A Colourful Introduction to the Wonderful World of Complex Numbers.

1. The real number system has been adequate in dealing with almost everything - but what about numbers with negative roots? (Un)Luckily, complex numbers were created. The concept of complex numbers first surfaced around the 16th century, in the midst of the Renaissance. The idea of of an "imaginary number" was a preposterous notion - in fact, the term "imaginary" was coined as a sign of disrespect. Nevertheless, the idea was gradually accepted by mathematicians around the world (headed by headed by a description of the geometric sequence created by Caspar Wessel). Complex numbers are a system of numbers that has been developed to deal with negative roots. A complex number (in the form a + bi), is actually an extension created by assigning a unit, i, which satisfies that i^2 = -1. In this form, the a and b are real numbers. Using complex numbers, we are now able to solve negative roots.


2. As ridiculous as it may sound, complex numbers actually have many uses in the real world. In some of the sciences, especially physics, complex numbers sneakily creep their way in. There are only a few cases where complex numbers are directly used. For example, in electrical circuits, imaginary numbers can be used to easily determine the changes in voltage and current.Also in electromagnetism, complex numbers are used. In most cases however, complex numbers aren't directly used, but instead, the concept of complex numbers is involved. Some examples of these can be found in engineering and architecture, where the concept of complex numbers is extremely useful.


3.1) (3+2i)^2 - 6 (3+2i) + 13 = 0
9 + 12i + 4i^2 - 18 - 12i + 13 = 0
4 + 4i^2 = 0 (i^2 = -1)
4 - 4 = 0

(3-2i)^2 - 6 (3-2i) + 13 = 0
9 - 12i + 4i^2 - 18 + 12i + 13 = 0
4 + 4i^2 = 0 (i^2 = -1)
4 - 4 = 0

2a) a = 1 b = 4 c = 5
-4 ± ((4)^2 - 4(1)(5))^1/2 / 2(1)
-4 ± (-4)^1/2 / 2
-4 ± ((4) (-1))^1/2 / 2
-4 ± ((4)^1/2 (-1)^1/2 )/ 2
-4 ± 2i / 2
-2 ± i

2b) a = 1 b =-2 c = 3
2 ± ((-2)^2 - 4(1)(3))^1/2 / 2(1)
2 ± (-16)^1/2 / 2
2 ± ((16) (-1))^1/2 / 2
2 ± ((16)^1/2 (-1)^1/2 )/ 2
2 ± 4 i / 2
± 2i

3) x^2 + 2x + 2
a = 1 b = 2 c = 2
-2 ± ((2)^2 - 4(1)(2))^1/2 / 2(1)
-2 ± (-4)^1/2 / 2
-2 ± ((4) (-1))^1/2 / 2
-2 ± ((4)^1/2 (-1)^1/2 )/ 2
-2 ± 2 i / 2
-1 ± i

Check:

(-1 + i)^2 + 2 (-1+i) + 2 = 0
1 - 2i + i^2 - 2 + 2i + 2 = 0
1 + i^2 = 0
1 - 1 = 0

(-1 - i)^2 + 2 (-1 - i) + 2 = 0
1 + 2i + i^2 - 2 - 2i + 2 = 0
1 + i^2 = 0
1 - 1 = 0

Sources:

http://mathforum.org/library/drmath/view/53844.htm
http://www.math.toronto.edu/mathnet/questionCorner/complexinlife.html
http://en.wikipedia.org/wiki/Complex_numbers
http://www.purplemath.com/modules/complex.htm