Complex Numbers Overview

The Complex Numbers

• A complex number ( z ) is represented as an ordered pair ( (x, y) ) where ( x ) and ( y ) are real numbers.
• The real part of ( z ) is denoted as ( \text{Re} z = x ) and the imaginary part as ( \text{Im} z = y ).
• The set of complex numbers is denoted by ( \mathbb{C} ).
• The number ( x ) corresponds to the complex form ( (x, 0) ) and ( i ) represents ( (0, 1) ).
• A field isomorphism exists between the real numbers ( \mathbb{R} ) and a subset of ( \mathbb{C} ).
• Addition of complex numbers is defined as:
  ( (a, b) + (c, d) = (a + c, b + d) )
• Multiplication of complex numbers is defined as:
  ( (a, b)(c, d) = (ac - bd, bc + ad) )
• Commutative Properties:
  • ( z_1 + z_2 = z_2 + z_1 )
  • ( z_1 z_2 = z_2 z_1 )
• Associative Property:
  ( z_1 (z_2 + z_3) = z_1 z_2 + z_1 z_3 )
• Complex numbers can be expressed in the form ( z = x + iy ) by associating ( (x, 0) ) with ( x ) and ( (0, y) ) with ( iy ).
• The imaginary unit ( i ) satisfies the equation ( i^2 = -1 ).
• Division of complex numbers, where ( z \neq 0 ), is defined as:
  ( \frac{1}{z} = \frac{1}{x + iy} = \frac{x - iy}{x^2 + y^2} )
• This operation leads to results of the form:
  ( \frac{x_1 + iy_1}{x_2 + iy_2} = \frac{x_1 x_2 + y_1 y_2}{x_2^2 + y_2^2} + i\frac{y_1 x_2 - x_1 y_2}{x_2^2 + y_2^2} )