Underappreciated Theorems: Grade School Math We Thought …
Irina Tyshkevich Professor AubertinWRIT 340: Revised Illumin Article 6 December 2013Underappreciated Theorems: Grade School Math We Thought We Could Live WithoutThroughout the years of Grade School, student voices unite in a common cry - “Why are we learning this? We won’t need this in the real world!” Outshined by the glamour and grandeur of a fast paced materialistic world focused on the bigger picture, the importance and relevance to our lives of basic mathematic theorems is quite often overlooked. Not only are the conclusions derived from these equations significant in mathematics, but they are also the essential foundation of many processes and inventions that our society has come to rely on – the very basic unit that can be credited for many developments around which our lives revolve. Introduction Consider that you are going on a road trip across the country or simply need to get from one location to the next. What once was a tedious and lengthy task of trying to decode unwieldy and complicated paper maps, which, more often than not, resulted in the accidental arrival to a wrong location, has now been made much easier and more accurate by the Global Positioning System (GPS). This small device sitting on the dashboard of our cars has made destinations significantly more reachable and the world a little bit smaller. Each device produced and sold in the market certainly necessitates advanced engineering to assure that the screen works and the device communicates effectively with positioning satellites. But, how does the GPS accomplish accurately finding locations, measuring distances and choosing a route - the very task that dictates its purpose??Let’s also take a look at our economy, which is largely reliant on the market interaction between buyers and sellers. Businesses focus on maintaining a balance between consumer savings and producer surplus, in order to achieve the highest amount of revenue. Without this market, the production, development and further innovation of all the services and products, which we have tightly integrated into our lives, such as the GPS we use to get from one place to another, would struggle to exist without a monetary motive. But, how do businesses ascertain that this balance is maintained and that they will make enough revenue to remain in operation – an integral process that makes our economy stay afloat?The answers to both of the questions above are the Pythagorean Theorem and the Fundamental Theorem of Calculus, respectively – two of the very mathematic equations that we learned in Grade School math, thought we’d never need again and constantly overlook in our daily lives.?The Pythagorean Theorem “The area of a square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides (See Figure 1).” – Pythagorus. 197739010350500Figure 1 - A Visual Representation of The Pythagorean Theorem [1].MeaningWhile the functionality of a GPS might take an intimidating few pages of explanation, contrarily, the Pythagorean Theorem outlines a quick calculation pertaining to right triangles. In the simplest of terms, this theorem claims that the length of the longest leg, also known as the hypotenuse, is equal to the square-root of the sum of the squares of the two shorter legs. Also noteworthy, are the standard triplets, multiples of which always satisfy the equation [2]. Perhaps the most difficult attribute of learning this theorem in Grade School was remembering how to say "hypotenuse". The concept is simple and the math is straight forward, so the most common complaint is about the seemingly trivial application, and the most perplexing related question ponders who is responsible for bringing hours of impractical math homework upon us. ?HistoryAlthough there is evidence that the concept behind this theorem dates back to the Babylonians, 1000 years earlier, the proof is credited to the Greek philosopher, Pythagoras, who lived around 500B.C. Pythagoras is responsible for important developments in mathematics, astronomy and the theory of music. In the mathematical world, the Pythagorean Theorem is most notable for the link it creates between algebra, the core of geometry and the foundation of trigonometry [3].?There have been many instances throughout Ancient History that evidence the knowledge and application of the Pythagorean Theorem. A Chinese book, written sometime between 500-200 B.C., shows the theorem’s logic through a geometrical demonstration. Indian mathematicians, in the ancient world of 800-600 B.C, integrated the theorem into their religion by using it to build alters that satisfied accurately the strict requirements for orientation, shape, and area. Ancient Mayans used the theorem to dictate their lives directly via the application of Pythagorean triplets in their Long Count calendar [4]. It was like this, hidden and overlooked within the structure of our daily lives, that the value and applications of Pythagoras’s discovery were passed on from generation to generation, until it found its way into our present day classrooms. Modern UseAlgebra math teachers are owed a huge round of applause for assigning all those Pythagorean Theorem practice problems. Without the knowledge and understanding of the concept and calculations behind this Theorem, we would be less adept in the areas of navigation, construction and virtual games. NavigationFinding the hypotenuse of a right triangular surface is one of the primary methods used in map-making and navigation. This is made possible by the Distance Formula, d =(x2-x1)2+(y2-y1)2which is derived from the Pythagorean Theorem. The connection between the two equations is simply the representation of points on a Cartesian Coordinate Plane. Maps online, such as MapQuest and Google Maps use this method to consider the distance between two cities and compare the lengths of different routes [5]. Another application of the Theorem, as it relates to navigation, direction and location, is the process of triangulation. Triangulation divides a surface into a set of adjacent triangles in order to find the coordinates of a point and the distance to that point. This process uses the law of sines and cosines, which can reduce to the Pythagorean Theorem, to find the lengths of the sides of the triangles in the divided up plan [6]. Triangulation is most prominently used in determining the location of where an Emergency Call was dialed. ConstructionThe formula behind the Pythagorean Theorem is unavoidably used during the construction of buildings – from finding missing side lengths to excavating putting in floor details. Since this is an aspect of only building construction, it can be argued that the knowledge of the Theorem isn’t a daily necessity. However, it is important to note that without construction, and thus, without this equation, buildings would not exist and we would not have homes. A very basic procedure that construction workers follow, in order to accurately create a right angle when the high-tech tools aren’t handy, is inspired entirely by the Pythagorean Theorem. Using a few pegs and a few pieces of string, a mock right triangle is created. Off of this triangle, the placement of the corner and length of the walls is determined [7].Video GamesIn addition to aiding in the calculation of distance between two characters or objects through the use of path finding algorithms, the Pythagorean Theorem is used heavily in Video Game environments because they are drawn using triangles and because the Theorem is a leading actor in the generation of 3-D virtual realities. One of the ways a game world can be created is through the use of the isometric perspective. This projection allows a 3D object to be embodied on a 2D plane. Isometric tiles, shaped like diamonds, are blanketed with an image and then positioned together to seamlessly produce a beautiful virtual game world [8]. The Fundamental Theorem of Calculus “(Part 1)Let f be a continuous function on [a, b] and define a function g:[a, b] R by g(x) := axf.Then g is differentiable on (a, b), and for every x € (a, b), g’(x) = f(x).At the end points, g has a one-sided derivative, and the same formula holds. That is, the right-handed derivative of g at a is f(a), and the left-handed derivative of f at b is f(b).(Part 2) Let f be a continuous function on [a, b]. Suppose that F is continuous on [a, b] and that F’ = f on (a, b). Then, abf=Fb-Fa” [9]. (See Figure 2).159385012065000Figure 2 – Combined Visual Representation of both parts of The Fundamental Theorem of Calculus [10].MeaningThe state of our economy is never constant, so it isn’t surprising that the related Fundamental Theorem of Calculus works within the realm of functions and limits, combining the two to calculate the instantaneous rate of change [3]. This Fundamental Theorem links two subfields of Calculus. The first of the two is derivatives, which are the primary actors in studies of the behavior and rate of how quantities change. Secondly, the Fundamental Theorem engulfs integration, which is the reverse process of differentiation most commonly applicable to shapes and areas [11]. Although, Calculus is a daunting math class to face and has a large chance of hurting Grade Point Averages, it can’t be denied that Calculus has had, starting from the most basic level, a large influence on many aspects of our lives, for many centuries. History Calculus, also known as the “language of numbers”, as a mathematical field, emerged in the late 17th Century. Both Isaac Newton and Gottfried Leibniz are credited for the logic and notation, although there is a debate over who actually discovered which aspects. Augustin Lois Cauchy later developed the mathematics of Calculus by providing proofs for the Fundamental Theorem of Calculus [12]. The Fundamental Theorem of Calculus provides the logic behind the graphical measurement of solids, curves and areas, for which reason it is an important mathematical technique [13]. Throughout the years, the Fundamental Theorem of Calculus has made its way into many walks of our life, including physics, electromagnetism motion, acoustics, astronomy and dynamics. Mathematical models using the theorem’s equation are used as proofs and help in the calculation of problems in these fields. Einstein’s infamous theory of relativity, also, heavily relies on this rather unnoticed mathematical field of Calculus [14]. Modern UseIt’s quite easy to overlook the importance of the Fundamental Theorem of Calculus because its influence impacts mostly smaller steps of large processes. But, although small, the conclusions provided by the calculations made during these small steps, are crucial to the function of common systems, such as medicine, economics and computer science. MedicineThe Theorem’s very basic contribution to biology is used to determine rates –from birth and death rates, to medicinal healing rates [14]. Footprints made by the Fundamental Theorem of Calculus can be found in multiples points of a medical treatment process. The equation is used to determine what concentration of medicine should be given to patients, considering frequency of intake and the patient’s speed of metabolism. This application, due to the equation’s stress on limit and rate of change, can give insight into the titrimetric composition of the medicinal solution. This mathematical is also consulted in the process of patient care by giving a method to represent response to treatment [15].EconomicsThe Fundamental Theorem of Calculus works hand in hand with many Economic concepts, especially considering that both fields are built around analyzing the relations between two or more variables. Differentiation is the process used by economists to measure average change in income, as it relates to education, experience, or some other factor that increased or decreased in that year. Because the Theorem is concerned with limits reached and change behavior, it is the driving force behind Marginal Analysis, which is significantly important to businesses, when it comes to making decisions about factory output and profitability; if the marginal revenue exceeds marginal cost, and this can be used in the analysis of even a small incremental change, the firm increases profits. On a more directly personal front, Calculus is used to determine interest that needs to be paid over the lifespan of a loan, whether the loan was for a home, car, business, or other personal endeavor [16]. Without businesses and the aid the Fundamental Theorem gives in the decision making process, our economy and trade networks would crumble. The lack of an economy would make a significant impact on our daily lives, which revolve around income and buying goods and services produced by businesses. Computer ScienceIn addition to the procedures of differentiation and integration, the presence of the Fundamental Theorem of Calculus in the real world is also linked to the processes used to carefully and accurately manage infinitely large and infinitesimally small data – important concepts relating to data management and algorithm building in computer programming. Considering the recursive nature of Computer Science, the formula behind the Theorem is important in organizing long strings of bits – units of information in computing communications - into “bytes” and “words” and “functions” that are meaningful to the processor and allow a program to come to life [17].Without the software that Computer Science produces, and the existence of the Fundamental Theorem of Calculus which aids in the creation of these programs, our computer screens would be little more than a screen that turns off and on. Conclusion Math, very often, is a victim of complaint during the years of Grade School; a dreaded lesson to learn and, consequently, also one whose importance it is easy to underestimate. We complain about having to do hundreds of Pythagorean Theorem practice problems, under the assumption that we’ll never need to see the formula again. We are terrified of Calculus, writing it off as a waste of time and a grade killer. However, as it turns out, the Pythagorean Theorem has been a part of our daily lives for hundreds of years and will remain to play a role for as long as the technologies of navigation, construction and virtual reality continue to grow. Likewise, the Fundamental Theorem of Calculus is fundamental to the continuation of our civilization due to its central role in medicine, economics and computer science. It is important to remember that sometimes we get so lost in the bigger picture, that we easily forget to appreciate the simple things which are the building blocks of our lives. References [1] (2013) Pythagorean Theorem [Online]. Available: [2] (2007, Oct. 24) Surprising Uses of the Pythagorean Theorem [Online]. Available: [3] I. Stewart. In Pursuit of the Unknown: 17 Equations That Changed the World. USA: Basic Books, 2012. [4] (2001, May) L. Smoller. Did You Know…? [Online]. Available: [5] (2013) R.G. Stone. The Pythagorean Theorem and Distance Formula [Online]. Available: [6] (2006, Aug. 25) Geocache Description [Online]. Available: [7] (1994 – 1998) Using the Pythagorean Theorem in Construction [Online]. Available: [8] (2013) Claire. Math Apprentice: Math is the path to anything you want to be [Online]. Available: [9] (2010, May 2) The Fundamental Theorem of Calculus [Online]. Available: [10] (2011, Oct 1) The Fundamental Theorem of Calculus Study Guide [Online] Available: [11] (2004) The Fundamental Theorem of Calculus [Online]. Available: [12] (2011) Cauchy and the Rigorous Development of Calculus [Online]. Available: [13] (2012) J. Kingsley and K. Shirley. Calculus: A Modern Approach [Online]. Available: [14] (1999 – 2012) What is Calculus? When Do You Use It In The Real World? [Online]. Available: [15] C. Neuhauser. Calculus For Biology and Medicine, 3rd Edition. USA: Prentice Hall, 2010. [16] (2001 – 2013) S. Hall. How is Calculus Used in Economics? [Online]. Available: [17] A. D. Hwang. Calculus for Mathematicians, Computer Scientists, and Physicists: An Introduction to Abstract Mathematics [Online]. Available: ................
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