The complex numbers are a natural extension of the real numbers: the real number line can be viewed as a subset of the complex number plane. Every complex number can be visualized as a point in the plane; using the definitions below, it becomes possible to add, subtract, multiply and divide such points.
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)
So defined, the complex numbers form a field, the complex number field, denoted by C (or more properly by the unicode character ℂ, if your browser supports unicode display). If we identify the real number a with the complex number (a, 0), then the field of real numbers R becomes a subfield of C. Furthermore, C forms 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
Geometry: Rectangular and Polar Coordinates
The complex number i = (0, 1) is called the imaginary unit. It is an imaginary number and it satisfies i2 = -1 (remember that we don't distinguish between (-1, 0) and -1). Every complex number z can then be written as z = a + i b, where a and b are real numbers uniquely determined by z. The number a is called the real part of z and b is the imaginary part of z.
Geometrically, the algebraic operations on complex numbers can be understood as follows: to add two complex numbers z1 = a1 + ib1 and z2 = a2 + ib2, we think of them as vectors in the x-y plane pointing from the origin to the point (a1, b1) and (a2, b2), respectively. Then we translate (move) the second vector, without changing its direction, so that its base point coincides with the first vector's tip; the second vector's tip will then correspond to the complex number z1 + z2. In order to multiply the two numbers z1 and z2, we first measure their respective counter clockwise angles with the positive x-axis and add these angles up: the resulting angle corresponds to the product vector z1 · z2. The length of this product vector is given by the product of the lengths of the two original vectors. Multiplication with a fixed complex number can
therefore be seen as a simultaneous rotation and stretching. Multiplication with i corresponds to a counter clockwise rotation by 90 degrees. Even the fact (-1) · (-1) = +1 from arithmetic can be understood geometrically as the combination of two 180 degree turns.
Sometimes, the representation of complex numbers in the form z = a + i b ("rectangular coordinates") is not as convenient as using polar coordinates: every non-zero complex number z can be written as z = r eiφ where r is a positive real number and the angle φ is a real number (see Euler's formula). Every pair (r, φ) of "polar coordinates" defines a unique non-zero complex number z in this fashion. The angle φ however is not uniquely defined by z since Euler's formula implies z = r ei(φ + 2πk) for any integer k. Because of this, one generally restricts φ to the interval (-π, π] and calls it the principal argument of z and writes φ = arg(z). With this convention, the polar coordinates are uniquely defined by z.
The multiplication of complex numbers is especially easy if the numbers are represented in polar coordinates: r eiφ · s eiψ = (rs) ei(φ + ψ). Division of complex numbers as well as exponentiation are also easy in polar coordinates. Addition however is cumbersome in this representation.
Absolute value, conjugation and distance
The absolute value or magnitude of a complex number z is its euclidean distance from the origin, if we think of z as a point in the plane. We denote it by |z|; it is always a non-negative real number. Algebraically, if z = a + ib, we can define |z| = sqrt(a2 + b2 ). If the complex number z is written in polar coordinates z = r eiφ, then |z| = r.
One can check readily that the absolute value has three important properties:
- |z + w| ≤ |z| + |w|
- |z w| = |z| |w|
- |z / w| = |z| / |w|
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 z* = a - ib. The following can be checked:
- (z + w)* = z* + w*
- (zw)* = z* w*
- (z/w)* = z* / w*
- z* = z if and only if z is real
- |z|2 = z z*
- z-1 = z* / |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.
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. Because of this, mathematicians sometimes consider the complex numbers to be more "natural" than the real numbers: all polynomial equations have solutions among the complex numbers, which is not true for the real numbers.
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 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 forth 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 negative numbers themselves were not 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 existance of complex numbers was not completely accepted until the above mentioned geometrical interpretation had been described by Wessel in 1799, 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.
Complex numbers are used in electrical engineering and other fields as a convenient description for periodically varying signals (see Fourier analysis). In an expression z = r eiφ one may think of r as the amplitude and φ as the phase of a sine wave of given frequency. Furthermore, when representing a sinusoidal current or voltage as the real part of a complex valued function of the form
- f(t) = z eiωt
where ω represents the angular frequency and the complex number z encodes the phase and amplitude, the treatment of resistors, capacitors and inductors can be unified by introducing imaginary resistances for the latter two (see electrical network). Electrical engineers and physicists use the letter j for the imaginary unit since i is typically reserved for varying currents.
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.
Alternative representations 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
/ a -b \ \ b a /
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
/ a -b \ / 1 0 \ / 0 -1 \ \ b a / = a \ 0 1 / + b \ 1 0 /
which suggests that we should identify the real number one with the matrix
/ 1 0 \ \ 0 1 /
and the imaginary unit with
/ 0 -1 \ \ 1 0 /
a rotation by 90 degrees. Note that the square of this latter matrix is indeed equal to -1.
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.
- complex geometry
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- Leonhard Euler
- the most remarkable formula in the world