Complex number Totally Explained

Everything about Complex Number totally explained

In mathematics, the complex numbers are the extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:
ť

i^2=-1.,

Every complex number can be written 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.
   Complex numbers have addition, subtraction, multiplication, and division operations which extend the corresponding operations on real numbers, although with a number of additional elegant and useful properties. Notably, negative real numbers can be obtained by squaring complex numbers.
   Complex numbers were first discovered by Cardan, who called them "fictitious", during his attempts to find solutions to cubic equations . The solution of a general cubic equation may require intermediate calculations containing the square roots of negative numbers, even when the final solutions are real numbers, a situation known as casus irreducibilis. This ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, it's always possible to find solutions to polynomial equations of degree one or higher.
   Complex numbers are used in many different fields including applications in engineering, electromagnetism, quantum physics, applied mathematics, and chaos theory. When the underlying field of numbers for a mathematical construct is the field of complex numbers, the name usually reflects that fact. Examples are complex analysis, complex matrix, complex polynomial and complex Lie algebra.

Definitions

Notation

The set of all complex numbers is usually denoted by C, or in blackboard bold by

mathbb

, 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 + bomega

, 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 numbers, 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 ť

F(x) = 0.

The late writers (from 1884) on the general theory include Weierstrass, Schwarz, Richard Dedekind, Otto Hölder, Bonaventure Berloty, Henri Poincaré, Eduard Study, and Alexander MacFarlane.

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