26 Ways to Fail

F is for FISH!!!!


Or Mrs. Fish to be more correct. Very few of the students in 9th grade have Mrs. Fish as a math teacher 3rd period.. There are 11 in fact! 6 boys and 5 girls... that's useless information though. Today we were learning about IMAGINARY NUMBERS... (ohhh....ahhhh) (Just as a side note Peter is an imaginary number, I'm a rational number and I think... oh never mind..)


Mrs. Fish was just saying the real use of imaginary numbers... None that she knows of. So here are some awesome facts about IMAGINARY NUMBERS:

An Imaginary number is the square root of a negative number

The name "imaginary number" was originally coined in the seventeenth century as a derogatory term as such numbers were regarded by some as fictitious or useless, but today they have essential, concrete applications in a variety of scientific and related areas.

Ooo.... A practical use for IMAGINARY NUMBERS is in THE CONTROL THEORY!!!!
It's as interesting as imagined.

(Control theory is an interdisciplinary branch of engineering and mathematics, that deals with the behavior of dynamical systems. The desired output of a system is called the reference. When one or more output variables of a system need to follow a certain reference over time, a controller manipulates the inputs to a system to obtain the desired effect on the output of the system. A related theory known as perceptual control theory has been used to model living systems on the premise that outputs are manipulated to obtain the desired effect on the input to the system.)
It is also used in Cartography. So that means we can talk to Mrs. Lippincott about it!!


The powers of i repeat in a cycle
i^-3 = i
i^-2=-1
i^-1=-i
i^0+=1
i^1=i
i^2=-1
i^3=-i
i^4=1
i^5=i
i^6=-1

i stands for Imaginary numbers by the way. So the Square Root of -49 would be 7i but DON'T write is as i7 other wise people are going to have headaches. (more so that if they read my random story)

Though it was first used in the 16th century it did not become popular or better known (it was still considered a just plain weird thing with math) in the 17th and 18th centuries. (By the way most of this is coming from Wikipedia) It is well known today though to graph quadratics. (better known tomorrow when we actually do it and learn how to. )

*By the way the contest is STILL going on because I still only have 1 entrant.

THIS IS MY FAVORITE QUOTE WHEN LOOKING FOR INFORMATION FOR THIS:

http://mathforum.org/dr.math/faq/faq.imag.num.html

Contrary to what some people might tell you, imaginary numbers are not numbers that only exist in the brains of weird people. Or maybe they are; all numbers in math are "imaginary" in the sense that you can't touch them or experience them directly.

There is some really cool stuff on this website that I do not wish to copy because I do not have enough time. So check this out: http://mathforum.org/library/drmath/view/55747.html
And to those of you wondering what the square root of "i" is:



The square root of i

We have i = cos(pi/2) + i.sin(pi/2)

sqrt(i) = (cos(pi/2) + i.sin(pi/2))^(1/2)

By DeMoivre's theorem:

= cos(pi/4) + i.sin(pi/4)

1 + i
= ----------
sqrt(2)

To learn more about DeMoivre and his theorem, look at:

http://mathforum.org/library/drmath/view/53975.html


This made a lot of sense to me and I hope it does to you!
-Jasmine Crespo