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 i^{2}= -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:
- (a+ib) + (c+id) = (a+c) + i(b+d)
- (a+ib) · (c+id) = ac-bd + i (bc+ad)
History
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.
Definition
Formally we may define complex numbers as ordered pairs of real numbers (a, b) together with the operations:
- (a, b) + (c, d) = (a + c, b + d)
- (a, b) · (c, d) = (ac - bd, bc + ad)
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.
Geometry
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
- z = x + iy = r (cos φ + i sin φ).
Multiplication with i corresponds to a counter clockwise rotation by 90 degrees. The geometric content of the equation i^{2} = -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.
Absolute value, conjugation and distance
Recall that the absolute value (or modulus or magnitude) of a complex number z = r e^{iφ} is defined as |z| = r. Algebraically, if z = a + ib, then |z| = &radic(a^{2} + b^{2} ).
One can check readily that the absolute value has three important properties:
- |z + w| ≤ |z| + |w|
- |z w| = |z| |w|
- |z / w| = |z| / |w|
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:
- if and only if z is real
- if z is non-zero
The complex argument of z=re^{iφ} is φ. Note that the complex argument is unique up to modulo 2π.
Matrix representation of complex numbers
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.
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.
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. It can be identified as the quotient ring of the polynomial ring R[X] by the ideal generated by the polynomial X^{2} + 1:
- C = R[X] / (X^{2} + 1).
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
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
- f(t) = z e^{iωt}
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) = e^{rt}.
In fluid dynamics, complex functions are used to describe potential flow in 2d.
See also
quaternions, complex geometry, local fields, phasors, Leonhard Euler, the most remarkable formula in the world, Hypercomplex number, De Moivre's formula, Complex numbers at Wikibooks