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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: | 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: | + | * Addition: [[Image:1_ECE301Fall2008mboutin.png]] |
| − | * Subtraction: | + | * Subtraction: [[Image:2_ECE301Fall2008mboutin.png]] |
| − | * Multiplication: | + | * Multiplication: [[Image:3_ECE301Fall2008mboutin.png]] |
| − | * Division: | + | * Division: [[Image:4_ECE301Fall2008mboutin.png]] |
where c and d are not both zero. | where c and d are not both zero. | ||
Revision as of 15:42, 4 September 2008
Complex Numbers
Definition
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:* Subtraction:
* Multiplication:
* Division:
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
