A number that is a pair of real numbers, (a,b). The algebra of complex numbers obeys these laws of addition: (a,b) + (c,d) = (a + c, b + d) and multiplication: (a,b)(c,d) = (ac ? bd, ad + bc). The real part, a, and the imaginary part, b, may be zero. The real numbers are the subset of complex numbers which arise on setting the imaginary part = 0: (a,0). Since (0,1)(0,1) = (?1,0), it is possible to regard (0,1) as a square root of ?1, often denoted i, and complex numbers can be written in the form a + bi. In the same way that real numbers are represented on the number-line, complex numbers are represented in the plane, so the complex number (a,b) is represented by the point with the co-ordinates (a,b). This representation is called the Argand diagram after the Swiss mathematician Jean Robert Argand (17681822).
In mathematics, a complex number is a number of the form
where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. The real number a is called the real part of the complex number, and the real number b is the imaginary part. When the imaginary part b is 0, the complex number is just the real number a.
Complex numbers can be added, subtracted, multiplied, and divided like real numbers, but they have additional elegant properties. For example, real numbers alone do not provide a solution for every polynomial algebraic equation with real coefficients, while complex numbers do (the fundamental theorem of algebra). It follows that complex numbers are written as a + bj.
Definitions
Equality
Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.
Notation and operations
The set of all complex numbers is usually denoted by C, or in blackboard bold by . 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:
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 complex number field
Formally, the complex numbers can be defined as ordered pairs of real numbers (a, b) together with the operations:
So defined, the complex numbers form a field, the complex number field, denoted by C.
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.
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):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
A complex number can be viewed as a point or a position vector on a two-dimensional Cartesian coordinate system called the complex plane or Argand diagram (named after Jean-Robert Argand).
The Cartesian coordinates of the complex number are the real part x and the imaginary part y, while the polar coordinates are r = |z|, called the absolute value or modulus, and φ = arg(z), called the complex argument of z (mod-arg form).
The complex argument of 0 is not defined by the equations above.
Note that for a non-zero complex number the complex argument is unique modulo 2π, that is, if any two values of the complex argument exactly differ by an integer multiple of 2π, they are considered equivalent.
By simple trigonometric identities, we see that
and that
Now the addition of two complex numbers is just the vector addition of two vectors, and the multiplication with a fixed complex number can be seen as a simultaneous rotation and stretching.
Absolute value, conjugation and distance
The absolute value (or modulus or magnitude) of a complex number z = r e is defined as |z| Algebraically, if z = a + ib, then
One can check readily that the absolute value has three important properties:
if and only if (triangle inequality)for all complex numbers z and w.
The complex conjugate of the complex number z = a + ib is defined to be a - ib, written as or .
The latter formula is the method of choice to compute the inverse of a complex number if it is given in rectangular coordinates.
Complex fractions
We can divide a complex number (a + bi) by another complex number (c + di) whose magnitude is non-zero in two ways. This causes the denominator to simplify into a real number:
Matrix representation of complex numbers
While usually not useful, alternative representations of complex fields can give some insight into their nature. Every such matrix has the form
with real numbers a and b. In fact, this is exactly the field of complex numbers. Every such matrix can be written as
which suggests that we should identify the real number 1 with the matrix
and the imaginary unit i with
a counter-clockwise rotation by 90 degrees.
The absolute value of a complex number expressed as a matrix is equal to the square root of the determinant of that matrix.
If the matrix is viewed as a transformation of a 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.
If the matrix elements are themselves complex numbers, then the resulting algebra is that of the quaternions.
Geometric interpretation of the operations on complex numbers
Consider a plane.
Addition
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.
Multiplication
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.
Conjugation
The complex conjugate of a point A is a point X = A* such that the triangles with vertices 0, 1, A and 0, 1, X are mirror image of each other.
Some properties
Real vector space
C is a two-dimensional real vector space. Unlike the reals, complex numbers cannot be 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
with complex coefficients a and b.
The function
corresponds to rotations combined with scaling, while the function
corresponds to reflections combined with scaling.
Solutions of polynomial equations
A root of the polynomial p is a complex number z such that p(z) = 0. A most striking result 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 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 complex numbers in this way.
Algebraic characterization
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 closedConsequently, C contains many proper subfields which are isomorphic to C.
Characterization as a topological field
As noted above, the algebraic characterization of C fails to capture some of its most important properties.
Given these properties, one can then define a topology on C by taking the sets
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. This gives another characterization of C as a topological field, since C can be distinguished from R by noting the nonzero complex numbers are connected whereas the nonzero real numbers are not.
Complex analysis
The study of functions of a complex variable is known as complex analysis and has enormous practical use in applied mathematics as well as in other branches of mathematics. Often, the most natural proofs for statements in real analysis or even number theory employ techniques from complex analysis (see prime number theorem for an example). Unlike real functions which are commonly represented as two dimensional graphs, complex functions have four dimensional graphs and may usefully be illustrated by color coding a three dimensional graph to suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane.
Applications
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.
Signal analysis
Complex numbers are used in signal analysis and other fields as a convenient description for periodically varying signals.
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
where ω represents the angular frequency and the complex number z encodes the phase and amplitude as explained above.
Improper integrals
In applied fields, the use of complex analysis is often used to compute certain real-valued improper integrals, by means of complex-valued functions.
Quantum mechanics
The complex number field is also of utmost importance in quantum mechanics since the underlying theory is built on (infinite dimensional) Hilbert spaces over C.
Applied mathematics
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.
Fluid dynamics
In fluid dynamics, complex functions are used to describe potential flow in 2d.
Fractals
Certain fractals are plotted in the complex plane e.g.
History
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 CE, when he considered the volume of an impossible frustum of a pyramid , 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). However formal calculations with complex numbers show that the equation z − 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 seemed to be capriciously inconsistent with the algebraic identity , 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. To De Moivre is due (1730) the well-known formula which bears his name, de Moivre's formula:
and to Euler (1748) Euler's formula of complex analysis:
The existence of complex numbers was not completely accepted until the geometrical interpretation (see below) had been described by Caspar Wessel in 1799; 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. 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 − 1 = 0. Other such classes (called cyclotomic fields) of complex numbers are derived from the roots of unity x − 1 = 0 for higher values of k. The general theory of fields was created by Évariste Galois, who studied the fields generated by the roots of any polynomial equation
The late writers (from 1884) on the general theory include Weierstrass, Schwarz, Richard Dedekind, Otto Hölder, Berloty, Henri Poincaré, Eduard Study, and Alexander MacFarlane.
The formally correct definition using pairs of real numbers was given in the 19th century. A gentle introduction to the history of complex numbers and the beginnings of complex analysis.
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