Mathematical Journeys - Saint Ann's School

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  43  Luckily it is possible to find the treasure using complex numbers. We began by placing the pine and the oak at points –1 and 1, respectively. Wherever the two trees actually are does not matter as the picture can be scaled and shifted to suit any location. If we determine the locations of each stake as a function of the unknown fixed point, we find that the midpoint of line between each stake is simply –i. Therefore we know the treasure is simply found by drawing a perpendicular line through the center of the line connecting the oak and the pine and extending it 1/2 the distance between the oak and the pine. This problem demonstrates another way in which it is much simpler and straightforward to use complex numbers rather than geometry. Another interesting application of complex numbers that we explored had to do with the translation of a three-dimensional object onto a two-dimensional plane. First, we read Drawing with Complex Numbers by Michael Eastwood and Roger Penrose. We immediately learned how to represent numbers of higher dimensions as a set of real numbers, and to identify which dimension these sets belong to. A triplet of numbers belongs to our familiar three-dimensional space, whereas four numbers would belong to the fourth-dimension, five numbers to the fifth, and so on. We also learned, that it is possible in standard projection to project a lower dimensional object onto a higher dimensional plane by simply putting a 0 in the remaining slots to be filled. We talked about Gauss’ revolutionary theorem of axonometry which states that if we project a 3-D cube onto the complex plane, with one corner projected onto the origin, the sum of each of the adjacent corners’ projected points squared will be 0. We found it rather shocking to see this complex physical transformation reduced to such a simple equation, and we attempted to reconcile our intuition with the argument set forth. We also attempted a few experiments of our own, using a poster board that we had constructed for the Math Fair. We found that our experiments checked out, very nearly equalling zero. It was surprising to us that our humble attempt at trying to project a cube to a plane, armed with a laser and an imperfect graph, worked out so well.
44    We both have learned so much about complex numbers this past year. Both of us went into the class having absolutely no idea what they were. We had never even heard the term before, and the idea of the square root of negative one was a dimly remembered concept from previous math classes. We learned that they are so useful, and very applicable to everyday life. Many of the complicated geometric maneuvers that we had performed in Geometry could be expressed much simpler using complex numbers. Even problems that we would not have previously thought would have tangible solutions (like the Treasure Map problem and the idea of projected a three dimensional object to a two dimensional plane) were relatively easily solved using complex numbers. As we became better versed in the complex plane, it became apparent that they would be useful in studying any number of things. Pleasant’s brother, an engineer, expressed surprise that she was only just learning about complex numbers; they had become such an integral part of his daily life that he had forgotten that relatively few people were exposed to them. It is clear that complex numbers are incredibly important and applicable to any number of different topics, from video game design to the field of medical imaging to engineering. We both now have a very strong foundation in the complex plane, a foundation on which we are eager to expand upon further in the years to come.      Works Cited Eastwood, Michael, and Roger Penrose. "Drawing with Complex Numbers." The Mathematical Intelligencer 22.4 (2000): 8-13. Print. Gamow, George. One Two Three... Infinity. New York: New American Library, 1953. Print. Theodosopoulos, Ted. "Complex Geometry." Saint Ann's School. Web. .     
Page 1 and 2: MATHEMATICAL JOURNEYS Saint Ann’s
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