Complex Number and Complex Conjugation
A complex number is represented as,
c = a + bi
where a and b are real numbers and i = √(-1)
The number a and b represent the real and the imaginary part of the complex number:
a = real part of c = Re(a + bi)
b = imaginary part of c = Im(a + bi)
Example: if c = 6 + 4i, then a = 6 and b = 4.
It should be clear from the above definition that real numbers and imaginary numbers are subsets of complex numbers. If Re(a + bi) = 0, then c is a pure imaginary number, and if Im(a + bi) = 0, then c is a real number.
A complex number has two real coordinates, namely its real and imaginary parts. Therefore, a complex number can be plotted as a point in a complex plane using the real part of the complex number as the x-co-ordinate of the point, and the imaginary part b as the y-coordinate in the above fig: The horizontal axis of the complex plane is called the real axis and the vertical axis is called the imaginary axis.
The operation of complex conjugation has a very important part in the theory of complex numbers. The complex conjugate of c above is obtained by replacing i with -i. Thus, the complex conjugate of c denoted as c* is
c* = a – ib
The above figure is a reflection of the real axis. The complex conjugate of algebraic expressions can be derived using the following two relations:
(a+b)* = a* + b*
(ab)* = a*b*
The modulus or absolute value of a complex number c = a + ib is denoted by |c| and is the distance from point c to the origin in the complex plane. Equivalently, |c| is the length of the vector corresponding to c.