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주해수용(籌解需用)[편집]

담헌 홍대용(洪大容)의 수학사상[편집]

담헌 홍대용(1731~1783)은 조선시대 후기 영 정조 시대를 산 인물로 선비 정신의 표상적인 인물이다.

북학파에 속하는 실학자이고 많은 제자를 길러낸 조선의 대표적인 선비라 할 수 있는 김원행의 제자였다. 영조 41년(1765) 35세 때 , 서장관으로 북경에 가는 숙부를 따라가서 중국의 선비들과 교류하여, 각종 서양의 문물을 접하였다.

담험은 과학 및 예술 계통에서 뚜렷한 저술을 남겼다. 이 저술들 중에서 <주해수용>은 <담헌서>에 포함되어 있는데

이 <담헌서>는 현재 완역되어 있다. <담헌서>중 수학 분야에 해당하는 외집 4, 5, 6권 부분은 번역이 되었으나, 그 내용 및 가치의 분석은 미미한 실정이다.

Notation and operations[편집]

The set of all complex numbers is usually denoted by C, or in blackboard bold by $\mathbb {C}$ (Unicode ℂ). The real numbers, R, may be regarded as "lying in" C by considering every real number as a complex: $a=a+0i$ .

Complex numbers are added, subtracted, and multiplied by formally applying the associative, commutative and distributive laws of algebra, together with the equation i² = −1:

\,(a+bi)+(c+di)=(a+c)+(b+d)i

\,(a+bi)-(c+di)=(a-c)+(b-d)i

\,(a+bi)(c+di)=ac+bci+adi+bdi^{2}=(ac-bd)+(bc+ad)i

Division of complex numbers can also be defined (see below). Thus, the set of complex numbers forms a field which, in contrast to the real numbers, is algebraically closed.

In mathematics, the adjective "complex" means that the field of complex numbers is the underlying number field considered, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra.

The field of complex numbers[편집]

Formally, the complex numbers can be defined as ordered pairs of real numbers (a, b) together with the operations:

(a,b)+(c,d)=(a+c,b+d)\,

(a,b)\cdot (c,d)=(ac-bd,bc+ad).\,

So defined, the complex numbers form a field, the complex number field, denoted by C (a field is an algebraic structure in which addition, subtraction, multiplication, and division are defined and satisfy certain algebraic laws. For example, the real numbers form a field).

Since a complex number a + bi is uniquely specified by an ordered pair (a, b) of real numbers, the complex numbers are in one-to-one correspondence with points on a plane, called the complex plane.

We identify the real number a with the complex number (a, 0), and in this way the field of real numbers R becomes a subfield of C. The imaginary unit i is the complex number (0, 1).

In C, we have:

additive identity ("zero"): (0, 0)
multiplicative identity ("one"): (1, 0)
additive inverse of (a,b): (−a, −b)
multiplicative inverse (reciprocal) of non-zero (a, b): $\left({a \over a^{2}+b^{2}},{-b \over a^{2}+b^{2}}\right).$

C can also be defined as the topological closure of the algebraic numbers or as the algebraic closure of R, both of which are described below.

The complex plane[편집]

**Geometric representation of z and its conjugate in the complex plane.**

A complex number z can be viewed as a point or a position vector in a two-dimensional Cartesian coordinate system called the complex plane or Argand diagram (named after Jean-Robert Argand – see figure at right). The point and hence the complex number z can be specified by Cartesian (rectangular) coordinates. The Cartesian coordinates of the complex number are the real part x = Re(z) and the imaginary part y = Im(z). The representation of a complex number by its Cartesian coordinates is called the Cartesian form or rectangular form or algebraic form of that complex number.

Polar form[편집]

Alternatively, the complex number z can be specified by polar coordinates. The polar coordinates are r = |z| ≥ 0, called the absolute value or modulus, and φ = arg(z), called the argument of z. For r = 0 any value of φ describes the same number. To get a unique representation, a conventional choice is to set arg(0) = 0. For r > 0 the argument φ is unique modulo 2π; that is, if any two values of the complex argument differ by an exact integer multiple of 2π, they are considered equivalent. To get a unique representation, a conventional choice is to limit φ to the interval (-π,π], i.e. −π < φ ≤ π. The representation of a complex number by its polar coordinates is called the polar form of the complex number.

Conversion from the polar form to the Cartesian form[편집]

x=r\cos \varphi

y=r\sin \varphi

Conversion from the Cartesian form to the polar form[편집]

r={\sqrt {x^{2}+y^{2}}}

\varphi ={\begin{cases}\arctan({\frac {y}{x}})&{\mbox{if }}x>0\\\arctan({\frac {y}{x}})+\pi &{\mbox{if }}x<0{\mbox{ and }}y\geq 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\\\mathrm {undefined} &{\mbox{if }}x=0{\mbox{ and }}y=0.\end{cases}}

The previous formula requires rather laborious case differentiations. However, many programming languages provide a variant of the arctangent function which is often named atan2 and processes these internally. A formula that uses the arccos function requires fewer case differentiations:

\varphi ={\begin{cases}+\arccos {\frac {x}{r}}&{\mbox{if }}y\geq 0{\mbox{ and }}r\neq 0\\-\arccos {\frac {x}{r}}&{\mbox{if }}y<0\\\mathrm {undefined} &{\mbox{if }}r=0.\end{cases}}

Notation of the polar form[편집]

The notation of the polar form as

z=r\,(\cos \varphi +i\sin \varphi )\,

is called trigonometric form. The notation cis φ is sometimes used as an abbreviation for cos φ + i sin φ. Using Euler's formula it can also be written as

z=r\,\mathrm {e} ^{i\varphi }\,,

which is called exponential form.

Multiplication, division, exponentiation, and root extraction in the polar form[편집]

Multiplication, division, exponentiation, and root extraction are much easier in the polar form than in the Cartesian form.

Using sum and difference identities its possible to obtain that

r_{1}\,e^{i\varphi _{1}}\cdot r_{2}\,e^{i\varphi _{2}}=r_{1}\,r_{2}\,e^{i(\varphi _{1}+\varphi _{2})}\,

and that

{\frac {r_{1}\,e^{i\varphi _{1}}}{r_{2}\,e^{i\varphi _{2}}}}={\frac {r_{1}}{r_{2}}}\,e^{i(\varphi _{1}-\varphi _{2})}.\,

Exponentiation with integer exponents; according to De Moivre's formula,

{\big (}r\,e^{i\varphi }{\big )}^{n}=r^{n}\,e^{in\varphi }.\,

Exponentiation with arbitrary complex exponents is discussed in the article on exponentiation.

The addition of two complex numbers is just the vector addition of two vectors, and multiplication by a fixed complex number can be seen as a simultaneous rotation and stretching.

Multiplication by i corresponds to a counter-clockwise rotation by 90 degrees (π/2 radians). The geometric content of the equation i² = −1 is that a sequence of two 90 degree rotations results in a 180 degree (π radians) rotation. Even the fact (−1) · (−1) = +1 from arithmetic can be understood geometrically as the combination of two 180 degree turns.

All the roots of any number, real or complex, may be found with a simple algorithm. The nth roots are given by

{\sqrt[{n}]{re^{i\varphi }}}={\sqrt[{n}]{r}}\ e^{i\left({\frac {\varphi +2k\pi }{n}}\right)}

for k = 0, 1, 2, …, n − 1, where ${\sqrt[{n}]{r}}$ represents the principal nth root of r.

Absolute value, conjugation and distance[편집]

The absolute value (or modulus or magnitude) of a complex number z = r e^iφ is defined as |z| = r. Algebraically, if z = a + bi, then $|z|={\sqrt {a^{2}+b^{2}}}.$

One can check readily that the absolute value has three important properties:

|z|=0\,

if and only if

z=0\,

|z+w|\leq |z|+|w|\,

(triangle inequality)

|z\cdot w|=|z|\cdot |w|\,

for all complex numbers z and w. It then follows, for example, that $|1|=1$ and $|z/w|=|z|/|w|$ . By defining the distance function d(z, w) = |z − w| we turn the set of complex numbers into a metric space and we can therefore talk about limits and continuity.

The complex conjugate of the complex number z = a + bi is defined to be a − bi, written as ${\bar {z}}$ or $z^{*}\,$ . As seen in the figure, ${\bar {z}}$ is the "reflection" of z about the real axis. The following can be checked:

{\overline {z+w}}={\bar {z}}+{\bar {w}}

{\overline {z\cdot w}}={\bar {z}}\cdot {\bar {w}}

{\overline {(z/w)}}={\bar {z}}/{\bar {w}}

{\bar {\bar {z}}}=z

{\bar {z}}=z

if and only if z is real

|z|=|{\bar {z}}|

|z|^{2}=z\cdot {\bar {z}}

z^{-1}={\bar {z}}\cdot |z|^{-2}

if z is non-zero.

The latter formula is the method of choice to compute the inverse of a complex number if it is given in rectangular coordinates.

That conjugation commutes with all the algebraic operations (and many functions; e.g. $\sin {\bar {z}}={\overline {\sin z}}$ ) is rooted in the ambiguity in choice of i (−1 has two square roots). It is important to note, however, that the function $f(z)={\bar {z}}$ is not complex-differentiable (see holomorphic function).

Complex fractions[편집]

We can divide a complex number (a + bi) by another complex number (c + di) ≠ 0 in two ways. The first way has already been implied: to convert both complex numbers into exponential form, from which their quotient is easily derived. The second way is to express the division as a fraction, then to multiply both numerator and denominator by the complex conjugate of the denominator. The new denominator is a real number.

{\begin{aligned}{a+bi \over c+di}&={(a+bi)(c-di) \over (c+di)(c-di)}={(ac+bd)+(bc-ad)i \over c^{2}+d^{2}}\\&=\left({ac+bd \over c^{2}+d^{2}}\right)+i\left({bc-ad \over c^{2}+d^{2}}\right).\,\end{aligned}}

Matrix representation of complex numbers[편집]

While usually not useful, alternative representations of the complex field can give some insight into its nature. One particularly elegant representation interprets each complex number as a 2×2 matrix with real entries which stretches and rotates the points of the plane. Every such matrix has the form

{\begin{bmatrix}a&-b\\b&\;\;a\end{bmatrix}}

where a and b are real numbers. The sum and product of two such matrices is again of this form. Every non-zero matrix of this form is invertible, and its inverse is again of this form. Therefore, the matrices of this form are a field. In fact, this is exactly the field of complex numbers. Every such matrix can be written as

{\begin{bmatrix}a&-b\\b&\;\;a\end{bmatrix}}=a{\begin{bmatrix}1&\;\;0\\0&\;\;1\end{bmatrix}}+b{\begin{bmatrix}0&-1\\1&\;\;0\end{bmatrix}}

which suggests that we should identify the real number 1 with the identity matrix

{\begin{bmatrix}1&\;\;0\\0&\;\;1\end{bmatrix}},

and the imaginary unit i with

{\begin{bmatrix}0&-1\\1&\;\;0\end{bmatrix}},

a counter-clockwise rotation by 90 degrees. Note that the square of this latter matrix is indeed equal to the 2×2 matrix that represents −1.

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}

If the matrix is viewed as a transformation of the plane, then the transformation rotates points through an angle equal to the argument of the complex number and scales by a factor equal to the complex number's absolute value. The conjugate of the complex number z corresponds to the transformation which rotates through the same angle as z but in the opposite direction, and scales in the same manner as z; this can be represented by the transpose of the matrix corresponding to z.

If the matrix elements are themselves complex numbers, the resulting algebra is that of the quaternions. In other words, this matrix representation is one way of expressing the Cayley-Dickson construction of algebras.

Geometric interpretation of the operations on complex numbers[편집]

Consider a plane. One point is the origin, 0. Another point is unity, or 1.

The sum of two points A and B is the point X = A + B such that the triangles with vertices 0, A, B, and X, B, A, are congruent.

The product of two points A and B is the point X = AB such that the triangles with vertices 0, 1, A, and 0, B, X, are similar.

The complex conjugate of a point A is the point X = A* such that the triangles with vertices 0, 1, A, and 0, 1, X, are mirror images of each other.

Interpretation of the roots of quadratics as complex numbers[편집]

A lot of Precalculus students usually ask questions as to the significance of the "non-real" roots of a quadratic equation on the Complex Plane. The solution to this is as follows: (r and s are the zeroes of the quadratic)

f(x)=x^{2}+Ax+B\,

-A=r+s\,

B=rs\,

r=a+bi\,

s=a-bi\,

Therefore....

$-A=2a$

$B=a^{2}+b^{2}$

So, if you draw the arbitrary points a + bi and a − bi on the complex plane, and you notice the right triangle, you can find what a and b actually are −A/2 and (4B − A²)^1/2/2 respectively. Thus we have the quadratic formula.

Note: First, remember to write it in a (plusorminus)bi form; second, this is not by any means an actual proof of the quadratic formula; it just shows the relevance of the complex zeroes of a quadratic on a complex plane. Lastly, this is only plausible whenever a and b are real numbers.

Some properties[편집]

Real vector space[편집]

C is a two-dimensional real vector space. Unlike the reals, the set of complex numbers cannot be totally ordered in any way that is compatible with its arithmetic operations: C cannot be turned into an ordered field.

R-linear maps C → C have the general form

f(z)=az+b{\overline {z}}

with complex coefficients a and b. Only the first term is C-linear, and only the first term is holomorphic; the second term is real-differentiable, but does not satisfy the Cauchy-Riemann equations.

The function

f(z)=az\,

corresponds to rotations combined with scaling, while the function

f(z)=b{\overline {z}}

corresponds to reflections combined with scaling.

Solutions of polynomial equations[편집]

A root of the polynomial p is a complex number z such that p(z) = 0. A surprising result in complex analysis is that all polynomials of degree n with real or complex coefficients have exactly n complex roots (counting multiple roots according to their multiplicity). This is known as the fundamental theorem of algebra, and it shows that the complex numbers are an algebraically closed field.

Indeed, the complex number field is the algebraic closure of the real number field, and Cauchy constructed the field of complex numbers in this way. It (C) can also be characterized as the quotient ring of the polynomial ring R[X] over the ideal generated by the polynomial X² + 1:

\mathbb {C} =\mathbb {R} [X]/(X^{2}+1).\,

This is indeed a field because X² + 1 is irreducible, hence generating a maximal ideal, in R[X]. The image of X in this quotient ring is the imaginary unit i.

Algebraic characterization[편집]

The field C is (up to field isomorphism) characterized by the following three facts:

its characteristic is 0
its transcendence degree over the prime field is the cardinality of the continuum
it is algebraically closed

Consequently, C contains many proper subfields which are isomorphic to C. Another consequence of this characterization is that the Galois group of C over the rational numbers is enormous, with cardinality equal to that of the power set of the continuum.

Characterization as a topological field[편집]

As noted above, the algebraic characterization of C fails to capture some of its most important properties. These properties, which underpin the foundations of complex analysis, arise from the topology of C. The following properties characterize C as a topological field:

C is a field.
C contains a subset P of nonzero elements satisfying:
- P is closed under addition, multiplication and taking inverses.
- If x and y are distinct elements of P, then either x-y or y-x is in P
- If S is any nonempty subset of P, then S+P=x+P for some x in C.
C has a nontrivial involutive automorphism x→x*, fixing P and such that xx* is in P for any nonzero x in C.

Given these properties, one can then define a topology on C by taking the sets

$B(x,p)=\{y|p-(y-x)(y-x)^{*}\in P\}$

as a base, where x ranges over C, and p ranges over P.

To see that these properties characterize C as a topological field, one notes that P ∪ {0} ∪ -P is an ordered Dedekind-complete field and thus can be identified with the real numbers R by a unique field isomorphism. The last property is easily seen to imply that the Galois group over the real numbers is of order two, completing the characterization.

Pontryagin has shown that 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 by noting that the nonzero complex numbers are connected, while the nonzero real numbers are not.

Complex analysis[편집]

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.

Applications[편집]

The words "real" and "imaginary" were meaningful when complex numbers were used mainly as an aid in manipulating "real" numbers, with only the "real" part directly describing the world. Later applications, and especially the discovery of quantum mechanics, showed that nature has no preference for "real" numbers and its most real descriptions often require complex numbers, the "imaginary" part being just as physical as the "real" part.

Control theory[편집]

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 system has poles that are

in the right half plane, it will be unstable,
all in the left half plane, it will be stable,
on the imaginary axis, it will have marginal stability.

If a system has zeros in the right half plane, it is a nonminimum phase system.

Signal analysis[편집]

Complex numbers are used in signal analysis and other fields as a convenient description for periodically varying signals. The absolute value |z| is interpreted as the amplitude and the argument arg(z) as the phase of a sine wave of given frequency.

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

f(t)=ze^{i\omega t}\,

where ω represents the angular frequency and the complex number z encodes the phase and amplitude as explained above.

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, frequency-dependent resistances for the latter two and combining all three in a single complex number called the impedance. (Electrical engineers and some physicists use the letter j for the imaginary unit since i is typically reserved for varying currents and may come into conflict with i.) 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.

Improper integrals[편집]

In applied fields, complex numbers are often used to compute certain real-valued improper integrals, by means of complex-valued functions. Several methods exist to do this; see methods of contour integration.

Quantum mechanics[편집]

The complex number field is relevant in the mathematical formulation 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 the Heisenberg's matrix mechanics -- make use of complex numbers.

Relativity[편집]

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.) Complex numbers are essential to spinors which are a generalization of the tensors used in relativity.

Applied mathematics[편집]

In differential equations, it is common to first find all complex roots r of the characteristic equation of a linear differential equation and then attempt to solve the system in terms of base functions of the form f(t) = e^rt.

Fluid dynamics[편집]

In fluid dynamics, complex functions are used to describe potential flow in two dimensions.

Fractals[편집]

Certain fractals are plotted in the complex plane e.g. Mandelbrot set and Julia set.

History[편집]

The earliest fleeting reference to square roots of negative numbers perhaps occurred in the work of the Greek mathematician and inventor Heron of Alexandria in the 1st century AD, when he considered the volume of an impossible frustum of a pyramid ^[1], though negative numbers were not conceived in the Hellenistic world.

Complex numbers became more prominent in the 16th century, when closed formulas 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. For example, Tartaglia's cubic formula gives the following solution to the equation x³ − x = 0:

{\frac {1}{\sqrt {3}}}\left({\sqrt {-1}}^{1/3}+{\frac {1}{{\sqrt {-1}}^{1/3}}}\right).

At first glance this looks like nonsense. However formal calculations with complex numbers show that the equation z³ = i has solutions −i, ${\scriptstyle {\frac {\sqrt {3}}{2}}}+{\scriptstyle {\frac {1}{2}}}i$ and ${\scriptstyle {\frac {-{\sqrt {3}}}{2}}}+{\scriptstyle {\frac {1}{2}}}i$ . Substituting these in turn for ${\scriptstyle {\sqrt {-1}}^{1/3}}$ in Tartaglia's cubic formula and simplifying, one gets 0, 1 and −1 as the solutions of x³ − x = 0.

This was doubly unsettling since not even negative numbers were considered to be on firm ground at the time. The term "imaginary" for these quantities was coined by René Descartes in 1637 and was meant to be derogatory (see imaginary number for a discussion of the "reality" of complex numbers). A further source of confusion was that the equation ${\sqrt {-1}}^{2}={\sqrt {-1}}{\sqrt {-1}}=-1$ seemed to be capriciously inconsistent with the algebraic identity ${\sqrt {a}}{\sqrt {b}}={\sqrt {ab}}$ , which is valid for positive real numbers a and b, and which was also used in complex number calculations with one of a, b positive and the other negative. The incorrect use of this identity (and the related identity ${\frac {1}{\sqrt {a}}}={\sqrt {\frac {1}{a}}}$ ) in the case when both a and b are negative even bedeviled Euler. This difficulty eventually led to the convention of using the special symbol i in place of ${\sqrt {-1}}$ to guard against this mistake.

The 18th century saw the labors of Abraham de Moivre and Leonhard Euler. To De Moivre is due (1730) the well-known formula which bears his name, de Moivre's formula:

(\cos \theta +i\sin \theta )^{n}=\cos n\theta +i\sin n\theta \,

and to Euler (1748) Euler's formula of complex analysis:

\cos \theta +i\sin \theta =e^{i\theta }.\,

The existence of complex numbers was not completely accepted until the geometrical interpretation (see below) had been described by Caspar Wessel in 1799; it was rediscovered several years later and popularized by Carl Friedrich Gauss, and as a result the theory of complex numbers received a notable expansion. The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in Wallis's De Algebra tractatus.

Wessel's memoir appeared in the Proceedings of the Copenhagen Academy for 1799, and is exceedingly clear and complete, even in comparison with modern works. He also considers the sphere, and gives a quaternion theory from which he develops a complete spherical trigonometry. In 1804 the Abbé Buée independently came upon the same idea which Wallis had suggested, that $\pm {\sqrt {-1}}$ should represent a unit line, and its negative, perpendicular to the real axis. Buée's paper was not published until 1806, in which year Jean-Robert Argand also issued a pamphlet on the same subject. It is to Argand's essay that the scientific foundation for the graphic representation of complex numbers is now generally referred. Nevertheless, in 1831 Gauss found the theory quite unknown, and in 1832 published his chief memoir on the subject, thus bringing it prominently before the mathematical world. Mention should also be made of an excellent little treatise by Mourey (1828), in which the foundations for the theory of directional numbers are scientifically laid. The general acceptance of the theory is not a little due to the labors of Augustin Louis Cauchy and Niels Henrik Abel, and especially the latter, who was the first to boldly use complex numbers with a success that is well known.

The common terms used in the theory are chiefly due to the founders. Argand called $\cos \phi +i\sin \phi$ the direction factor, and $r={\sqrt {a^{2}+b^{2}}}$ the modulus; Cauchy (1828) called $\cos \phi +i\sin \phi$ the reduced form (l'expression réduite); Gauss used i for ${\sqrt {-1}}$ , introduced the term complex number for $a+bi$ , and called $a^{2}+b^{2}$ the norm.

The expression direction coefficient, often used for $\cos \phi +i\sin \phi$ , is due to Hankel (1867), and absolute value, for modulus, is due to Weierstrass.

Following Cauchy and Gauss have come a number of contributors of high rank, of whom the following may be especially mentioned: Kummer (1844), Leopold Kronecker (1845), Scheffler (1845, 1851, 1880), Bellavitis (1835, 1852), Peacock (1845), and De Morgan (1849). Möbius must also be mentioned for his numerous memoirs on the geometric applications of complex numbers, and Dirichlet for the expansion of the theory to include primes, congruences, reciprocity, etc., as in the case of real numbers.

A complex ring or field is a set of complex numbers which is closed under addition, subtraction, and multiplication. Gauss studied complex numbers of the form $a+bi$ , where a and b are integral, or rational (and i is one of the two roots of $x^{2}+1=0$ ). His student, Ferdinand Eisenstein, studied the type $a+b\omega$ , where $\omega$ is a complex root of $x^{3}-1=0$ . Other such classes (called cyclotomic fields) of complex numbers are derived from the roots of unity $x^{k}-1=0$ for higher values of $k$ . This generalization is largely due to Kummer, who also invented ideal numbers, which were expressed as geometrical entities by Felix Klein in 1893. The general theory of fields was created by Évariste Galois, who studied the fields generated by the roots of any polynomial equation