Complex number

 ( Composition Algebras ) 

The argument of (sometimes called the "phase" ) is the angle of the radius with the positive real axis, and is written as , expressed in in this article. The angle is defined only up to adding integer multiples of $2\backslash pi$, since a rotation by $2\backslash pi$ (or 360°) around the origin leaves all points in the complex plane unchanged. One possible choice to uniquely specify the argument is to require it to be within the interval $(-\backslash pi,\backslash pi]$, which is referred to as the principal value.Other authors, including , chose the argument to be in the interval $[0,\; 2\backslash pi)$. The argument can be computed from the rectangular form by means of the arctan (inverse tangent) function.

$z=r(\backslash cos\backslash varphi\; +i\backslash sin\backslash varphi)$

If two complex numbers are given in polar form, i.e., and , the product and division can be computed as $$z\_1\; z\_2\; =\; r\_1\; r\_2\; (\backslash cos(\backslash varphi\_1\; +\; \backslash varphi\_2)\; +\; i\; \backslash sin(\backslash varphi\_1\; +\; \backslash varphi\_2)).$$ $$\backslash frac\{z\_1\}\{z\_2\}\; =\; \backslash frac\{r\_1\}\{r\_2\}\; \backslash left(\backslash cos(\backslash varphi\_1\; -\; \backslash varphi\_2)\; +\; i\; \backslash sin(\backslash varphi\_1\; -\; \backslash varphi\_2)\backslash right),\; \backslash text\{if\; \}z\_2\; \backslash ne\; 0.$$ (These are a consequence of the trigonometric identities for the sine and cosine function.) In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product. The picture at the right illustrates the multiplication of $$(2+i)(3+i)=5+5i.$$ Because the real and imaginary part of are equal, the argument of that number is 45 degrees, or (in radian). On the other hand, it is also the sum of the angles at the origin of the red and blue triangles are arctan(1/3) and arctan(1/2), respectively. Thus, the formula $$\backslash frac\{\backslash pi\}\{4\}\; =\; \backslash arctan\backslash left(\backslash frac\{1\}\{2\}\backslash right)\; +\; \backslash arctan\backslash left(\backslash frac\{1\}\{3\}\backslash right)$$ holds. As the arctan function can be approximated highly efficiently, formulas like this – known as Machin-like formulas – are used for high-precision approximations of :

The nth root of a complex number are given by $$z^\{1/n\}\; =\; \backslash sqrtnr\; \backslash left(\; \backslash cos\; \backslash left(\backslash frac\{\backslash varphi+2k\backslash pi\}\{n\}\backslash right)\; +\; i\; \backslash sin\; \backslash left(\backslash frac\{\backslash varphi+2k\backslash pi\}\{n\}\backslash right)\backslash right)$$ for . (Here $\backslash sqrtnr$ is the usual (positive) th root of the positive real number .) Because sine and cosine are periodic, other integer values of do not give other values. For any $z\; \backslash ne\; 0$, there are, in particular n distinct complex n-th roots. For example, there are 4 fourth roots of 1, namely

$z\_1\; =\; 1,\; z\_2\; =\; i,\; z\_3\; =\; -1,\; z\_4\; =\; -i.$

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

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

Many mathematicians contributed to the development of complex numbers. The rules for addition, subtraction, multiplication, and root extraction of complex numbers were developed by the Italian mathematician Rafael Bombelli.

The earliest fleeting reference to of can perhaps be said to occur in the work of the Greek mathematician Hero of Alexandria in the 1st century AD, where in his Stereometrica he considered, apparently in error, the volume of an impossible frustum of a pyramid to arrive at the term $\backslash sqrt\{81\; -\; 144\}$ in his calculations, which today would simplify to $\backslash sqrt\{-63\}\; =\; 3i\backslash sqrt\{7\}$. Negative quantities were not conceived of in Hellenistic mathematics and Hero merely replaced the negative value by its positive $\backslash sqrt\{144\; -\; 81\}\; =\; 3\backslash sqrt\{7\}.$

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

The term "imaginary" for these quantities was coined by René Descartes in 1637, who was at pains to stress their unreal nature:

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 non-negative real numbers and , and which was also used in complex number calculations with one of , positive and the other negative. The incorrect use of this identity in the case when both and are negative, and the related identity $\backslash frac\{1\}\{\backslash sqrt\{a\}\}\; =\; \backslash sqrt\{\backslash frac\{1\}\{a\}\}$, even bedeviled Leonhard Euler. This difficulty eventually led to the convention of using the special symbol in place of $\backslash sqrt\{-1\}$ to guard against this mistake.

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

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

In 1748, Euler went further and obtained Euler's formula of complex analysis:

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

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

The idea of a complex number as a point in the complex plane was first described by Denmark–Norway mathematician Caspar Wessel in 1799, although it had been anticipated as early as 1685 in John Wallis A Treatise of Algebra.

Wessel's memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed. In 1806 Jean-Robert Argand independently issued a pamphlet on complex numbers and provided a rigorous proof of the fundamental theorem of algebra. Carl Friedrich Gauss had earlier published an essentially topology proof of the theorem in 1797 but expressed his doubts at the time about "the true metaphysics of the square root of −1".Gauss, Carl Friedrich (1799) "Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse." New Ph.D. thesis, University of Helmstedt, (Germany). (in Latin) It was not until 1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane,

If one formerly contemplated this subject from a false point of view and therefore found a mysterious darkness, this is in large part attributable to clumsy terminology. Had one not called +1, −1, $\backslash sqrt\{-1\}$ positive, negative, or imaginary (or even impossible) units, but instead, say, direct, inverse, or lateral units, then there could scarcely have been talk of such darkness.

In the beginning of the 19th century, other mathematicians discovered independently the geometrical representation of the complex numbers: Buée, Mourey, 1861 reprint of 1828 original. Warren, Français and his brother, Bellavitis.

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

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

The common terms used in the theory are chiefly due to the founders. Argand called the direction factor, and $r\; =\; \backslash sqrt\{a^2\; +\; b^2\}$ the modulus; + \tfrac{b}{\sqrt{a^2 + b^2}} \sqrt{-1} ], whose module is unity 1, would represent its direction.]}} Cauchy (1821) called the reduced form (l'expression réduite) and apparently introduced the term argument; Gauss used for $\backslash sqrt\{-1\}$, introduced the term complex number for , and called the norm. The expression direction coefficient, often used for , is due to Hankel (1867), From p. 71: "Wir werden den Factor (cos φ + i sin φ) haüfig den Richtungscoefficienten nennen." (We will often call the factor (cos φ + i sin φ) the "coefficient of direction".) and absolute value, for modulus, is due to Weierstrass.

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

$\backslash RX\; \backslash to\; \backslash C$

$\backslash RX\; /\; (X^2\; +\; 1)\; \backslash stackrel\; \backslash cong\; \backslash to\; \backslash C$

Accepting that $\backslash Complex$ is algebraically closed, because it is an algebraic extension of $\backslash mathbb\{R\}$ in this approach, $\backslash Complex$ is therefore the algebraic closure of $\backslash R.$

$$\backslash begin\{pmatrix\}$$

 a &   -b  \\
 b & \;\; a
     

A simple computation shows that the map $$a+ib\backslash mapsto\; \backslash begin\{pmatrix\}$$

 a &   -b  \\
 b & \;\; a
     

The polar form representation of complex numbers explicitly gives these matrices as scaled rotation matrix. $$r\; (\backslash cos\; \backslash theta\; +\; i\; \backslash sin\; \backslash theta)\backslash mapsto\; \backslash begin\{pmatrix\}$$

 r \cos \theta &   -r \sin \theta  \\
 r \sin \theta & \;\; r \cos \theta
     

graph of the function . Darker spots mark moduli near zero, brighter spots are farther away from the origin. The color encodes the argument. The function has zeros for and poles at $\backslash pm\; \backslash sqrt.$]] Unlike real functions, which are commonly represented as two-dimensional graphs, 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.

$\backslash exp(z+2\backslash pi\; i)\; =\; \backslash exp\; z\; \backslash exp\; (2\; \backslash pi\; i)\; =\; \backslash exp\; z$

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

$\backslash exp\; z\; =\; w$

If $z\; \backslash in\; \backslash Complex\; \backslash setminus\; \backslash left(\; -\backslash R\_\{\backslash ge\; 0\}\; \backslash right)$ is not a non-positive real number (a positive or a non-real number), the resulting principal value of the complex logarithm is obtained with . It is an analytic function outside the negative real numbers, but it cannot be prolongated to a function that is continuous at any negative real number $z\; \backslash in\; -\backslash R^+$, where the principal value is .

Complex exponentiation is defined as $$z^\backslash omega\; =\; \backslash exp(\backslash omega\; \backslash ln\; z),$$ and is multi-valued, except when is an integer. For , for some natural number , this recovers the non-uniqueness of th roots mentioned above. If is real (and an arbitrary complex number), one has a preferred choice of $\backslash ln\; x$, the real logarithm, which can be used to define a preferred exponential function.

Complex numbers, unlike real numbers, do not in general satisfy the unmodified power and logarithm identities, particularly when naïvely treated as single-valued functions; see failure of power and logarithm identities. For example, they do not satisfy $$a^\{bc\}\; =\; \backslash left(a^b\backslash right)^c.$$ 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.

The value of a trigonometric or hyperbolic function of a complex number can be expressed in terms of those functions evaluated on real numbers, via angle-addition formulas. For ,

$$\backslash sin\{z\}\; =\; \backslash sin\{x\}\; \backslash cosh\{y\}\; +\; i\; \backslash cos\{x\}\; \backslash sinh\{y\}$$

$$\backslash cos\{z\}\; =\; \backslash cos\{x\}\; \backslash cosh\{y\}\; -\; i\; \backslash sin\{x\}\; \backslash sinh\{y\}$$

$$\backslash tan\{z\}\; =\; \backslash frac\{\backslash tan\{x\}\; +\; i\; \backslash tanh\{y\}\}\{1\; -\; i\; \backslash tan\{x\}\; \backslash tanh\{y\}\}$$

$$\backslash cot\{z\}\; =\; -\backslash frac\{1\; +\; i\; \backslash cot\{x\}\; \backslash coth\{y\}\}\{\backslash cot\{x\}\; -i\; \backslash coth\{y\}\}$$

$$\backslash sinh\{z\}\; =\; \backslash sinh\{x\}\; \backslash cos\{y\}\; +\; i\; \backslash cosh\{x\}\; \backslash sin\{y\}$$

$$\backslash cosh\{z\}\; =\; \backslash cosh\{x\}\; \backslash cos\{y\}\; +\; i\; \backslash sinh\{x\}\; \backslash sin\{y\}$$

$$\backslash tanh\{z\}\; =\; \backslash frac\{\backslash tanh\{x\}\; +\; i\; \backslash tan\{y\}\}\{1\; +\; i\; \backslash tanh\{x\}\; \backslash tan\{y\}\}$$

$$\backslash coth\{z\}\; =\; \backslash frac\{1\; -\; i\; \backslash coth\{x\}\; \backslash cot\{y\}\}\{\backslash coth\{x\}\; -\; i\; \backslash cot\{y\}\}$$

Where these expressions are not well defined, because a trigonometric or hyperbolic function evaluates to infinity or there is division by zero, they are nonetheless correct as limits.

$\backslash lim\_\{z\; \backslash to\; z\_0\}\; \{f(z)\; -\; f(z\_0)\; \backslash over\; z\; -\; z\_0\; \}$

$f(z)\; =\; \backslash overline\; z$

$\backslash frac\{\backslash partial\; f\}\{\backslash partial\; \backslash overline\; z\}\; =\; 0.$

Complex analysis shows some features not apparent in real analysis. For example, the identity theorem asserts that two holomorphic functions and agree if they agree on an arbitrarily small open subset of $\backslash mathbb\{C\}$. Meromorphic functions, functions that can locally be written as with a holomorphic function , still share some of the features of holomorphic functions. Other functions have essential singularities, such as at .

Complex conjugation is also employed in inversive geometry, a branch of geometry studying reflections more general than ones about a line. In the network analysis of electrical circuits, the complex conjugate is used in finding the equivalent impedance when the maximum power transfer theorem is looked for.

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

Complex numbers often generalize concepts originally conceived in the real numbers. For example, the conjugate transpose generalizes the transpose, Hermitian matrix generalize Symmetric matrix, and Unitary matrix generalize orthogonal matrices.

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

• in the right half plane, it will be unstable,
• all in the left half plane, it will be BIBO stability,
• 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.

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

$$x(t)\; =\; \backslash operatorname\{Re\}\; \backslash \{X(\; t\; )\; \backslash \}$$

and

$$X(\; t\; )\; =\; A\; e^\{i\backslash omega\; t\}\; =\; a\; e^\{\; i\; \backslash phi\; \}\; e^\{i\backslash omega\; t\}\; =\; a\; e^\{i\; (\backslash omega\; t\; +\; \backslash phi)\; \}$$

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 use digital versions of Fourier analysis (and wavelet analysis) to transmit, Data compression, restore, and otherwise process Digital data Sound signals, still images, and video signals.

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

$$\backslash begin\{align\}$$

 \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(\left(e^{i\alpha t} + e^{-i\alpha t}\right) \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).
     

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

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

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

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

$$v(t)\; =\; \backslash operatorname\{Re\}(V)\; =\; \backslash operatorname\{Re\}\backslash left\; =\; V\_0\; \backslash cos\; \backslash omega\; t.$$

The complex-valued signal is called the analytic signal representation of the real-valued, measurable signal .

• First, it has characteristic 0. This means that for any number of summands (all of which equal one).
• Second, its transcendence degree over $\backslash Q$, the prime field of $\backslash Complex,$ is the cardinality of the continuum.
• Third, it is algebraically closed (see above).

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

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

The only connected space locally compact topological ring are $\backslash R$ and $\backslash Complex.$ This gives another characterization of $\backslash Complex$ as a topological field, because $\backslash Complex$ can be distinguished from $\backslash R$ because the nonzero complex numbers are connected space, while the nonzero real numbers are not.