle scriptstyle sqrt -1 ^ 2 = sqrt -1 sqrt -1 =-1 seemed to be capriciously inconsistent with the algebraic identity a b = a b displaystyle scriptstyle sqrt a sqrt b = sqrt ab , 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 (and the related identity 1 a = 1 a displaystyle scriptstyle frac 1 sqrt a = sqrt frac 1 a ) in the case when both a and b are negative even bedeviled Euler.
This difficulty eventually led to the convention of using the special
symbol i in place of √−1 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
( cos θ + i sin θ ) n = cos n θ + i sin n θ . displaystyle (cos theta +isin theta )^ n =cos ntheta +isin ntheta ., In 1748
cos θ + i sin θ = e i θ displaystyle cos theta +isin theta =e^ itheta , 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
cos ϕ + i sin ϕ displaystyle scriptstyle cos phi +isin phi the direction factor, and r = a 2 + b 2 displaystyle scriptstyle r= sqrt a^ 2 +b^ 2 the modulus; Cauchy (1828) called cos ϕ + i sin ϕ displaystyle cos phi +isin phi the reduced form (l'expression réduite) and apparently introduced the term argument; Gauss used i for − 1 displaystyle scriptstyle sqrt -1 , introduced the term complex number for a + bi, and called a2 + b2 the norm. The expression direction coefficient, often used for cos ϕ + i sin ϕ displaystyle cos phi +isin phi , is due to Hankel (1867), 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,
C → C , z ↦ w z displaystyle mathbb C rightarrow mathbb C ,zmapsto wz for some fixed complex number w can be represented by a 2 × 2 matrix (once a basis has been chosen). With respect to the basis (1, i), this matrix is ( Re ( w ) − Im ( w ) Im ( w ) Re ( w ) ) displaystyle begin pmatrix operatorname Re (w)&-operatorname Im (w)\operatorname Im (w)&;;operatorname Re (w)end pmatrix i.e., 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 matrix J = ( p q r − p ) , p 2 + q r + 1 = 0 displaystyle J= begin pmatrix p&q\r&-pend pmatrix ,quad p^ 2 +qr+1=0 has the property that its square is the negative of the identity matrix: J2 = −I. Then z = a I + b J : a , b ∈ R displaystyle z=aI+bJ:a,bin R is also isomorphic to the field C, and gives an alternative complex structure on R2. 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]/(x2 − 1) (as opposed to R[x]/(x2 + 1)). In this ring, the equation a2 = 1 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 Qp of p-adic numbers (for any prime number p), which are thereby analogous to R. There are no other nontrivial ways of completing Q than R and Qp, by Ostrowski's theorem. The algebraic closures Q p ¯ displaystyle overline mathbf Q _ p of Qp still carry a norm, but (unlike C) are not complete with respect to it. The completion C p displaystyle mathbf C _ p of Q p ¯ displaystyle overline mathbf Q _ p turns out to be algebraically closed. This field is called p-adic complex numbers by analogy. The fields R and Qp and their finite field extensions, including C, are local fields. See also[edit] Wikimedia Commons has media related to Complex numbers. Algebraic surface
Circular motion using complex numbers
Complex-base system
Complex geometry
Complex square root
Eisenstein integer
Euler's identity
Gaussian integer
Notes[edit] ^ An extensive account of the history, from initial skepticism to
ultimate acceptance, can be found in Nicolas Bourbaki, "1. Foundations
of mathematics; logic; set theory", Elements of the history of
mathematics, Springer, pp. 18–24 .
^ Penrose, Roger (2016). The Road to Reality: A Complete Guide to the
Laws of the Universe (reprinted ed.). Random House. pp. 72–73.
ISBN 978-1-4464-1820-8. Extract of page 73: "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."
^ Nicolas Bourbaki. "VIII.1". General topology. Springer-Verlag.
^ Burton (1995, p. 294)
^ Sheldon Axler (2010). College algebra. Wiley. p. 262.
^ Nicolas Bourbaki. "VIII.1". General topology. Springer-Verlag.
^ Complex Variables (2nd Edition), M.R. Spiegel, S. Lipschutz, J.J.
Schiller, D. Spellman, Schaum's Outline Series, Mc Graw Hill (USA),
ISBN 978-0-07-161569-3
^ Aufmann, Richard N.; Barker, Vernon C.; Nation, Richard D. (2007),
"Chapter P", College Algebra and Trigonometry (6 ed.), Cengage
Learning, p. 66, ISBN 0-618-82515-0
^
( u 3 + v 3 ) 3 = 3 u v 3 ( u 3 + v 3 ) + u + v displaystyle scriptstyle left( sqrt[ 3 ] u + sqrt[ 3 ] v right)^ 3 =3 sqrt[ 3 ] uv left( sqrt[ 3 ] u + sqrt[ 3 ] v right)+u+v With x = u 3 + v 3 displaystyle scriptstyle x= sqrt[ 3 ] u + sqrt[ 3 ] v , p = 3 u v 3 displaystyle scriptstyle p=3 sqrt[ 3 ] uv , q = u + v displaystyle scriptstyle q=u+v , u and v can be expressed in terms of p and q as u = q / 2 + ( q / 2 ) 2 − ( p / 3 ) 3 displaystyle scriptstyle u=q/2+ sqrt (q/2)^ 2 -(p/3)^ 3 and v = q / 2 − ( q / 2 ) 2 − ( p / 3 ) 3 displaystyle scriptstyle v=q/2- sqrt (q/2)^ 2 -(p/3)^ 3 , respectively. Therefore, x = q / 2 + ( q / 2 ) 2 − ( p / 3 ) 3 3 + q / 2 − ( q / 2 ) 2 − ( p / 3 ) 3 3 displaystyle scriptstyle x= sqrt[ 3 ] q/2+ sqrt (q/2)^ 2 -(p/3)^ 3 + sqrt[ 3 ] q/2- sqrt (q/2)^ 2 -(p/3)^ 3 . When ( q / 2 ) 2 − ( p / 3 ) 3 displaystyle scriptstyle (q/2)^ 2 -(p/3)^ 3 is negative (casus irreducibilis), the second cube root should be
regarded as the complex conjugate of the first one.
^ Descartes, René (1954) [1637], La Géométrie The Geometry of
References[edit] Mathematical references[edit] Ahlfors, Lars (1979),
Historical references[edit] Burton, David M. (1995), The History of Mathematics (3rd ed.), New York: McGraw-Hill, ISBN 978-0-07-009465-9 Katz, Victor J. (2004), A History of Mathematics, Brief Version, Addison-Wesley, ISBN 978-0-321-16193-2 Nahin, Paul J. (1998), An Imaginary Tale: The Story of − 1 displaystyle scriptstyle sqrt -1 , Princeton University Press, ISBN 0-691-02795-1 A gentle introduction to the history of complex numbers and the beginnings of complex analysis. H. D. Ebbinghaus; H. Hermes; F. Hirzebruch; M. Koecher; K. Mainzer; J. Neukirch; A. Prestel; R. Remmert (1991), Numbers (hardcover ed.), Springer, ISBN 0-387-97497-0 An advanced perspective on the historical development of the concept of number. Further reading[edit] The Road to Reality: A Complete Guide to the Laws of the Universe, by Roger Penrose; Alfred A. Knopf, 2005; ISBN 0-679-45443-8. Chapters 4–7 in particular deal extensively (and enthusiastically) with complex numbers. Unknown Quantity: A Real and Imaginary History of Algebra, by John Derbyshire; Joseph Henry Press; ISBN 0-309-09657-X (hardcover 2006). A very readable history with emphasis on solving polynomial equations and the structures of modern algebra. Visual Complex Analysis, by Tristan Needham; Clarendon Press; ISBN 0-19-853447-7 (hardcover, 1997). History of complex numbers and complex analysis with compelling and useful visual interpretations. Conway, John B., Functions of One Complex Variable I (Graduate Texts in Mathematics), Springer; 2 edition (12 September 2005). ISBN 0-387-90328-3. External links[edit] Wikiversity has learning resources about Complex Numbers Wikibooks has a book on the topic of: Calculus/Complex numbers
Hazewinkel, Michiel, ed. (2001) [1994], "Complex number", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 Introduction to Complex Numbers from Khan Academy Imaginary Numbers on In Our Time at the BBC. Euler's Investigations on the Roots of Equations at Convergence. MAA Mathematical Sciences Digital Library. John and Betty's Journey Through Complex Numbers Dimensions: a math film. Chapter 5 presents an introduction to complex arithmetic and stereographic projection. Chapter 6 discusses transformations of the complex plane, Julia sets, and the Mandelbrot set. v t e Complex numbers Complex conjugate Complex plane Imaginary number Real number Unit complex number v t e
Countable sets Natural numbers ( N displaystyle mathbb N ) Integers ( Z displaystyle mathbb Z ) Rational numbers ( Q displaystyle mathbb Q ) Constructible numbers Algebraic numbers ( A displaystyle mathbb A 𝔸) Periods Computable numbers Definable real numbers Arithmetical numbers Gaussian integers Division algebras Real numbers ( R displaystyle mathbb R ) Complex numbers ( C displaystyle mathbb C ) Quaternions ( H displaystyle mathbb H ) Octonions ( O displaystyle mathbb O 𝕆) Split Composition algebras over R displaystyle mathbb R : Split-complex numbers Split-quaternions Split-octonions over C displaystyle mathbb C : Bicomplex numbers Biquaternions Bioctonions Other hypercomplex Dual numbers Dual quaternions Hyperbolic quaternions Sedenions ( S displaystyle mathbb S 𝕊) Split-biquaternions Multicomplex numbers Other types Cardinal numbers Irrational numbers Fuzzy numbers Hyperreal numbers Levi-Civita field Surreal numbers Transcendental numbers Ordinal numbers p-adic numbers Supernatural numbers Superreal numbers Classification List Authority control GND: 41286 |