#785 Dr. Roger Roybal MATH October 2017 Project 1 A proof is a deliberate process where a mathematical concept is proven to be true using a

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1 #785 Dr. Roger Roybal MATH October 2017 Project 1 A proof is a deliberate process where a mathematical concept is proven to be true using a step-by-step explanation and a lettered diagram. The first step in the construction of a proof is to have a mathematical concept that you are trying to prove. Next, a lettered diagram that will be used to explain the steps to arriving at the final proof of the concept is designed. Following the construction of the diagram, the step-by-step reasoning is done to explain how to arrive at truthfulness of the concept. The use of the steps and the lettered diagram was the revolutionary geometrical system that the Greeks developed to prove mathematical concepts to be true (Swanson). Proof based mathematics differs from non-proof based mathematics in that prior to the Greeks developing the system of using proofs, non-proof based mathematics relied on practical observation (Evans, 78). An example of non-proof based mathematics is the Egyptian method for arriving at the measurements of a (3,4,5) right triangle (Netz, 22). The Egyptians used a very practical approach that involved taking a length of rope and tying 12 knots at equal intervals down the length of the rope (Evans, 78). For the short leg of the triangle, they would use three of the knot segments, followed by the long leg of the triangle using four segments, and finally the hypotenuse would have the remaining five knot segment (Evans, 78). The Egyptians discovered that this worked from a system of practical observation that they had made through trial and error (Shuttleworth). On the other hand, the Greek s proof based mathematics took the idea of the Egyptian (3,4,5) right triangle and used the geometric proof system to develop a concept that applied to all right triangles (Violatti). Pythagoras developed the idea that if the squares of the two shorter legs were added together, that this value would be equal to the square of the hypotenuse (Violatti). Using the proof system of step-by-step deductive process, accompanied by lettered diagram, Pythagoras proved this to be true for all right triangles (Violatti). The proof based system allowed the Greeks to go beyond what the Egyptians had discovered through observation and take mathematics to the next conceptual level (Swanson). The Greek use of proofs helped them to make great leaps forward in their focus on studying geometrical concepts in mathematics. They used simple concepts and added to them using deductive reasoning to arrive at more complex ideas. An example would be Pythagoras proof of a^2 + b^2 = c^2 as illustrated in the following proof and diagrams (Turcin, Chapter 11):

2 The area of the first square is given by (a+b)^2 or 4(1/2ab)+ a^2 + b^2. The area of the second square is given by (a+b)^2 or 4(1/2ab) + c^2. Since the squares have equal areas we can set them equal to another and subtract equals. 4(1/2ab) + a^2 + b^2 = 4(1/2ab)+ c^2. Subtracting equals from both sides gives us a^2 + b^2 = c^2 (Turcin, Chapter 11). In this proof, Pythagoras explained each step of his thought process and used a diagram to illustrate everything that he was explaining (Turcin, Chapter 11). He used the basic properties of triangles and squares and manipulated these geometric shapes to prove his theorem of right triangles to be true. This method of using proofs gave the Greek mathematicians a structured basis from which they could continue to build into more complex geometric ideas (Violatti). Once a concept was proven as true, other Greek mathematicians could then build on this concept and this continued to help the advancement of Greek mathematics for hundreds of years (Turcin, Chapter 11). However, the very rigid structure of proofs and the Greek s narrow focus on everything being proven geometrically, was a hindrance to even further advancements of mathematics (Turcin, Chapter 11). For example the equation (a+b)(a-b) = a^2 b^2 is fairly easy to prove algebraically but for the Greek s to prove this using geometry was far more complex as shown in the following diagram from the second book of Euclid s Elements (Heath):

3 The theorem is: "If a straight line be cut into equal and unequal segments, the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of the section is equal to the square on the half. The steps of the proof are: Rectangle ABFE is equal to rectangle BDHF. Rectangle BCGF is equal to rectangle GHKJ. If square FGJI is added to these two rectangles (which together form rectangle ACGE which is ''contained by the unequal segments of the whole'') what we end up with is precisely rectangle BDKI, which is constructed ''on the half.'' Thus we have the equality (a+b)(a--b) + b 2 = a 2 which is equivalent to the equality above (Turcin, Chapter 11). In the case of this simple algebraic identity, the use of a proof is much more complicated and requires a high conceptual ability to understand the subtraction of areas (Turcin, Chapter 11). The Greeks were masters of geometry but the use of arithmetic algebra, which uses letters and symbols rather than the language of figures, was mostly neglected by the Greeks (Swanson). This meant that the Greek s could not expand into mathematical concepts such as negative, irrational, and imaginary numbers because these concepts were difficult to prove with geometry alone (Violatti). For my proof, I am going to prove that: The sum of the angles of a triangle are equal to a straight angle (Eves, 159 #58(a)). I will use the following lettered diagram: The steps of this proof are: 1) Draw a triangle with point and interior angles of (X,Y,Z). 2) Draw two parallel lines through points (X,Y) and point Z and extend them out (Red lines). 3) Extend out line (X,Z) (blue line) and this will be a transversal of the other two parallel lines. 4) Since the line that goes through Z and X is a transversal, X has a corresponding angle at point Z. 5) At the opposite of the intersection, there is a vertical angle that is also equal to X. 6) Now extend the line going through points (Y,Z) (blue line) and this line will also be a transversal of the parallel lines.

4 7) Now mark the corresponding and vertical angles that are equal to the value of Y. 8) The measure of X + Z at the top of the triangle is supplementary to the Y angle. 9) Therefore X + Z + Y (green arc) = 180 degrees because they are supplementary. 10) Now I have proven that the interior angles of triangle (X,Y,Z) are equal to the supplementary angles of the straight angle (X,Z,Y). The advantage of using a proof in the case of my problem is that it makes explaining where the corresponding and vertical angles are for X and Y more easily understood. Once these are visible, it is easy to arrive at adding the supplementary angles of X,Z, and Y to get the 180 degrees of the straight angle. The lettered diagram in the proof allows for easy explanation of where the parallel lines and the transverse lines are. Also because this proof uses the concepts of corresponding angles, vertical angles, and supplementary angles if a student got stuck somewhere in the middle of the process, they could be shown step-by-step how to arrive at the end. One disadvantage of using a proof in the case of my problem are that it takes ten steps. Another drawback is that constructing a diagram to come to the conclusion that a straight angle has 180 degrees and the sum of the interior angles of a triangle are also equal 180 degrees seems unnecessary. If the typical student knows that a straight angle equals 180 degrees and that the interior angles of a triangle always equal 180 degrees, I don t think that a proof is necessary. It is a lot of work to prove a concept that should be fairly easy for the typical student to intuit.

5 Works Cited Evans, Brian R. The Development of Mathematics throughout the Centuries: A Brief History in a Cultural Context. N.P.: Wiley, 2014, Print. Eves, Howard Whitley. An Introduction to the History of Mathematics. Saunders College Publishing, 1992, Print. Heath, T.L., Euclid: The Thirteen Books of The Elements, Dover, Shuttleworth, Martyn. "Egyptian Mathematics." History of Mathematics. Explorable, 10 Jan Swanson, Mark. "Comparing and Contrasting Ancient Number Systems." University of Colorado Denver, 17 Sept Turcin, Valentin Fedorovic. The Phenomenon of Science. Columbia U.P.,1977, Print. Violatti, Christian. "Greek Mathematics." Ancient History Encyclopedia. Ancient History Encyclopedia, 24 Sept