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 the imaginary unit, which satisfies the equation i^{2} = −1.^{[1]} In this expression, a is the real part and b is the imaginary part of the complex number. Complex numbers extend the concept of the onedimensional number line to the twodimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point (a, b) in the complex plane. A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way the complex numbers contain the ordinary real numbers while extending them in order to solve problems that cannot be solved with real numbers alone.
As well as their use within mathematics, complex numbers have practical applications in many fields, including physics, chemistry, biology, economics, electrical engineering, and statistics. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers. He called them "fictitious" during his attempts to find solutions to cubic equations in the 16th century.^{[2]}
Overview
Complex numbers allow for solutions to certain equations that have no real solutions: the equation
 $(x+1)^2\; =\; 9\; \backslash ,$
has no real solution, since the square of a real number is either 0 or positive. Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the imaginary unit i where i^{2} = −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 i^{2} = −1:
 $((1+3i)+1)^2\; =\; (3i)^2\; =\; (3^2)(i^2)\; =\; 9(1)\; =\; 9$
 $((13i)+1)^2\; =\; (3i)^2\; =\; (3)^2(i^2)\; =\; 9(1)\; =\; 9$
In fact not only quadratic equations, but all polynomial equations with real or complex coefficients in a single variable can be solved using complex numbers.
Definition
A complex number is a number that can be expressed in the form z = a + bi, where Template:Mvar and Template:Mvar are real numbers and i is the imaginary unit, satisfying i^{2} = −1. For example, −3.5 + 2i is a complex number.
The real number Template:Mvar in the complex number z = a + bi is called the real part of Template:Mvar, and the real number Template:Mvar is often called the imaginary part. By this convention the imaginary part does not include the imaginary unit: hence Template:Mvar, not bi, is the imaginary part.^{[3]}^{[4]} The real part Template:Mvar is denoted by Re(z) or ℜ(z), and the imaginary part Template:Mvar is denoted by Im(z) or ℑ(z). For example,
 $\backslash begin\{align\}$
\operatorname{Re}(3.5 + 2i) &= 3.5 \\
\operatorname{Im}(3.5 + 2i) &= 2
\end{align}
Any complex number z, may be formally defined in terms of its real and imaginary parts as follows (this is sometimes known as the "Cartesian" form):
 $z\; =\; \backslash operatorname\{Re\}(z)\; +\; \backslash operatorname\{Im\}(z)\; \backslash cdot\; i$
A real number Template:Mvar can be regarded as a complex number a + 0i with an imaginary part of zero. A pure imaginary number bi is a complex number 0 + bi whose real part is zero. It is common to write Template:Mvar 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.
The set of all complex numbers is denoted by ℂ, $\backslash mathbf\{C\}$ or $\backslash mathbb\{C\}$.
Notation
Some authors write a + ib instead of a + bi. In some disciplines, in particular electromagnetism and electrical engineering, j is used instead of i,^{[5]} since Template:Mvar is frequently used for electric current. In these cases complex numbers are written as a + bj or a + jb.
Complex plane
Main article:
Complex plane
A complex number can be viewed as a point or position vector in a twodimensional Cartesian coordinate system called the complex plane or Argand diagram (see Pedoe 1988 and Solomentsev 2001), named after JeanRobert Argand. The numbers are conventionally plotted using the real part as the horizontal component, and imaginary part as vertical (see Figure 1). These two values used to identify a given complex number are therefore called its Cartesian, rectangular, or algebraic form.
A position vector may also be defined in terms of its magnitude and direction relative to the origin. These are emphasized in a complex number's polar form. Using the polar form of the complex number in calculations may lead to a more intuitive interpretation of mathematical results. Notably, the operations of addition and multiplication take on a very natural geometric character when complex numbers are viewed as position vectors: addition corresponds to vector addition while multiplication corresponds to multiplying their magnitudes and adding their arguments (i.e. the angles they make with the x axis). Viewed in this way the multiplication of a complex number by i corresponds to rotating the position vector counterclockwise by a quarter turn (90°) about the origin: $(a+bi)i\; =\; ai+bi^2\; =\; b+ai$.
History in brief
 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 socalled casus irreducibilis). This conundrum led Italian mathematician Gerolamo Cardano to conceive of complex numbers in around 1545, though his understanding was rudimentary.
Work on the problem of general polynomials ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher. Complex numbers thus form an algebraically closed field, where any polynomial equation has a root.
Many mathematicians contributed to the full development of complex numbers. The rules for addition, subtraction, multiplication, and division of complex numbers were developed by the Italian mathematician Rafael Bombelli.^{[6]} A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton, who extended this abstraction to the theory of quaternions.
Relations
Equality
Two complex numbers are equal if and only if both their real and imaginary parts are equal. In other words:
 $z\_\{1\}\; =\; z\_\{2\}\; \backslash ,\; \backslash ,\; \backslash leftrightarrow\; \backslash ,\; \backslash ,\; (\; \backslash operatorname\{Re\}(z\_\{1\})\; =\; \backslash operatorname\{Re\}(z\_\{2\})\; \backslash ,\; \backslash and\; \backslash ,\; \backslash operatorname\{Im\}\; (z\_\{1\})\; =\; \backslash operatorname\{Im\}\; (z\_\{2\}))$
Ordering
Because complex numbers are naturally thought of as existing on a twodimensional plane, there is no natural linear ordering on the set of complex numbers.^{[7]}
Elementary operations
Conjugation
The complex conjugate of the complex number z = x + yi is defined to be x − yi. It is denoted $\backslash bar\{z\}$ or z*.
Formally, for any complex number z:
 $\backslash bar\{z\}\; =\; \backslash operatorname\{Re\}(z)\; \; \backslash operatorname\{Im\}(z)\; \backslash cdot\; i$
Geometrically, $\backslash bar\{z\}$ is the "reflection" of Template:Mvar about the real axis. In particular, conjugating twice gives the original complex number: $\backslash bar\{\backslash bar\{z\}\}=z$.
The real and imaginary parts of a complex number Template:Mvar can be extracted using the conjugate:
 $\backslash operatorname\{Re\}\backslash ,(z)\; =\; \backslash tfrac\{1\}\{2\}(z+\backslash bar\{z\}),\; \backslash ,$
 $\backslash operatorname\{Im\}\backslash ,(z)\; =\; \backslash tfrac\{1\}\{2i\}(z\backslash bar\{z\}).\; \backslash ,$
Moreover, a complex number is real if and only if it equals its conjugate.
Conjugation distributes over the standard arithmetic operations:
 $\backslash overline\{z+w\}\; =\; \backslash bar\{z\}\; +\; \backslash bar\{w\},\; \backslash ,$
 $\backslash overline\{zw\}\; =\; \backslash bar\{z\}\; \; \backslash bar\{w\},\; \backslash ,$
 $\backslash overline\{z\; w\}\; =\; \backslash bar\{z\}\; \backslash bar\{w\},\; \backslash ,$
 $\backslash overline\{(z/w)\}\; =\; \backslash bar\{z\}/\backslash bar\{w\}.\; \backslash ,$
The reciprocal of a nonzero complex number z = x + yi is given by
 $\backslash frac\{1\}\{z\}=\backslash frac\{\backslash bar\{z\}\}\{z\; \backslash bar\{z\}\}=\backslash frac\{\backslash bar\{z\}\}\{x^2+y^2\}.$
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.
Addition and subtraction
Complex numbers are added by adding the real and imaginary parts of the summands. That is to say:
 $(a+bi)\; +\; (c+di)\; =\; (a+c)\; +\; (b+d)i.\backslash $
Similarly, subtraction is defined by
 $(a+bi)\; \; (c+di)\; =\; (ac)\; +\; (bd)i.\backslash $
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
The multiplication of two complex numbers is defined by the following formula:
 $(a+bi)\; (c+di)\; =\; (acbd)\; +\; (bc+ad)i.\backslash $
In particular, the square of the imaginary unit is −1:
 $i^2\; =\; i\; \backslash times\; i\; =\; 1.\backslash $
The preceding definition of multiplication of general complex numbers follows naturally from this fundamental property of the imaginary unit. Indeed, if i is treated as a number so that di means Template:Mvar times i, the above multiplication rule is identical to the usual rule for multiplying two sums of two terms.
 $(a+bi)\; (c+di)\; =\; ac\; +\; bci\; +\; adi\; +\; bidi\; \backslash $ (distributive law)
 $=\; ac\; +\; bidi\; +\; bci\; +\; adi\; \backslash $ (commutative law of addition—the order of the summands can be changed)
 $=\; ac\; +\; bdi^2\; +\; (bc+ad)i\; \backslash $ (commutative and distributive laws)
 $=\; (acbd)\; +\; (bc\; +\; ad)i\; \backslash $ (fundamental property of the imaginary unit).
The division of two complex numbers is defined in terms of complex multiplication, which is described above, and real division. Where at least one of Template:Mvar and Template:Mvar is nonzero:
 $\backslash ,\backslash frac\{a\; +\; bi\}\{c\; +\; di\}\; =\; \backslash left(\{ac\; +\; bd\; \backslash over\; c^2\; +\; d^2\}\backslash right)\; +\; \backslash left(\; \{bc\; \; ad\; \backslash over\; c^2\; +\; d^2\}\; \backslash right)i.$
Division can be defined in this way because of the following observation:
 $\backslash ,\backslash frac\{a\; +\; bi\}\{c\; +\; di\}\; =\; \backslash frac\{\backslash left(a\; +\; bi\backslash right)\; \backslash cdot\; \backslash left(c\; \; di\backslash right)\}\{\backslash left\; (c\; +\; di\backslash right)\; \backslash cdot\; \backslash left\; (c\; \; di\backslash right)\}\; =\; \backslash left(\{ac\; +\; bd\; \backslash over\; c^2\; +\; d^2\}\backslash right)\; +\; \backslash left(\; \{bc\; \; ad\; \backslash over\; c^2\; +\; d^2\}\; \backslash right)i.$
As shown earlier, c − di is the complex conjugate of the denominator c + di. The real part Template:Mvar and the imaginary part Template:Mvar of the denominator must not both be zero for division to be defined.
Square root
The square roots of a + bi (with b ≠ 0) are $\backslash pm\; (\backslash gamma\; +\; \backslash delta\; i)$, where
 $\backslash gamma\; =\; \backslash sqrt\{\backslash frac\{a\; +\; \backslash sqrt\{a^2\; +\; b^2\}\}\{2\}\}$
and
 $\backslash delta\; =\; \backslash sgn\; (b)\; \backslash sqrt\{\backslash frac\{a\; +\; \backslash sqrt\{a^2\; +\; b^2\}\}\{2\}\},$
where sgn is the signum function. This can be seen by squaring $\backslash pm\; (\backslash gamma\; +\; \backslash delta\; i)$ to obtain a + bi.^{[8]}^{[9]} Here $\backslash sqrt\{a^2\; +\; b^2\}$ is called the modulus of a + bi, and the square root with nonnegative real part is called the principal square root.
Polar form
Absolute value and argument
An alternative way of defining a point P in the complex plane, other than using the x and ycoordinates, is to use the distance of the point from O, the point whose coordinates are (0, 0) (the origin), together with the angle subtended between the positive real axis and the line segment OP in a counterclockwise direction. This idea leads to the polar form of complex numbers.
The absolute value (or modulus or magnitude) of a complex number z = x + yi is
 $\backslash textstyle\; r=z=\backslash sqrt\{x^2+y^2\}.\backslash ,$
If Template:Mvar is a real number (i.e., y = 0), then r =  x . In general, by Pythagoras' theorem, Template:Mvar is the distance of the point P representing the complex number Template:Mvar to the origin.
The argument or phase of Template:Mvar is the angle of the radius OP with the positive real axis, and is written as $\backslash arg(z)$. As with the modulus, the argument can be found from the rectangular form $x+yi$:^{[10]}
 $\backslash varphi\; =\; \backslash arg(z)\; =$
\begin{cases}
\arctan(\frac{y}{x}) & \mbox{if } x > 0 \\
\arctan(\frac{y}{x}) + \pi & \mbox{if } x < 0 \mbox{ and } y \ge 0\\
\arctan(\frac{y}{x})  \pi & \mbox{if } x < 0 \mbox{ and } y < 0\\
\frac{\pi}{2} & \mbox{if } x = 0 \mbox{ and } y > 0\\
\frac{\pi}{2} & \mbox{if } x = 0 \mbox{ and } y < 0\\
\mbox{indeterminate } & \mbox{if } x = 0 \mbox{ and } y = 0.
\end{cases}
The value of Template:Mvar must always be expressed in radians. It can increase by any integer multiple of 2π and still give the same angle. Hence, the arg function is sometimes considered as multivalued. Normally, as given above, the principal value in the interval Template:Openclosed is chosen. Values in the range Template:Closedopen are obtained by adding 2π if the value is negative. The polar angle for the complex number 0 is indeterminate, but arbitrary choice of the angle 0 is common.
The value of Template:Mvar equals the result of atan2: $\backslash varphi\; =\; \backslash mbox\{atan2\}(\backslash mbox\{imaginary\},\; \backslash mbox\{real\})$.
Together, Template:Mvar and Template:Mvar 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 coordinates from the polar form is done by the formula called trigonometric form
 $z\; =\; r(\backslash cos\; \backslash varphi\; +\; i\backslash sin\; \backslash varphi\; ).\backslash ,$
Using Euler's formula this can be written as
 $z\; =\; r\; e^\{i\; \backslash varphi\}.\backslash ,$
Using the cis function, this is sometimes abbreviated to
 $z\; =\; r\; \backslash \; \backslash operatorname\{cis\}\; \backslash \; \backslash varphi.\; \backslash ,$
In angle notation, often used in electronics to represent a phasor with amplitude Template:Mvar and phase Template:Mvar, it is written as^{[11]}
 $z\; =\; r\; \backslash ang\; \backslash varphi\; .\; \backslash ,$
Multiplication and division in polar form
Formulas for multiplication, division and exponentiation are simpler in polar form than the corresponding formulas in Cartesian coordinates. Given two complex numbers z_{1} = r_{1}(cos φ_{1} + i sin φ_{1}) and z_{2} =r_{2}(cos φ_{2} + i sin φ_{2}), because of the wellknown trigonometric identities
 $\backslash cos(a)\backslash cos(b)\; \; \backslash sin(a)\backslash sin(b)\; =\; \backslash cos(a\; +\; b)$
 $\backslash cos(a)\backslash sin(b)\; +\; \backslash cos(b)\backslash sin(a)\; =\; \backslash sin(a\; +\; b)$
we may derive
 $z\_1\; z\_2\; =\; r\_1\; r\_2\; (\backslash cos(\backslash varphi\_1\; +\; \backslash varphi\_2)\; +\; i\; \backslash sin(\backslash varphi\_1\; +\; \backslash varphi\_2)).\backslash ,$
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 quarterturn counterclockwise, which gives back i^{2} = −1. The picture at the right illustrates the multiplication of
 $(2+i)(3+i)=5+5i.\; \backslash ,$
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
 $\backslash frac\{\backslash pi\}\{4\}\; =\; \backslash arctan\backslash frac\{1\}\{2\}\; +\; \backslash arctan\backslash frac\{1\}\{3\}$
holds. As the arctan function can be approximated highly efficiently, formulas like this—known as Machinlike formulas—are used for highprecision approximations of π.
Similarly, division is given by
 $\backslash frac\{z\_1\}\{\; z\_2\}\; =\; \backslash frac\{r\_1\}\{\; r\_2\}\; \backslash left(\backslash cos(\backslash varphi\_1\; \; \backslash varphi\_2)\; +\; i\; \backslash sin(\backslash varphi\_1\; \; \backslash varphi\_2)\backslash right).$
Exponentiation
Euler's formula
Euler's formula states that, for any real number x,
 $e^\{ix\}\; =\; \backslash cos\; x\; +\; i\backslash sin\; x\; \backslash $
where e is the base of the natural logarithm. This can be proved by observing that
 $\backslash begin\{align\}$
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{align}
and so on, and by considering the Taylor series expansions of e^{ix}, cos(x) and sin(x):
 $\backslash begin\{align\}$
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) + i\left( x  \frac{x^3}{3!} + \frac{x^5}{5!}  \frac{x^7}{7!} + \cdots \right) \\[8pt]
&{}= \cos x + i\sin x \ .
\end{align}
The rearrangement of terms is justified because each series is absolutely convergent.
Natural logarithm
Euler's formula allows us to observe that, for any complex number
 $z\; =\; r(\backslash cos\; \backslash varphi\; +\; i\backslash sin\; \backslash varphi\; ).\backslash ,$
where r is a nonnegative real number, one possible value for z's natural logarithm is
 $\backslash ln(r)\; +\; \backslash varphi\; i$
Because cos and sin are periodic functions, the natural logarithm may be considered a multivalued function, with:
 $\backslash ln(z)\; =\; \backslash left\backslash \{\; \backslash ln(r)\; +\; (\backslash varphi\; +\; 2\backslash pi\; k)i\; \backslash ;\backslash ;\; k\; \backslash in\; \backslash mathbb\{Z\}\backslash right\backslash \}$
Integer and fractional exponents
We may use the identity
 $\backslash ln(a^\{b\})\; =\; b\; \backslash ln(a)$
to define complex exponentiation, which is likewise multivalued:
 $\backslash ln((r(\backslash cos\; \backslash varphi\; +\; i\backslash sin\; \backslash varphi\; ))^\{n\})$
 $=\; n\; \backslash ln(r(\backslash cos\; \backslash varphi\; +\; i\backslash sin\; \backslash varphi))$
 $=\; \backslash \{\; n\; (\backslash ln(r)\; +\; (\backslash varphi\; +\; k2\backslash pi)\; i)\; \; k\; \backslash in\; \backslash mathbb\{Z\}\; \backslash \}$
 $=\; \backslash \{\; n\; \backslash ln(r)\; +\; n\; \backslash varphi\; i\; +\; nk2\backslash pi\; i\; \; k\; \backslash in\; \backslash mathbb\{Z\}\; \backslash \}$
Where n is an integer, this simplifies to de Moivre's formula:
 $(r(\backslash cos\; \backslash varphi\; +\; i\backslash sin\; \backslash varphi\; ))^\{n\}\; =\; r^n\backslash ,(\backslash cos\; n\backslash varphi\; +\; i\; \backslash sin\; n\; \backslash varphi).$
The Template:Mvarth roots of Template:Mvar are given by
 $\backslash sqrt[n]\{z\}\; =\; \backslash sqrt[n]r\; \backslash left(\; \backslash cos\; \backslash left(\backslash frac\{\backslash varphi+2k\backslash pi\}\{n\}\backslash right)\; +\; i\; \backslash sin\; \backslash left(\backslash frac\{\backslash varphi+2k\backslash pi\}\{n\}\backslash right)\backslash right)$
for any integer k satisfying 0 ≤ k ≤ n − 1. Here ^{n}√r is the usual (positive) Template:Mvarth root of the positive real number Template:Mvar. While the Template:Mvarth root of a positive real number Template:Mvar is chosen to be the positive real number Template:Mvar satisfying c^{n} = x there is no natural way of distinguishing one particular complex Template:Mvarth root of a complex number. Therefore, the Template:Mvarth root of Template:Mvar is considered as a multivalued function (in Template:Mvar), as opposed to a usual function Template:Mvar, for which f(z) is a uniquely defined number. Formulas such as
 $\backslash sqrt[n]\{z^n\}\; =\; z$
(which holds for positive real numbers), do in general not hold for complex numbers.
Properties
Field structure
The set C of complex numbers is a field. 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 Template:Mvar, 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 z_{1} and z_{2}:
 $z\_1+\; z\_2\; =\; z\_2\; +\; z\_1,$
 $z\_1\; z\_2\; =\; z\_2\; z\_1.$
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 z_{1} < z_{2} that is compatible with the addition and multiplication. In fact, in any ordered field, the square of any element is necessarily positive, so i^{2} = −1 precludes the existence of an ordering on C.
When the underlying field for a mathematical topic or construct is the field of complex numbers, the thing'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
Given any complex numbers (called coefficients) a_{0}, …, a_{n}, the equation
 $a\_n\; z^n\; +\; \backslash dotsb\; +\; a\_1\; z\; +\; a\_0\; =\; 0$
has at least one complex solution z, provided that at least one of the higher coefficients a_{1}, …, a_{n} is nonzero. This is the statement of the fundamental theorem of algebra. 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 x^{2} − 2 does not have a rational root, since √2 is not a rational number) nor the real numbers R (the polynomial x^{2} + a does not have a real root for a > 0, since the square of Template:Mvar is positive for any real number Template:Mvar).
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 Galois theory and the fact that any real polynomial of odd degree has at least one root.
Because of this fact, theorems that hold for any algebraically closed field, apply to C. For example, any nonempty complex square matrix has at least one (complex) eigenvalue.
Algebraic characterization
The field C has the following three properties: first, it has characteristic 0. This means that 1 + 1 + ⋯ + 1 ≠ 0 for any number of summands (all of which equal one). Second, its transcendence degree over Q, the prime field of C, is the cardinality of the continuum. Third, it is algebraically closed (see above). It can be shown that any field having these properties is isomorphic (as a field) to C. For example, the algebraic closure of Q_{p} also satisfies these three properties, so these two fields are isomorphic. Also, C is isomorphic to the field of complex Puiseux series. However, specifying an isomorphism requires the axiom of choice. Another consequence of this algebraic characterization is that C contains many proper subfields that are isomorphic to C.
Characterization as a topological field
The preceding characterization of C describes the algebraic aspects of C, only. That is to say, the properties of nearness and continuity, which matter in areas such as analysis and topology, are not dealt with. The following description of C as a topological field (that is, a field that is equipped with a topology, which allows the notion of convergence) does take into account the topological properties. C contains a subset P (namely the set of positive real numbers) of nonzero elements satisfying the following three conditions:
 P is closed under addition, multiplication and taking inverses.
 If Template:Mvar and Template:Mvar are distinct elements of P, then either x − y or y − x is in P.
 If Template:Mvar is any nonempty subset of P, then S + P = x + P for some Template:Mvar 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 Template:Mvar in C.
Any field Template:Mvar 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 Template:Mvar ranges over the field and Template:Mvar ranges over P. With this topology Template:Mvar 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.
Formal construction
Formal development
Above, complex numbers have been defined by introducing i, the imaginary unit, as a symbol. More rigorously, the set C of complex numbers can be defined as the set R^{2} of ordered pairs (a, b) of real numbers. In this notation, the above formulas for addition and multiplication read
 $(a,\; b)\; +\; (c,\; d)\; =\; (a\; +\; c,\; b\; +\; d)\backslash ,$
 $(a,\; b)\; \backslash cdot\; (c,\; d)\; =\; (ac\; \; bd,\; bc\; +\; ad).\backslash ,$
It is then just a matter of notation to express (a, b) as a + bi.
Though this lowlevel 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 an addition, subtraction, multiplication and division operations that behave as is familiar from, say, rational numbers. For example, the distributive law
 $(x+y)\; z\; =\; xz\; +\; yz$
must hold for any three elements Template:Mvar, Template:Mvar and Template:Mvar 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
 $a\_nX^n+\backslash dotsb+a\_1X+a\_0$,
where the a_{0}, …, a_{n} 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 polynomial ring.
The quotient ring R[X]/(X ^{2} + 1) can be shown to be a field.
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. Moreover, the above formulas for addition etc. correspond to the ones yielded by this abstract algebraic approach – the two definitions of the field C are said to be isomorphic (as fields). Together with the abovementioned fact that C is algebraically closed, this also shows that C is an algebraic closure of R.
Matrix representation of complex numbers
Complex numbers a + ib can also be represented by 2 × 2 matrices that have the following form:
 $$
\begin{pmatrix}
a & b \\
b & \;\; a
\end{pmatrix}.
Here the entries Template:Mvar and Template:Mvar 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 geometric description of the multiplication of complex numbers can also be phrased 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\; =$
\begin{vmatrix}
a & b \\
b & a
\end{vmatrix}
= (a^2)  ((b)(b)) = a^2 + b^2.
The conjugate $\backslash 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 $\backslash begin\{pmatrix\}0\; \&\; 1\; \backslash \backslash 1\; \&\; 0\; \backslash end\{pmatrix\}$ that square to the negative of the identity matrix. See the article on 2 × 2 real matrices for other representations of complex numbers.
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 twodimensional graphs, complex functions have fourdimensional graphs and may usefully be illustrated by colorcoding a threedimensional graph to suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane.
Complex exponential and related functions
The notions of convergent series and continuous functions in (real) analysis have natural analogs in complex analysis. A sequence of complex numbers is said to converge if and only if its real and imaginary parts do. This is equivalent to the (ε, δ)definition of limits, where the absolute value of real numbers is replaced by the one of complex numbers. From a more abstract point of view, C, endowed with the metric
 $\backslash operatorname\{d\}(z\_1,\; z\_2)\; =\; z\_1\; \; z\_2\; \backslash ,$
is a complete metric space, which notably includes the triangle inequality
 $z\_1\; +\; z\_2\; \backslash le\; z\_1\; +\; z\_2$
for any two complex numbers z_{1} and z_{2}.
Like in real analysis, this notion of convergence is used to construct a number of elementary functions: the exponential function exp(z), also written e^{z}, is defined as the infinite series
 $\backslash exp(z):=\; 1+z+\backslash frac\{z^2\}\{2\backslash cdot\; 1\}+\backslash frac\{z^3\}\{3\backslash cdot\; 2\backslash cdot\; 1\}+\backslash cdots\; =\; \backslash sum\_\{n=0\}^\{\backslash infty\}\; \backslash frac\{z^n\}\{n!\}.\; \backslash ,$
and the series defining the real trigonometric functions sine and cosine, as well as hyperbolic functions such as sinh also carry over to complex arguments without change. Euler's identity states:
 $\backslash exp(i\backslash varphi)\; =\; \backslash cos(\backslash varphi)\; +\; i\backslash sin(\backslash varphi)\; \backslash ,$
for any real number φ, in particular
 $\backslash exp(i\; \backslash pi)\; =\; 1\; \backslash ,$
Unlike in the situation of real numbers, there is an infinitude of complex solutions Template:Mvar of the equation
 $\backslash exp(z)\; =\; w\; \backslash ,$
for any complex number w ≠ 0. It can be shown that any such solution Template:Mvar—called complex logarithm of Template:Mvar—satisfies
 $\backslash log(x+iy)=\backslash lnw\; +\; i\backslash arg(w),\; \backslash ,$
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 Template:Openclosed.
Complex exponentiation z^{ω} is defined as
 $z^\backslash omega\; =\; \backslash exp(\backslash omega\; \backslash log\; z).\; \backslash ,$
Consequently, they are in general multivalued. For ω = 1 / n, for some natural number Template:Mvar, this recovers the nonuniqueness of Template:Mvarth roots mentioned above.
Complex numbers, unlike real numbers, do not in general satisfy the unmodified power and logarithm identities, particularly when naïvely treated as singlevalued functions; see failure of power and logarithm identities. For example they do not satisfy
 $\backslash ,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
A function f : C → C is called holomorphic if it satisfies the Cauchy–Riemann equations. For example, any Rlinear map C → C can be written in the form
 $f(z)=az+b\backslash overline\{z\}$
with complex coefficients Template:Mvar and Template:Mvar. This map is holomorphic if and only if b = 0. The second summand $b\; \backslash overline\; z$ is realdifferentiable, but does not satisfy the Cauchy–Riemann equations.
Complex analysis shows some features not apparent in real analysis. For example, any two holomorphic functions Template:Mvar and Template:Mvar that agree on an arbitrarily small open subset of C necessarily agree everywhere. Meromorphic functions, functions that can locally be written as f(z)/(z − z_{0})^{n} with a holomorphic function Template:Mvar, still share some of the features of holomorphic functions. Other functions have essential singularities, such as sin(1/z) at z = 0.
Applications
Complex numbers have essential concrete applications in a variety of scientific and related areas such as signal processing, control theory, electromagnetism, fluid dynamics, quantum mechanics, cartography, and vibration analysis. Some applications of complex numbers are:
Control theory
In control theory, systems are often transformed from the time domain to the frequency domain using the Laplace transform. The system's poles and zeros are then analyzed in the complex plane. The root locus, Nyquist plot, and Nichols plot techniques all make use of the complex plane.
In the root locus method, it is especially important whether the poles and zeros are in the left or right half planes, i.e. have real part greater than or less than zero. If a linear, timeinvariant (LTI) system has poles that are
If a system has zeros in the right half plane, it is a nonminimum phase system.
Improper integrals
In applied fields, complex numbers are often used to compute certain realvalued improper integrals, by means of complexvalued functions. Several methods exist to do this; see methods of contour integration.
Fluid dynamics
In fluid dynamics, complex functions are used to describe potential flow in two dimensions.
Dynamic equations
In differential equations, it is common to first find all complex roots Template:Mvar 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) = e^{rt}. Likewise, in difference equations, the complex roots Template:Mvar 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) = r ^{t}.
Electromagnetism and electrical engineering
In electrical engineering, the Fourier transform is used to analyze varying voltages and currents. The treatment of resistors, capacitors, and inductors can then be unified by introducing imaginary, frequencydependent resistances for the latter two and combining all three in a single complex number called the impedance. This approach is called phasor calculus.
In electrical engineering, the imaginary unit is denoted by j, to avoid confusion with Template:Mvar which is generally in use to denote electric current.
Since the voltage in an AC circuit is oscillating, it can be represented as
 $V\; =\; V\_0\; e^\{j\; \backslash omega\; t\}\; =\; V\_0\; \backslash left\; (\backslash cos\; \backslash omega\; t\; +\; j\; \backslash sin\backslash omega\; t\; \backslash right\; ),$
To obtain the measurable quantity, the real part is taken:
 $\backslash mathrm\{Re\}(V)\; =\; \backslash mathrm\{Re\}\backslash left\; [\; V\_0\; e^\{j\; \backslash omega\; t\}\; \backslash right\; ]\; =\; V\_0\; \backslash cos\; \backslash omega\; t.$
See for example.^{[12]}
Signal analysis
Complex numbers are used in signal analysis and other fields for a convenient description for periodically varying signals. For given real functions representing actual physical quantities, often in terms of sines and cosines, corresponding complex functions are considered of which the real parts are the original quantities. For a sine wave of a given frequency, the absolute value  z  of the corresponding Template:Mvar is the amplitude and the argument arg(z) the phase.
If Fourier analysis is employed to write a given realvalued signal as a sum of periodic functions, these periodic functions are often written as complex valued functions of the form
 $f\; (\; t\; )\; =\; z\; e^\{i\backslash omega\; t\}\; \backslash ,$
where ω represents the angular frequency and the complex number z 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:
 $$
\begin{align}
\cos((\omega+\alpha)t)+\cos\left((\omega\alpha)t\right) & = \operatorname{Re}\left(e^{i(\omega+\alpha)t} + e^{i(\omega\alpha)t}\right) \\
& = \operatorname{Re}\left((e^{i\alpha t} + e^{i\alpha t})\cdot e^{i\omega t}\right) \\
& = \operatorname{Re}\left(2\cos(\alpha t) \cdot e^{i\omega t}\right) \\
& = 2 \cos(\alpha t) \cdot \operatorname{Re}\left(e^{i\omega t}\right) \\
& = 2 \cos(\alpha t)\cdot \cos\left(\omega t\right)\,.
\end{align}
Quantum mechanics
The complex number field is intrinsic to the mathematical formulations of quantum mechanics, where complex Hilbert spaces provide the context for one such formulation that is convenient and perhaps most standard. The original foundation formulas of quantum mechanics – the Schrödinger equation and Heisenberg's matrix mechanics – make use of complex numbers.
Relativity
In special and general relativity, some formulas for the metric on spacetime become simpler if one takes the time variable to be imaginary. (This is no longer standard in classical relativity, but is used in an essential way in quantum field theory.) Complex numbers are essential to spinors, which are a generalization of the tensors used in relativity.
Geometry
Fractals
Certain fractals are plotted in the complex plane, e.g. the Mandelbrot set and Julia sets.
Triangles
Every triangle has a unique Steiner inellipse—an ellipse inside the triangle and tangent to the midpoints of the three sides of the triangle. The foci of a triangle's Steiner inellipse can be found as follows, according to Marden's theorem:^{[13]}^{[14]} Denote the triangle's vertices in the complex plane as a = x_{A} + y_{A}i, b = x_{B} + y_{B}i, and c = x_{C} + y_{C}i. Write the cubic equation $(xa)(xb)(xc)=0$, take its derivative, and equate the (quadratic) derivative to zero. Marden's Theorem says that the solutions of this equation are the complex numbers denoting the locations of the two foci of the Steiner inellipse.
Algebraic number theory
As mentioned above, any nonconstant polynomial equation (in complex coefficients) has a solution in C. A fortiori, the same is true if the equation has rational coefficients. The roots of such equations are called algebraic numbers – they are a principal object of study in algebraic number theory. Compared to 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 Pythagorean triples (a, b, c), that is to say integers satisfying
 $a^2\; +\; b^2\; =\; c^2\; \backslash ,$
(which implies that the triangle having side lengths Template:Mvar, Template:Mvar, and Template:Mvar is a right triangle). They can be studied by considering Gaussian integers, that is, numbers of the form x + iy, where Template:Mvar and Template:Mvar are integers.
Analytic number theory
Analytic number theory studies numbers, often integers or rationals, by taking advantage of the fact that they can be regarded as complex numbers, in which analytic methods can be used. This is done by encoding numbertheoretic information in complexvalued functions. For example, the Riemann zetafunction ζ(s) is related to the distribution of prime numbers.
History
The earliest fleeting reference to square roots of negative numbers can perhaps be said to occur in the work of the Greek mathematician Hero of Alexandria in the 1st century AD, where in his Stereometrica he considers, apparently in error, the volume of an impossible frustum of a pyramid to arrive at the term √81 − 144 in his calculations, although negative quantities were not conceived of in Hellenistic mathematics and Heron merely replaced it by its positive.^{[15]}
The impetus to study complex numbers proper first arose in the 16th century when algebraic solutions for the roots of cubic and quartic polynomials were discovered by Italian mathematicians (see Niccolo 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 $x^3\; =\; px\; +\; q$^{[16]} gives the solution to the equation x^{3} = x as
 $\backslash frac\{1\}\{\backslash sqrt\{3\}\}\backslash left((\backslash sqrt\{1\})^\{1/3\}+\backslash frac\{1\}\{(\backslash sqrt\{1\})^\{1/3\}\}\backslash right).$
At first glance this looks like nonsense. However formal calculations with complex numbers show that the equation z^{3} = i has solutions −i, $\{\backslash scriptstyle\backslash frac\{\backslash sqrt\{3\}\}\{2\}\}+\{\backslash scriptstyle\backslash frac\{1\}\{2\}\}i$ and $\{\backslash scriptstyle\backslash frac\{\backslash sqrt\{3\}\}\{2\}\}+\{\backslash scriptstyle\backslash frac\{1\}\{2\}\}i$. Substituting these in turn for $\{\backslash scriptstyle\backslash sqrt\{1\}^\{1/3\}\}$ in Tartaglia's cubic formula and simplifying, one gets 0, 1 and −1 as the solutions of x^{3} − 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. Rafael Bombelli was the first to explicitly address these seemingly paradoxical solutions of cubic equations and developed the rules for complex arithmetic trying to resolve these issues.
The term "imaginary" for these quantities was coined by
René Descartes in 1637, although he was at pains to stress their imaginary nature
^{[17]}[...] quelquefois seulement imaginaires c’estàdire 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.
([...] 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.)
A further source of confusion was that the equation
$\backslash sqrt\{1\}^2=\backslash sqrt\{1\}\backslash sqrt\{1\}=1$ seemed to be capriciously inconsistent with the algebraic identity
$\backslash sqrt\{a\}\backslash sqrt\{b\}=\backslash sqrt\{ab\}$, which is valid for nonnegative real numbers
Template:Mvar and
Template:Mvar, and which was also used in complex number calculations with one of
Template:Mvar,
Template:Mvar positive and the other negative. The incorrect use of this identity (and the related identity
$\backslash scriptstyle\; 1/\backslash sqrt\{a\}=\backslash sqrt\{1/a\}$) in the case when both
Template:Mvar and
Template:Mvar 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.
Even so Euler considered it natural to introduce students to complex numbers much earlier than we do today. In his elementary algebra text book,
Elements of Algebra, he introduces these numbers almost at once and then uses them in a natural way throughout.
In the 18th century complex numbers gained wider use, as it was noticed that formal manipulation of complex expressions could be used to simplify calculations involving trigonometric functions. For instance, in 1730 Abraham de Moivre noted that the complicated identities relating trigonometric functions of an integer multiple of an angle to powers of trigonometric functions of that angle could be simply reexpressed by the following wellknown formula which bears his name, de Moivre's formula:
 $(\backslash cos\; \backslash theta\; +\; i\backslash sin\; \backslash theta)^\{n\}\; =\; \backslash cos\; n\; \backslash theta\; +\; i\backslash sin\; n\; \backslash theta.\; \backslash ,$
In 1748 Leonhard Euler went further and obtained Euler's formula of complex analysis:
 $\backslash cos\; \backslash theta\; +\; i\backslash sin\; \backslash theta\; =\; e\; ^\{i\backslash theta\; \}\; \backslash ,$
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 Caspar Wessel in 1799, although it had been anticipated as early as 1685 in Wallis's De Algebra tractatus.
Wessel's memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed. In 1806 JeanRobert Argand independently issued a pamphlet on complex numbers and provided a rigorous proof of the fundamental theorem of algebra. Gauss had earlier published an essentially topological proof of the theorem in 1797 but expressed his doubts at the time about "the true metaphysics of the square root of −1". It was not until 1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane, largely establishing modern notation and terminology. The English mathematician G. H. Hardy remarked that Gauss was the first mathematician to use complex numbers in 'a really confident and scientific way' although mathematicians such as Niels Henrik Abel and Carl Gustav Jacob Jacobi were necessarily using them routinely before Gauss published his 1831 treatise.^{[18]} Augustin Louis Cauchy and Bernhard Riemann together brought the fundamental ideas of complex analysis to a high state of completion, commencing around 1825 in Cauchy's case.
The common terms used in the theory are chiefly due to the founders. Argand called $\backslash cos\; \backslash phi\; +\; i\backslash sin\; \backslash phi$ the direction factor, and $r\; =\; \backslash sqrt\{a^2+b^2\}$ the modulus; Cauchy (1828) called $\backslash cos\; \backslash phi\; +\; i\backslash sin\; \backslash phi$ the reduced form (l'expression réduite) and apparently introduced the term argument; Gauss used i for $\backslash sqrt\{1\}$, introduced the term complex number for a + bi, and called a^{2} + b^{2} the norm. The expression direction coefficient, often used for $\backslash cos\; \backslash phi\; +\; i\backslash sin\; \backslash phi$, 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, Karl Weierstrass and many others.
Generalizations and related notions
The process of extending the field R of reals to C is known as Cayley–Dickson construction. It can be carried further to higher dimensions, yielding the quaternions H and octonions O which (as a real vector space) are of dimension 4 and 8, respectively. However, with increasing dimension, the algebraic properties familiar from real and complex numbers vanish: the quaternions are only a skew field, i.e. for some x, y: x·y ≠ y·x for two quaternions, the multiplication of octonions fails (in addition to not being commutative) to be associative: for some x, y, z: (x·y)·z ≠ x·(y·z). However, all of these are normed division algebras over R. By Hurwitz's theorem they are the only ones. The next step in the Cayley–Dickson construction, the sedenions fail to have this structure.
The Cayley–Dickson construction is closely related to the regular representation of C, thought of as an Ralgebra (an Rvector space with a multiplication), with respect to the basis (1, i). This means the following: the Rlinear map
 $\backslash mathbb\{C\}\; \backslash rightarrow\; \backslash mathbb\{C\},\; z\; \backslash mapsto\; wz$
for some fixed complex number Template:Mvar can be represented by a 2 × 2 matrix (once a basis has been chosen). With respect to the basis (1, i), this matrix is
 $$
\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
 $J\; =\; \backslash begin\{pmatrix\}p\; \&\; q\; \backslash \backslash \; r\; \&\; p\; \backslash end\{pmatrix\},\; \backslash quad\; p^2\; +\; qr\; +\; 1\; =\; 0$
has the property that its square is the negative of the identity matrix: J^{2} = −I. Then
 $\backslash \{\; z\; =\; a\; I\; +\; b\; J:\; a,b\; \backslash in\; R\; \backslash \}$
is also isomorphic to the field C, and gives an alternative complex structure on R^{2}. This is generalized by the notion of a linear complex structure.
Hypercomplex numbers also generalize R, C, H, and O. For example this notion contains the splitcomplex numbers, which are elements of the ring R[x]/(x^{2} − 1) (as opposed to R[x]/(x^{2} + 1)). In this ring, the equation a^{2} = 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 Q_{p} of padic numbers (for any prime number p), which are thereby analogous to R. There are no other nontrivial ways of completing Q than R and Q_{p}, by Ostrowski's theorem. The algebraic closure $\backslash overline\; \{\backslash mathbf\{Q\}\_p\}$ of Q_{p} still carry a norm, but (unlike C) are not complete with respect to it. The completion $\backslash mathbf\{C\}\_p$ of $\backslash overline\; \{\backslash mathbf\{Q\}\_p\}$ turns out to be algebraically closed. This field is called padic complex numbers by analogy.
The fields R and Q_{p} and their finite field extensions, including C, are local fields.
See also
Notes
References
Mathematical references
Historical references



 A gentle introduction to the history of complex numbers and the beginnings of complex analysis.

 An advanced perspective on the historical development of the concept of number.
Further reading
 The Road to Reality: A Complete Guide to the Laws of the Universe, by Roger Penrose; Alfred A. Knopf, 2005; ISBN 0679454438. 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 030909657X (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 0198534477 (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 (September 12, 2005). ISBN 0387903283.
External links
 Template:Springer
 Introduction to Complex Numbers from Khan Academy

 BBC.
 Euler's work on Complex Roots of Polynomials at Convergence. MAA Mathematical Sciences Digital Library.
 John and Betty's Journey Through Complex Numbers
 Mandelbrot set.
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