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Number with a real and an imaginary part From Wikipedia, the free encyclopedia

In mathematics, a **complex number** is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and satisfying the equation ; every complex number can be expressed in the form , where a and b are real numbers. Because no real number satisfies the above equation, i was called an imaginary number by René Descartes. For the complex number , a is called the **real part**, and b is called the **imaginary part**. The set of complex numbers is denoted by either of the symbols or **C**. Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world.^{[1]}^{[2]}

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Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equation has no real solution, because the square of a real number cannot be negative, but has the two nonreal complex solutions and .

Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule along with the associative, commutative, and distributive laws. Every nonzero complex number has a multiplicative inverse. This makes the complex numbers a field with the real numbers as a subfield.

The complex numbers also form a real vector space of dimension two, with as a standard basis. This standard basis makes the complex numbers a Cartesian plane, called the complex plane. This allows a geometric interpretation of the complex numbers and their operations, and conversely some geometric objects and operations can be expressed in terms of complex numbers. For example, the real numbers form the real line, which is pictured as the horizontal axis of the complex plane, while real multiples of are the vertical axis. A complex number can also be defined by its geometric polar coordinates: the radius is called the absolute value of the complex number, while the angle from the positive real axis is called the argument of the complex number. The complex numbers of absolute value one form the unit circle. Adding a fixed complex number to all complex numbers defines a translation in the complex plane, and multiplying by a fixed complex number is a similarity centered at the origin (dilating by the absolute value, and rotating by the argument). The operation of complex conjugation is the reflection symmetry with respect to the real axis.

The complex numbers form a rich structure that is simultaneously an algebraically closed field, a commutative algebra over the reals, and a Euclidean vector space of dimension two.

A complex number is an expression of the form *a* + *bi*, where a and b are real numbers, and *i* is an abstract symbol, the so-called imaginary unit, whose meaning will be explained further below. For example, 2 + 3*i* is a complex number.^{[3]}

For a complex number *a* + *bi*, the real number a is called its *real part* , and the real number b (not the complex number *bi*) is its *imaginary part*.^{[4]}^{[5]} The real part of a complex number z is denoted Re(*z*), , or ; the imaginary part is Im(*z*), , or : for example,, .

A complex number z can be identified with the ordered pair of real numbers , which may be interpreted as coordinates of a point in a Euclidean plane with standard coordinates, which is then called the *complex plane* or *Argand diagram,*^{[6]}^{[lower-alpha 1]}.^{[7]} The horizontal axis is generally used to display the real part, with increasing values to the right, and the imaginary part marks the vertical axis, with increasing values upwards.

A real number a can be regarded as a complex number *a* + 0*i*, whose imaginary part is 0. A purely imaginary number *bi* is a complex number 0 + *bi*, whose real part is zero. As with polynomials, it is common to write *a* + 0*i* = *a*, 0 + *bi* = *bi*, and *a* + (−*b*)*i* = *a* − *bi*; for example, 3 + (−4)*i* = 3 − 4*i*.

The set of all complex numbers is denoted by (blackboard bold) or **C** (upright bold).

In some disciplines such as electromagnetism and electrical engineering, j is used instead of i, as i frequently represents electric current,^{[8]}^{[9]} and complex numbers are written as *a* + *bj* or *a* + *jb*.

Two complex numbers and are added by separately adding their real and imaginary parts. That is to say:

Similarly, subtraction can be performed as

The addition can be geometrically visualized as follows: the sum of two complex numbers a and b, interpreted as points in the complex plane, is the point obtained by building a parallelogram from the three vertices O, and the points of the arrows labeled a and b (provided that they are not on a line). Equivalently, calling these points A, B, respectively and the fourth point of the parallelogram X the triangles OAB and XBA are congruent.

The product of two complex numbers is computed as follows:

For example, In particular, this includes as a special case the fundamental formula

This formula distinguishes the complex number *i* from any real number, since the square of any (negative or positive) real number is always a non-negative real number.

With this definition of multiplication and addition, familiar rules for the arithmetic of rational or real numbers continue to hold for complex numbers. More precisely, the distributive property, the commutative properties (of addition and multiplication) hold. Therefore, the complex numbers form an algebraic structure known as a *field*, the same way as the rational or real numbers do.^{[10]}

The *complex conjugate* of the complex number *z* = *x* + *yi* is defined as
^{[11]} It is also denoted by some authors by . Geometrically, z is the "reflection" of z about the real axis. Conjugating twice gives the original complex number: A complex number is real if and only if it equals its own conjugate. The unary operation of taking the complex conjugate of a complex number cannot be expressed by applying only their basic operations addition, subtraction, multiplication and division.

For any complex number *z* = *x* + *yi* , the product

is a *non-negative real* number. This allows to define the *absolute value* (or *modulus* or *magnitude*) of *z* to be the square root ^{[12]}
By Pythagoras' theorem, is the distance from the origin to the point representing the complex number *z* in the complex plane. In particular, the circle of radius one around the origin consists precisely of the numbers *z* such that . If is a real number, then : its absolute value as a complex number and as a real number are equal.

Using the conjugate, the reciprocal of a nonzero complex number can be computed to be

More generally, the division of an arbitrary complex number by a non-zero complex number equals
This process is sometimes called "rationalization" of the denominator (although the denominator in the final expression might be an irrational real number), because it resembles the method to remove roots from simple expressions in a denominator.^{[citation needed]}

The *argument* of z (sometimes called the "phase" φ)^{[7]} is the angle of the radius Oz with the positive real axis, and is written as arg *z*, expressed in radians in this article. The angle is defined only up to adding integer multiples of , since a rotation by (or 360°) around the origin leaves all points in the complex plane unchanged. One possible choice to uniquely specify the argument is to require it to be within the interval , which is referred to as the principal value.^{[13]}
The argument can be computed from the rectangular form x + yi by means of the arctan (inverse tangent) function.^{[14]}

For any complex number *z*, with absolute value and argument , the equation

holds. This identity is referred to as the polar form of *z*. It is sometimes abbreviated as .
In electronics, one represents a phasor with amplitude r and phase φ in angle notation:^{[15]}

If two complex numbers are given in polar form, i.e., *z*_{1} = *r*_{1}(cos *φ*_{1} + *i* sin *φ*_{1}) and *z*_{2} = *r*_{2}(cos *φ*_{2} + *i* sin *φ*_{2}), the product and division can be computed as
(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
Because the real and imaginary part of 5 + 5*i* 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
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.^{[citation needed]}

The *n*-th power of a complex number can be computed using de Moivre's formula, which is obtained by repeatedly applying the above formula for the product:
For example, the first few powers of the imaginary unit *i* are .

The n nth roots of a complex number z are given by
for 0 ≤ *k* ≤ *n* − 1. (Here is the usual (positive) nth root of the positive real number r.) Because sine and cosine are periodic, other integer values of k do not give other values. For any , there are, in particular *n* distinct complex *n*-th roots. For example, there are 4 fourth roots of 1, namely

In general there is *no* natural way of distinguishing one particular complex nth root of a complex number. (This is in contrast to the roots of a positive real number *x*, which has a unique positive real *n*-th root, which is therefore commonly referred to as *the* *n*-th root of *x*.) One refers to this situation by saying that the nth root is a n-valued function of z.

The fundamental theorem of algebra, of Carl Friedrich Gauss and Jean le Rond d'Alembert, states that for any complex numbers (called coefficients) *a*_{0}, ..., *a*_{n}, the equation
has at least one complex solution *z*, provided that at least one of the higher coefficients *a*_{1}, ..., *a*_{n} is nonzero.^{[16]} This property does not hold for the field of rational numbers (the polynomial *x*^{2} − 2 does not have a rational root, because √2 is not a rational number) nor the real numbers (the polynomial *x*^{2} + 4 does not have a real root, because the square of x is positive for any real number x).

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

The solution in radicals (without trigonometric functions) of a general cubic equation, when all three of its roots are real numbers, contains the square roots of negative numbers, a situation that cannot be rectified by factoring aided by the rational root test, if the cubic is irreducible; this is the so-called *casus irreducibilis* ("irreducible case"). This conundrum led Italian mathematician Gerolamo Cardano to conceive of complex numbers in around 1545 in his *Ars Magna*,^{[17]} though his understanding was rudimentary; moreover, he later described complex numbers as being "as subtle as they are useless".^{[18]} Cardano did use imaginary numbers, but described using them as "mental torture."^{[19]} This was prior to the use of the graphical complex plane. Cardano and other Italian mathematicians, notably Scipione del Ferro, in the 1500s created an algorithm for solving cubic equations which generally had one real solution and two solutions containing an imaginary number. Because they ignored the answers with the imaginary numbers, Cardano found them useless.^{[20]}

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.^{[21]} 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.^{[22]}

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 considered, apparently in error, the volume of an impossible frustum of a pyramid to arrive at the term in his calculations, which today would simplify to .^{[lower-alpha 2]} Negative quantities were not conceived of in Hellenistic mathematics and Hero merely replaced it by its positive ^{[24]}

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 (Niccolò Fontana Tartaglia and Gerolamo Cardano). It was soon realized (but proved much later)^{[25]} 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.^{[lower-alpha 3]} 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:^{[26]}

... sometimes only imaginary, that is one can imagine as many as I said in each equation, but sometimes there exists no quantity that matches that which we imagine.

[... quelquefois seulement imaginaires c'est-à-dire que l'on peut toujours en imaginer autant que j'ai dit en chaque équation, mais qu'il n'y a quelquefois aucune quantité qui corresponde à celle qu'on imagine.]

A further source of confusion was that the equation seemed to be capriciously inconsistent with the algebraic identity , which is valid for non-negative real numbers a and b, and which was also used in complex number calculations with one of a, b positive and the other negative. The incorrect use of this identity in the case when both a and b are negative, and the related identity , even bedeviled Leonhard Euler. This difficulty eventually led to the convention of using the special symbol *i* in place of to guard against this mistake.^{[citation needed]} Even so, Euler considered it natural to introduce students to complex numbers much earlier than we do today. In his elementary algebra text book, *Elements of Algebra*, he introduces these numbers almost at once and then uses them in a natural way throughout.

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

In 1748, Euler went further and obtained Euler's formula of complex analysis:^{[27]}

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

The idea of a complex number as a point in the complex plane (above) was first described by Danish–Norwegian mathematician Caspar Wessel in 1799,^{[28]} although it had been anticipated as early as 1685 in Wallis's *A Treatise of Algebra*.^{[29]}

Wessel's memoir appeared in the Proceedings of the 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.^{[30]} Carl Friedrich 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".^{[31]} It was not until 1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane,^{[32]} largely establishing modern notation and terminology:^{[33]}

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

In the beginning of the 19th century, other mathematicians discovered independently the geometrical representation of the complex numbers: Buée,^{[34]}^{[35]} Mourey,^{[36]} Warren,^{[37]}^{[38]}^{[39]} Français and his brother, Bellavitis.^{[40]}^{[41]}

The English mathematician 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.^{[42]}

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

The common terms used in the theory are chiefly due to the founders. Argand called cos *φ* + *i* sin *φ* the *direction factor*, and the *modulus*;^{[lower-alpha 4]}^{[43]} Cauchy (1821) called cos *φ* + *i* sin *φ* the *reduced form* (l'expression réduite)^{[44]} and apparently introduced the term *argument*; Gauss used *i* for ,^{[lower-alpha 5]} introduced the term *complex number* for *a* + *bi*,^{[lower-alpha 6]} and called *a*^{2} + *b*^{2} the *norm*.^{[lower-alpha 7]} The expression *direction coefficient*, often used for cos *φ* + *i* sin *φ*, is due to Hankel (1867),^{[48]} and *absolute value,* for *modulus,* is due to Weierstrass.

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

While the above low-level definitions, including the addition and multiplication, accurately describes the complex numbers, there are other, equivalent approaches that reveal the abstract algebraic structure of the complex numbers more immediately.

One approach to is via polynomials, i.e., expressions of the form
where the coefficients *a*_{0}, ..., *a*_{n} are real numbers. The set of all such polynomials is denoted by . Since sums and products of polynomials are again polynomials, this set forms a commutative ring, called the polynomial ring (over the reals). To every such polynomial *p*, one may assign the complex number , i.e., the value obtained by setting . This defines a function

This function is surjective since every complex number can be obtained in such a way: the evaluation of a linear polynomial at is . However, the evaluation of polynomial at *i* is 0, since This polynomial is irreducible, i.e., cannot be written as a product of two linear polynomials. Basic facts of abstract algebra then imply that the kernel of the above map is an ideal generated by this polynomial, and that the quotient by this ideal is a field, and that there is an isomorphism

between the quotient ring and . Some authors take this as the definition of .^{[49]}

Accepting that is algebraically closed, because it is an algebraic extension of in this approach, is therefore the algebraic closure of

Complex numbers *a* + *bi* can also be represented by 2 × 2 matrices that have the form
Here the entries a and b are real numbers. As the sum and product of two such matrices is again of this form, these matrices form a subring of the ring of 2 × 2 matrices.

A simple computation shows that the map is a ring isomorphism from the field of complex numbers to the ring of these matrices, proving that these matrices form a field. This isomorphism associates the square of the absolute value of a complex number with the determinant of the corresponding matrix, and the conjugate of a complex number with the transpose of the matrix.

The geometric description of the multiplication of complex numbers can also be expressed in terms of rotation matrices by using this correspondence between complex numbers and such matrices. The action of the matrix on a vector (*x*, *y*) corresponds to the multiplication of *x* + *iy* by *a* + *ib*. In particular, if the determinant is 1, there is a real number t such that the matrix has the form

In this case, the action of the matrix on vectors and the multiplication by the complex number are both the rotation of the angle t.

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.

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, , endowed with the metric
is a complete metric space, which notably includes the triangle inequality
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, which can be shown to converge for any *z*:
For example,