Complex Numbers

In Mathematics , a complex number is an expression of the form
:

a + bi \,

where ''a'' and ''b'' are Real Number s, and ''i'' is a specific Imaginary Number , called 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''.

For example, 3 + 2''i'' is a ''complex number'', with real part 3 and imaginary part 2.

Complex numbers can be added, subtracted, multiplied, and divided in a similar way to real numbers; but they have additional elegant properties. For example, every Polynomial algebraic equation has a complex number as a solution, not just some, as in the real numbers.

In some fields (in particular, Electrical Engineering , where ''i'' is a symbol for Current ), complex numbers are written as ''a'' + ''bj''.

DEFINITIONS

Notation and operations

The Set of all complex numbers is usually denoted by C, or in Blackboard Bold by

\mathbb{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 ''i'' ² = −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'' + ''bd i'' 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 complex number field

Formally, the complex numbers can be defined as Ordered Pair s 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.

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''): $\left({a\over a^2+b^2},{-b\over a^2+b^2}
ight).$

C can also be defined as the Topological Closure of the Algebraic Number s 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 ).

{r_2 e^{i\phi_2}}
= rac{r_1}{r_2} e^{i (\phi_1 - \phi_2)}. \,

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 (

\pi/2

Radian s). The geometric content of the equation ''i''² = −1 is that a sequence of two 90 degree rotations results in a 180 degree (

\pi

radians) 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

:<math> Z 0 \,</math> Iff <math> z = 0 \,</math>
:<math> Z W z \ w \,</math>
For All Complex Numbers ''z'' And ''w'' It Then Follows, For Example, That <math> 1 1 </math> and <math>z/w=z/w</math> By defining the distance function ''d''(''z'', ''w'') = ''z'' &minus ''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

:

\bar{z}=z

If And Only If ''z'' is real

: <math>z^2 z\bar{z}</math>
: <math>z^{-1} \bar{z}z^{-2}</math> &nbsp if ''z'' is non-zero
<math>B(x,p) \{y p - (y-x)(y-x)^\in P\}</math>
Complex Numbers Are Used In "http://wwwinformationdelightinfo/encyclopedia/entry/signal_analysis" class="copylinks">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

At first glance this looks like nonsense. However formal calculations with complex numbers show that the equation

z^3=i

has solutions −''i'',

rac{\sqrt{3}}{2}+rac{1}{2}i

and

rac{-\sqrt{3}}{2}+rac{1}{2}i

. Substituting these in turn for

\sqrt{-1}^{1/3}

into the cubic formula and simplifying, one gets 0, 1 and −1 as the solutions of

x^3-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

rac{1}{\sqrt{a}}=\sqrt{rac{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 	heta + i\sin 	heta)^{n} = \cos n 	heta + i\sin n 	heta \,

and to Euler (1748) Euler's Formula of Complex Analysis :

:

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

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 Number s, 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

:

\ F(x) = 0.

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 .

SEE ALSO

Circular Motion#Using Complex Numbers

Complex Geometry

De Moivre's Formula

Euler's Identity

Hypercomplex Number

Leonhard Euler

Local Field

Mandelbrot Set

Phasor (physics)

Phasor (electronics)

Quaternion

Riemann Sphere (extended complex plane)

Split-complex Number

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Information About

Complex Numbers