Complex Numbers

Definition

ComplexNumberArgand 1000 ECE301Fall2008mboutin.gif

The complex numbers are the field C of numbers of the form x+iy, where x and y are real numbers and i is the imaginary unit equal to the square root of -1, sqrt(-1). A complex number can be visually represented as a pair of numbers forming a vector on a diagram A complex number can be visually represented as a pair of numbers forming a vector on a diagram

Operations

Complex numbers are added, subtracted, multiplied, and divided by formally applying the associative, commutative and distributive laws of algebra, together with the equation i 2 = −1:

       * Addition: 1 ECE301Fall2008mboutin.png
       * Subtraction: 2 ECE301Fall2008mboutin.png
       * Multiplication: 3 ECE301Fall2008mboutin.png
       * Division: 4 ECE301Fall2008mboutin.png

where c and d are not both zero.

It is also possible to represent complex numbers as ordered pairs of real numbers, so that the complex number a + ib corresponds to (a, b). In this representation, the algebraic operations have the following formulas:

   (a, b) + (c, d) = (a + c, b + d) 
   (a, b)(c, d) = (ac − bd, bc + ad) 

Since the complex number a + bi is uniquely specified by the ordered pair (a, b), the complex numbers are in one-to-one correspondence with points on a plane. This complex plane is described below.

Sources : wikipedia

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