Option 2: Is there any good reason to have complex solutions? Are there any practical applications of complex solutions to a quadratic?
With complex solutions, we will no longer have to write “no solution” as our answer. I think complex solutions are great because it allows you to go more in depth with the equation, yet it adds more work to find the answer.
Complex numbers are comprised of real numbers and imaginary numbers, a+bi where a is the real part and bi is the imaginary part (contains the real number combined with the imaginary number). Imaginary numbers can be stated with i (which is called iota) and they are equaled to √-1. This is how it works:
i = √-1
i²= i x i = -1
i³= i x i x i = -i
i²²= (i²)¹¹ = (-1)¹¹ = -1
According to picomonster.com, the complex numbers are convenient when when it comes to describing two dimensional variables. The ‘real number’ is expresses one dimension and the ‘imaginary number’ quantifies on a different dimension because of its special properties. Without complex numbers, it will be hard to solve certain types of equations like x²-2=0. With the imaginary numbers we can solve x=±√-2 by using i to replace the number.
Stated in the Colombia University website, complex numbers allows engineers and physicists to analyze stresses in structures (bridges/buildings) and to study the flow of liquids. Engineers often use the complex numbers when they want a visual snapshot of their system, by plotting the two dimensional variables on the complex plane. Also complex numbers are needed to analyze electrical and mechanical engineering for it often describes physical characteristics which could be measured and detected, “Imaginary numbers are used in real world applications to quantify physical characteristics. They are typically paired with real numbers to form a complex number, which is useful for describing two dimensional characteristics.”
Complex numbers are helpful when dealing with airplanes, designing/testing the strength of buildings, physics, cellphones or designing stereos. When designing stereos to play louder, the electrical current measurements are important to use in order to use electricity creatively. Complex numbers are needed whenever operating the electrical current as AC (alternating current), which is the alternative to DC (direct current). Also according to the Colombia University website, “Scientists who do experiments on ways to make energy using fuel cells, batteries, and solar cells use complex numbers. Complex numbers are also used for generating fractals, which are geometric objects created by making a repeating pattern.” In short, complex numbers are required to be used by engineers, scientists, physicists, and etc.
Overall, I feel that complex numbers have many good reasons to be used because it can solve many certain types of equations until the very end and help engineers/inventors to create and observe the visual ‘2 dimensional physical characteristic’ image of their system. Complex numbers are useful in solving certain equations like x²-2=0, without their presence those types of equation will end up with an typical boring answer of ‘no solution’.
Azeem. “Evolution of Complex Numbers: History and Applications.” Scienceray.
…..N.p., n.d. Web. 20 Sept. 2011. <http://scienceray.com/physics/
Chung, Bradley. “Applications of Imaginary Numbers in the Real World.”
…..Picomonster. Picomonster, n.d. Web. 20 Sept. 2011.
Rios, Brenda. “Application of Complex Numbers.”Technology Integration Partnerships.
….Colombia University, n.d. <http://tip.columbia.edu/index.php?option=