The complex numbers are an extension of the real numbers, in which all polynomials have roots. The complex numbers contain a number i, the imaginary unit, with i2= -1. Every complex number can be represented in the form x+iy, where x and y are real numbers called the real part and the imaginary part of the complex number respectively.
The sum and product of two complex numbers are:
The earliest fleeting reference to square roots of negative numbers
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. They became more
prominent when in the 16th century closed formulas for the roots
of third and fourth degree polynomials were discovered by Italian
mathematicians (see Tartaglia, 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. 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
the 17th century
and was meant to be derogatory. 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. The
formally correct definition using pairs of real numbers was given in
the 19th century.
Formally we may define complex numbers as ordered pairs of real numbers (a, b) together with the operations:
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).
C could also be defined as the topological closure of algebraic numbers and the algebraic closure of R.
A complex number can also be viewed as a point or a position vector on the two dimensional Cartesian coordinate system. This representation is sometimes called an Argand diagram. In the figure, we have
History
Definition
So defined, the complex numbers form a field, the complex number
field, denoted by C (or in blackboard bold).Geometry
The latter expression is sometimes shorthanded as r cis φ, where r is called the absolute value of z and φ is called the complex argument of z.
However, Euler's formula states that ei φ = cisφ. The exponential form gives us a better insight then the shorthand rcisφ, which is almost never used in serious mathematical articles.
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.
Multiplication with i corresponds to a counter clockwise rotation by 90 degrees. The geometric content of the equation i2 = -1 is that a sequence of two 90 degree rotation results in a 180 degree rotation. Even the fact (-1) · (-1) = +1 from arithmetic can be understood geometrically as the combination of two 180 degree turns.
Recall that the absolute value (or modulus or magnitude) of a complex number z = r eiφ is defined as |z| = r. Algebraically, if z = a + ib, then |z| = &radic(a2 + b2 ).
One can check readily that the absolute value has three important properties:
Absolute value, conjugation and distance
for all complex numbers z and w. By defining the distance function d(z, w) = |z - w| we turn the complex numbers into a metric space and we can therefore talk about limits and continuity. The addition, subtraction, multiplication and division of complex numbers are then continuous operations. Unless anything else is said, this is always the metric being used on the complex numbers.
The complex conjugate of the complex number z = a + ib is defined to be a - ib, written as or z*. As seen in the figure, is the "reflection" of z about the real axis. The following can be checked:
The complex argument of z=reiφ is φ. Note that the complex argument is unique up to modulo 2π.
While usually not useful, alternative representations of complex field can give some insight into their nature. One particularly elegant representation interprets every complex number as 2x2 matrix with real entries which stretches and rotates the points of the plane. Every such matrix has the form
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. 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 described by the transpose of the matrix corresponding to z.
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.
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. It can be identified as the quotient ring of the polynomial ring R[X] by the ideal generated by the polynomial X2 + 1:
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.
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 the real part of complex valued functions of the form
In electrical engineering, this is done for varying voltages and currentss. 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.)
The residue theorem of complex analysis is often used in applied fields to compute certain improper integrals.
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.
In Special and general relativity, some formulas for the metric on spacetime become simpler if one takes the time variable to be imaginary.
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) = ert.
In fluid dynamics, complex functions are used to describe potential flow in 2d.
quaternions, complex geometry, local fields, phasors, Leonhard Euler, the most remarkable formula in the world, Hypercomplex number, De Moivre's formula, Complex numbers at WikibooksMatrix representation of complex numbers
with real numbers a and b. The sum and product of two such matrices is again of this form. Every non-zero such matrix 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
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. Note that the square of this latter matrix is indeed equal to -1.Some properties
Real vector space
Solutions of polynomial equations
This is indeed a field because X2 + 1 is irreducible. The image of X in this quotient ring becomes the imaginary unit i.Complex analysis
Applications
where ω represents the angular frequency and the complex
number z encodes the phase and amplitude as explained above.See also