Math Number 842 Professor R. Roybal MATH History of Mathematics 24th October, Project 1 - Proofs

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1 Math Number 842 Professor R. Roybal MATH History of Mathematics 24th October, 2017 Project 1 - Proofs Mathematical proofs are an important concept that was integral to the development of modern mathematics. They helped mathematics evolve in a logical fashion and is now considered a fundamental aspect of mathematics, but what is a proof? At its most basic form, we can say that a proof is an argument that provides evidence in order to establish a certain idea as a fact. If we apply this definition to mathematics, a mathematical proof is an argument that provides evidence in order to establish a certain mathematical idea as fact. By taking other established, proven statements, we can apply those statements in a variety of situations in order to further prove other, different mathematical statements and expand our knowledge in mathematics in a logical and consistent way. The development of the mathematical proof happened over the course of history. Proof-based mathematics relied on having definitions and a list of axioms and postulates we can use in our proofs (Krantz 7). By having these lists of definitions and already established axioms, proof-based mathematics often withstand the test of time and history, allowing those in the future to understand the reasoning behind the math. Because of the new discoveries proven through the use of proofs, ancient Greek mathematics, which was proof-based, evolved over time and became a basis for modern mathematical thinking. On the other hand, non-proof based mathematics, while still pertinent, was based more on intuition and tended to be more practical. For example, ancient Egyptian mathematics did not have many theoretical contribution and contained little evidence of abstract thought (Allen - Egypt 20). According to Allen, Egyptian mathematics, while extremely practical, did not venture much into abstract mathematics and therefore remained stagnant throughout their history. Their approaches to similar mathematical problems were also different. One famous example is the Pythagorean theorem. Most ancient civilizations knew about the relationship between the sides of a right triangle. For example, the Babylonians knew about the existence of Pythagorean triples (Allen - Babylon), however, they had no evidence of a written proof, and thus their knowledge was limited to a list of Pythagorean triples. The ancient Greeks, on the other hand, not only knew about the existence of the relationship between the sides of a right triangle, they also provided a proof about it, one of which could be found in Euclid s Elements. 1

2 While ancient Greek mathematics benefited greatly from the use of mathematical proofs, there were also several disadvantages. Here, we still explore both advantages and disadvantages of proof-based mathematics from ancient Greece. Some of the advantages of using mathematical proofs include the ability to expand upon other people s works. By having a foundation in a different proof, ancient Greek mathematicians could expand upon past work and derive new ideas. For example, Pythagoras had proven the relationship between the sides of a right triangle as a 2 + b 2 = c 2, with a and b being the legs of a right triangle and c the length of the hypotenuse, the longest side of the right triangle. By using this theorem, Pythagoras also proved the existence of irrational numbers. By setting the length of a and b to be one each, c 2 must equal two; however, Pythagoras found no such rational number exists and, therefore, the 2 must be an irrational number. By using his previous theorem, Pythagoras could prove the existence of another different mathematical concept, irrational numbers. While proof-based mathematics has its clear advantages, it also had its disadvantages. Proof-based mathematics often led to shocking conclusions, such as the one Pythagoras made with irrational numbers, which led to more unanswered questions in the mathematician and philosopher circles. According to Krantz, the discovery of irrational numbers created much upset and confusion when discovered, and took thousands of years of mathematicians to implement them into mathematics (11-12). Let us look at an example of a proof. If we assume the equality of alternate interior angles formed by a transversal cutting a pair of parallel lines, prove the following: a) the sum of angles of a triangle is equal to a straight angle b) the sum of interior angles of a convex polygon of n-sides is equal to (n 2) straight angles. Let us look at part a first. Let us cut two parallel lines with two transversal lines, such so that they intersect at one point with one of the parallel lines. Here, we can label the three angles that make a straight line at the three point intersection A, B, and C. Because it is given that alternate interior angles are equal, let us label the other angles of the triangles A and B as well, since they have equal measure to those angles. 2

3 Here, we can see from the top parallel line that angles A, B, and C make a straight angle, or 180. Thus, the interior angles of ABC must also equal 180. Now that we have proven part a, let us look at part b. Let us look at a convex, five-sided polygon, where n = 5. Let us divide the pentagon into triangles, as seen in the figure below. If we split up the pentagon in this fashion, we can only create three triangles, or 5 2 sides. Because each triangle contains 180, as seen in part a, the total sum of the interior angles of this polygon must be (5 2) 180 or (n 2) straight angles, where n = 5 in this case. This situation will occur for any convex polygon, as we can see in the figure below. 3

4 We can see in this seven-sided figure that it can only create five triangles, or n 2 triangles, where n = 7 in this case. Solving this problem by proof helped ensure that the reasoning and logic behind this mathematical idea was indeed correct and factual. Approaching this problem through a proof also helped with making connections between the information that was given to the end goal. For example, by looking at the alternate interior angles, one could see one half of a triangle and figure out the next step in the proof. Some disadvantages of using a proof-based approach could be the difficulty translating one s thought and connections into language and mathematical thought. One might also not know how to explain how to divide the polygon without the aid of pictures. Sometimes, one cannot make the connections between different trains of thought. For example, in this problem, one may not know to divide the polygons into triangles in a particular fashion in order to reach the conclusion. Solving a problem intuitively or by proof could be both advantageous and disadvantageous to a student. If the student is familiar with mathematical terms and the premises given for the proof, then that student should solve by proof. Having a step-by-step proof could benefit the student and help them apply that proof to other situations. Proofs could also be used to augment their learning and help them appreciate the underlying mathematics behind the idea. Otherwise, it is better for a student to learn intuitively. Some students are intimidated by the language and terms used in proofs. It could dissuade them from attempting the problem and become even more confusing. The student needs a good foundation in mathematics if they are to learn from proofs. 4

5 References Allen, Donald. Babylonian Mathematics. ~dallen/masters/egypt_babylon/babylon.pdf Allen, Donald. Egyptian Mathematics. masters/egypt_babylon/egypt.pdf Krantz, Steven. The History and Concept of Mathematical Proof. February 5,

P1-763.PDF Why Proofs?

P1-763.PDF Why Proofs? During the Iron Age men finally started questioning mathematics which eventually lead to the creating of proofs. People wanted to know how and why is math true, rather than just