P1-763.PDF Why Proofs?

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Or would you want an explanation of how they came upon that fact? People enjoy having answers and knowing what is accurate and what is not.

1 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 assuming. Imagine in today's 21st society if someone told you a fact, but they did not have any proof. Would you just believe them? Or would you want an explanation of how they came upon that fact? People enjoy having answers and knowing what is accurate and what is not. Euclid is a well known Greek mathematician who began working with proofs and ended up perfecting the way to write them. A proof can be defined as an argument that is used to demonstrate if something is true or false. Mathematical Proofs provide solid foundations for testing theories and answering the question as to why. They are specific rules that are used to prove a theorem to be true or they may find out it is false. Euclid recorded all of his Proofs into a series of 13 books called The Elements. Euclid s 13 books explain the development of the postulational way of thinking. (Storyofmathematics.com/greek) Proof based mathematics allows for further development and problem-solving. Once a proof has been created people can use it for an explanation of other problems as well. Mathematical Proofs enable us to think and create new formulas and equations, being able to relate one proof to other mathematical problems is a significant advantage. It saves time and allows for further development and studies to continue. We don t have to start all over when we use proofs; instead, we can use previous work on future problems. With the help from Proofs, we can further our understanding of math and figure out which theorems are true. Non-proof mathematics are based on intuition, and although many people believed in them, they did not provide evidence other than coincidence to show that they are true. An excellent example of non-proof based mathematics is the Egyptian s Pythagorean Triples Theorem. We can compare it to the Greeks Proof based Pythagorean Triples Theorem. The Egyptians believed in the mystique of numbers and used intuition as their logical reasoning for Pythagorean Triples. They

2 believed in a particular triangle that is known as the 3, 4, 5 triangle. They thought that this triangle was this way because the gods made it like that. Ancient Egyptians understood that everything in the universe is animated by life forces. Therefore, each particle is in constant movement and has interactions due to the effect of these life forces. (Francini, 2009) The Egyptians compared everything they knew about triangles with the Pythagorean triple 3,4,5 triangle however, their calculations did not enable them to use the triples theory on other triangles. Such as; a 5, 12, 13 triangle, their calculations were not adding up and were not able to continue any new development on other triangles. The Greeks used proof to explain their Pythagorean Triple theorem rather than intuition. They wanted to know how and why, so they used proofs as a way of visible rules to understand the triangles. These rules were able to apply to all triangles not just one type, like the Egyptians theory. It lead to further development in math and the continue of proof based mathematics. (Francini, 2009) Euclid s books The Elements, have shaped foundations in modern mathematics as well as answered many how and why questions. Since people started asking questions about the absolutes of math, we have been able to find out new developments. Proofs allowed us to understand and learn different things than we were told to believe in the past. Mathematics during this time almost became its own religion. People started becoming obsessed with creating Proofs and wanted answers to everything. It began to cause separation between people, and although the Greeks brought many advantages in math, there were also complications. Many people were angry and upset that what they had believed for so long is now considered wrong. For example; proofs allowed Pythagoreans to understand irrational numbers. As a fraction, an irrational number can t be expressed as a ratio of integers. Interestingly, these figures when expressed as a decimal do not repeat or terminate. This changed their entire idea of whole numbers, many found it interesting and wanted to learn more. However, others didn t like being told that what they knew to be true was wrong, and people started fighting and getting angry over this. It started to create a divide between those who wanted to base mathematics on intuition and those who wanted to base it on Proofs. It lead to physical fights where people were beaten and some died. All this violence was over mathematics. Many Greeks wanted to use Proofs for everything because it saved time and they didn t have to start all over when moving on to a new mathematical problem. This contributed to new concepts and development of mathematics. However, another disadvantage of using proofs that hindered Greek mathematics is that they now felt for something to exist it must be proven. It started limiting them and created distress when they couldn t prove certain things. (Krantz, 2007)

The Angle-Sum of a Triangle Proof: Assuming the equality of alternative interior angles formed by a transversal cutting a pair of parallel lines, Prove the following: a) The sum of the angles of a

The top point of the triangle should intersect line d. Both bottom points of the triangle should intersect line e. Step 3 : Label each point as A, B, C.

3 The Angle-Sum of a Triangle Proof: Assuming the equality of alternative interior angles formed by a transversal cutting a pair of parallel lines, Prove the following: a) The sum of the angles of a triangle is equal to a straight angle. Step 1 : Draw two parallel lines. Now label the top line d and the bottom line e. Step 2 : Now make a triangle between the two parallel lines. The top point of the triangle should intersect line d. Both bottom points of the triangle should intersect line e. Step 3 : Label each point as A, B, C. Step 4 : Place a point to the left of point C, on line d and label it X. Place a point to the right of point C on line d, and label it Y. Step 5 : Label the line AC as f, Label line BC as g. Now take notice of all the angles in the diagram. Step 6 : Since lines d and e are parallel, lines f and g are transversal because they cut a set of parallel lines. Step 7 : We are told: Assuming the equality of alternative interior angles formed by a transversal cutting a pair of parallel lines. This means angle ABC is equal to angle BCY, and angle BAC is equal to angle ACX. Step 8 : Observe that angle ACX plus angle ACB plus angle BCY equals 180 degrees. This is because they are all on the same line. Step 9 : Therefore, the measurements of angle ACB plus angle BAC plus angle ABC is equal to 180 degrees. Step 10 : The sum of the interior angles are equal to a straight angle.

4 b) The sum of the interior angles of a convex polygon of n sides is equal to (n-2) straight angles. Step 1 : Draw a complex polygon with n number of sides. (I choose seven.) Step 2 : Start be dividing the polygon into triangles that share a given point. Step 3 : Label the vertices A1, A2, A3, A4, A5, A6, and A n. Step 4 : Notice the A1 constructs diagonals to all the other vertices. This convex polygon has seven sides and five triangles. The five triangles are produced after completing step 2. Step 5 : Notice we have ( n -2) triangles. n = number of sides. So ( n -2) triangles were produced in the interior part of the polygon. Explanation/analysis: We know that the sum of the angles of a triangle is equal to 180 degrees, this is because of the Proof we solved for part a above. Since we have produced five triangles in the interior part of the convex polygon, we can conclude that the sum of the interior angles as (5 X 180). This is because the sum of the interior angles of a convex polygon is ( n -2), n = number of sides. Therefore (7-2) = 5. So, (5 X 180) = 900. Solving this problem by proof can be an advantage because we can answer the how and why questions. Using this proof we can use it for other triangles, not just this one. It allows for further development with other equations and mathematical problems. However, a disadvantage may be that this specific proof relies on other past proven knowledge. We need the past knowledge to solve this proof. If someone doesn't have that, then this proof wouldn t be much help to them. For a typical student, I believe it would it be better to solve this problem by proof. It would be difficult to explain how the sum of the angles of a triangle is equal to a straight angle by using only intuition. Without the proof, we would be going off of what someone just told us not knowing whether it is accurate or not. This could lead to many other complications and errors in other mathematics when trying to use intuition to continue further studies and theorems. (Eves, 1992)

5 References Eves, H. W. (1992). An introduction to the history of mathematics: with cultural connections. (6th ed.). Forth Worth, USA: Saunders. Euclid - Google Search. (n.d.). Retrieved March 20, 2017, from Francini, A. N. (2009). Egyptian Numerology: the Pythagorean Triangle and Its Esoteric Meaning. Rosicrucian Digest, No. 1. Greek Mathematics - The Story of Mathematics. (n.d.). Retrieved March 04, 2017, from Krantz, S.G. (2007). The History and Concept of Mathematical Proof. Retrieved March 22, 2017, from Pythagoras - Greek Mathematics - The Story of Mathematics. (n.d.). Retrieved March 24, 2017, from Sum of Interior angles of an n-sided polygon. (n.d.). Retrieved March 10, 2017, from The sum of the angles in a triangle is 180 degrees. (n.d.). Retrieved March 10, 2017, from

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

Math Number 842 Professor R. Roybal MATH 331 - History of Mathematics 24th October, 2017 Project 1 - Proofs Mathematical proofs are an important concept that was integral to the development of modern mathematics.