Complex Numbers

Complex Numbers

In mathematics, the complex numbers are an extension of the real numbers obtained by
adjoining an imaginary unit, denoted i, which satisfies: i2 = ?1.

Every complex number can be written in the form a + bi, where a and b are real numbers
called the real part and the imaginary part of the complex number, respectively.

Complex numbers are a field, and thus have addition, subtraction, multiplication, and division operations. These operations extend the corresponding operations on real numbers, although with a number of additional elegant and useful properties, e.g., negative real numbers can be obtained by squaring complex (imaginary) numbers.

Complex numbers were first conceived and defined by the Italian mathematician Girolamo Cardano, 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 irreducible. This ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, it is 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.
The set of all complex numbers is usually denoted by C, or in blackboard bold by C. Although other notations can be used, complex numbers are very often written in the Form a+bi

where a and b are real numbers, and i is the imaginary unit, which has the property

i2 = ?1

The real number a is called the real part of the complex number, and the real number
b is the imaginary part.
For example, 3 + 2i is a complex number, with real part 3 and imaginary part 2. If z = a +
ib, the real part a is denoted Re(z) or R(z), and the imaginary part b is denoted Im(z) or J(z). a+bj.

Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. In other words, if the two complex numbers are written as a + hi and c + di with a, h, c, and d real, then they are equal if and only if a = c and h = d.



Complex numbers are added, subtracted, multiplied, and divided by formally applying
the associative, commutative and distributive laws of algebra, together with the equation
i2=-I:
Addition: ( a + bi ) + ( c + di) = ( a + c) + ( b + d)i

Subtraction: (a + bi) -(c+ di) = (a -c) + (b -d)i

Multiplication:
(a + bi)(c+ di) = ac + bci + adi + bdi2 = (ac -bd) + (bc+ ad)i