This document discusses complex numbers. It begins by providing a brief history, noting that Gerolamo Cardano first conceived of complex numbers in 1545 while solving cubic equations. It then states that complex numbers form an algebraically closed field where every polynomial equation has a root. Many mathematicians contributed to fully developing the rules for addition, subtraction, multiplication and division of complex numbers. Complex numbers are used in fields like engineering, electromagnetism, quantum physics and more. They can represent AC voltage with potential as the real part and phase angle as the imaginary part. Complex numbers provide a way to solve equations like x2 = -4 by introducing an imaginary unit i where i2 = -1.

1. Complex Numbers
History in brief
Gerolamo Cardano was first to conceive idea of complex numbers
in around 1545, while finding the solution in radicals (without
trigonometric functions) of a cubic equation in some special
cases. Work on the problem of general polynomials ultimately led
to the fundamental theorem of algebra, which shows that with
complex numbers, a solution exists to every polynomial equation
of degree one or higher.
Complex numbers thus form an algebraically closed field, where
any polynomial equation has a root.
Many mathematicians contributed to the full development of
complex
numbers.
The
rules
for
addition,
subtraction,
multiplication, and division of complex numbers were developed
by the Italian mathematician Rafael Bombelli. A more abstract
formalism for the complex numbers was further developed by the
Irish mathematician William Rowan Hamilton.
Applications
Complex numbers are used in a number of fields, including:
engineering,
electromagnetism,
quantum
physics,
applied
mathematics, and chaos theory. Some fields of complex numbers
are complex analysis, complex matrix, complex polynomial, and

2. complex Lie algebra (In mathematics, Lie algebras are algebraic
structures which
were
introduced
to
study
the
concept
of infinitesimal transformations.
The term "Lie algebra" (after Sophus Lie) was introduced
by Hermann Weyl in the 1930s. In older texts, the name
"infinitesimal group" is used ) Some applications of complex
numbers are for representing AC Voltage which contains Potential
as the real part and Phase angle as the imaginary part. It is also
used to represent sinusoidal varying signals in Signal analysis.
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 complex valued functions.
Introduction
Consider the equation x2 + 4 = 0. This is the same as x2 = – 4.
Solving this equation means finding a number whose square is –
4. Clearly x cannot be a real number, since the square of a real
number cannot be negative, i.e., we cannot find a real number
whose square is – 4. Similarly, the equations of the type x2 + 1 =
0, x2 + 5 = 0, x2 + 9 = 0, etc. cannot have their roots as real

3. numbers. In order to overcome this difficulty, we have to introduce
a new type of numbers, whose square is negative.
 1 is
a non-real number. Let us denote it by the symbol i and call
it the imaginary unit.
Thus,
i  1
and hence i2 = – 1.
Definition
A number of the form x + iy, where x, y are real numbers and
i  1
is defined as a complex number.
If Z = x + iy, then x is called real part of z and y is called the
imaginary part of z and are denoted by Re (z) = x, Im (z) = y,
If x = 0 and y ≠ 0, then z = 0 + iy = iy is called purely imaginary
number.
If x ≠ 0 and y = 0, then z = x + i0 = x is a real number. Because of
this property the complex number system is considered as an
extension of real number system.
Algebra Of Complex Numbers
In performing operations with complex numbers, we can proceed
as in the algebra of real numbers, replacing i2 by – 1 when it
occurs.

4. If z1 = x1 + iy1 and z2 = x2 + iy2 are any two complex
numbers, we define following rules and laws of operations.
1.
Equality: Two complex numbers z1 and z2 are
said to be
equal if the only if their corresponding real and imaginary parts
are equal, i.e. x1 = x2 and y1 = y2.
2.
Addition:
z1 + z2 = (x1+iy1) + (x2 + iy2) = (x1 + x2) + i (y1+y2)
3.
Subtraction:
z1 – z2 = (x1+ iy1) – (x2+iy2) = (x1 – x2) + i (y1 – y2)
4.
Multiplication:
z1. z2 = (x1 + iy1). (x2 + iy2) = (x1 x2 – y1 y2) + i (x1 y2 + y1 x2)
5.
Division:
z1  x1  iy1   x1  iy1   x 2  iy 2 


.
z2  x 2  iy 2   x 2  iy 2   x 2  iy 2 

6.
x x
x
1
2
 y1 y 2 
2
2
y
2
2

i
 y1 x2  x1 y2 
x
2
2
 y2
2

Commutative law of Addition and Multiplication:
z1 + z = z2 + z1,
z1 z2 = z2 z1

5. 7.
Associative law of Addition and Multiplication:
z1 + (z2 + z3) = (z1 + z2) + (z3)
(Where z3 = x3 + iy3)
z1 (z2 z3) = (z1 z2) z3
8.
Distributive law:
z1 (z2 + z3) = z1 z2 + z1 z3
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