:
where ''a'' and ''b'' are Real Number s, 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 ''. Real numbers may be considered to be complex numbers with an imaginary part of zero; that is, the real number ''a'' is equivalent to the complex number ''a''+0''i''.
For example, 3 + 2''i'' is a ''complex number'', with real part 3 and imaginary part 2. If ''z'' = ''a'' + ''bi'', the real part (''a'') is denoted Re(''z''), or ℜ(''z''), and the imaginary part (''b'') is denoted Im(''z''), or ℑ(''z'').
Complex numbers can be added, subtracted, multiplied, and divided like real numbers and have other elegant properties. For example, real numbers alone do not provide a solution for every Polynomial algebraic equation with real Coefficient s, while complex numbers do (this is the Fundamental Theorem Of Algebra ). 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 .
In some fields (in particular, Electrical Engineering , where ''i'' is a symbol for Current ), the Imaginary Unit ''i'' is instead written as ''j'', so complex numbers are sometimes written as ''a'' + ''jb''.
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''r e''<sup>''i''φ</sup> is defined as ''z'' = ''r'' Algebraically, if ''z'' = ''a'' + ''bi'', then <math> z = \sqrt{a^2+b^2}</math><!--keep sentence-terminator within math element to make it align better with the formula-->
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0 \,</math> If And Only If <math> z = 0 \,</math>
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"http://wwwinformationdelightinfo/information/entry/triangle_inequality" class="copylinks">Triangle Inequality )
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z \cdot w \,</math>
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1 </math> and <math>z/w=z/w</math> By defining the '''distance''' function ''d''(''z'', ''w'') = ''z'' &minus ''w'' we turn the set of complex numbers into a Metric Space and we can therefore talk about Limits and Continuity
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:
if and only if ''z'' is real
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z\cdot\bar{z}</math>
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\bar{z}\cdotz^{-2}</math>   if ''z'' is non-zero
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''a''+''ib'', the complex number ''z'' can be specified by Polar Coordinates The polar coordinates are ''r'' =  ''z'' ≥ 0, called the ''' Absolute Value ''' or '''modulus''', and φ = arg(''z''), called the '''argument''' of ''z'' For ''r'' = 0 any value of φ describes the same number To get a unique representation, a conventional choice is to set arg(0) = 0 For ''r'' > 0 the argument φ is unique Modulo 2π that is, if any two values of the complex argument differ by an exact Integer multiple of 2π, they are considered equivalent To get a unique representation, a conventional choice is to limit φ to the interval (-π,π], ie &minusπ < φ ≤ π The representation of a complex number by its polar coordinates is called the ''polar form'' of the complex number
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= rac{r_1}{r_2}\,e^{i ( arphi_1 - arphi_2)}. \,
Exponentiation with integer exponents; according to
De Moivre's Formula ,
:
Exponentiation with arbitrary complex exponents is discussed in the article on
Exponentiation .
The addition of two complex numbers is just the
Vector Addition of two vectors, and multiplication by a fixed complex number can be seen as a simultaneous rotation and stretching.
Multiplication by ''i'' corresponds to a counter-clockwise rotation by 90
Degrees (π/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 (π radians) rotation. Even the fact (−1) · (−1) = +1 from arithmetic can be understood geometrically as the combination of two 180 degree turns.
All the roots of any number, real or complex, may be found with a simple
Algorithm . The ''n''th roots are given by
:
for ''k'' = 0, 1, 2, …, ''n'' − 1, where
represents the principal ''n''th root of ''r''.
While usually not useful, alternative representations of the complex field can give some insight into its nature. One particularly elegant representation interprets each complex number as a 2×2
Matrix with
Real entries which stretches and rotates the points of the plane. Every such matrix has the form
:
where ''a'' and ''b'' are real numbers. The sum and product of two such matrices is again of this form. Every non-zero matrix of this form 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 identity 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 the 2×2 matrix that represents −1.
The square of the absolute value of a complex number expressed as a matrix is equal to the
Determinant of that matrix.
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\{y p - (y-x)(y-x)^\in P\}</math>
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"http://wwwinformationdelightinfo/information/entry/signal_analysis" class="copylinks">Signal Analysis and other fields for a convenient description for periodically varying signals For given real functions representing actual physical quantities, often in terms of sines and cosines, corresponding complex functions are considered of which the real parts are the original quantities For a Sine Wave of a given Frequency , the absolute value ''z'' of the corresponding ''z'' is the Amplitude and the argument arg(''z'') the Phase
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At first glance this looks like nonsense. However formal calculations with complex numbers show that the equation ''z''³ = ''i'' has solutions −''i'',
and
. Substituting these in turn for
in Tartaglia's 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
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. The incorrect use of this identity (and the related identity
) 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
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 :
:
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 ; 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
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
the ''direction factor'', and
the ''modulus''; Cauchy (1828) called
the ''reduced form'' (l'expression réduite); Gauss used ''i'' for
, introduced the term ''complex number'' for
, and called
the ''norm''.
The expression ''direction coefficient'', often used for
, 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
, where ''a'' and ''b'' are integral, or rational (and ''i'' is one of the two roots of
). His student,
Ferdinand Eisenstein , studied the type
, where
is a complex root of
. Other such classes (called
Cyclotomic Fields ) of complex numbers are derived from the
Roots Of Unity for higher values of
. 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
:
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 .
- ''An Imaginary Tale: The Story of '', by Paul J. Nahin; Princeton University Press; ISBN 0-691-02795-1 (hardcover, 1998). A gentle introduction to the history of complex numbers and the beginnings of complex analysis.
- ''Numbers'', by H.-D. Ebbinghaus, H. Hermes, F. Hirzebruch, M. Koecher, K. Mainzer, J. Neukirch, A. Prestel, R. Remmert; Springer; ISBN 0-387-97497-0 (hardcover, 1991). An advanced perspective on the historical development of the concept of number.
- ''The Road to Reality: A Complete Guide to the Laws of the Universe'', by Roger Penrose ; Alfred A. Knopf, 2005; ISBN 0-679-45443-8. Chapters 4-7 in particular deal extensively (and enthusiastically) with complex numbers.
- ''Unknown Quantity: A Real and Imaginary History of Algebra'', by John Derbyshire; Joseph Henry Press; ISBN 0-309-09657-X (hardcover 2006). A very readable history with emphasis on solving polynomial equations and the structures of modern algebra.
- ''Visual Complex Analysis'', by Tristan Needham ; Clarendon Press; ISBN 0-198-53447-7 (hardcover, 1997). History of complex numbers and complex analysis with compelling and useful visual interpretations.
