Math Project 1 We haven t always been a world where proof based mathematics existed and in fact, the

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1 336 Math Project 1 We haven t always been a world where proof based mathematics existed and in fact, the 1 use of proofs emerged in ancient Greek mathematics sometime around 300 BC. It was essentially the Greeks greatest mathematical accomplishment and it became an important tool in the further development of mathematics to this day. Now, what is a proof and why do we use proofs? A proof is an argument to state that a mathematical statement is true. A proof is created by previously established statements such as theorems, axioms, and postulates. Given that every mathematical statement is considered to be limited to true or false, any and all statements and claims made in a proof must always be considered to be true. As stated above, proof based mathematics has not always existed and there are great differences in proof based and non-proof based mathematics. Proof based mathematics is clear and concise do it s nature of truth. Proof based mathematics proves any and every mathematical statement to be either true or false. Proofs require step by step instruction, proving one statement to be true before moving on to the next. It also creates a pathway to understanding and those who use it typically have thorough understanding of a problem. Non-proof based mathematics creates no reasoning or process to which a statement is true or false. Non-proof based mathematics is usually solved intuitively and this sometimes lead to mistakes. There is not such a large frame of reference to fall back on and it s also harder to identify where mistakes are made due to lack of detail which proofs provide. The difference between ancient Babylonian and ancient Greek mathematics is a great example of the difference between proof and non-proof based mathematics. For example, the Babylonians had to be somewhat familiar with what later came as the Pythagorean theorem because they were able to identify lengths of three sides of a right triangle through some sort of 2 algorithm or process. This is seen through a tablet created by Babylonian s called Plimpton 322; a tablet which contains sets of three positive integers which can be sides of a right triangle and 3 labeled today as a Pythagorean triple. Babylonians were known for their extensive mathematical 4 tables that showed they probably used empirical mathematics to prove something to be true. However, There is no evidence of use of previously true statements as a foundation as to why something is considered to be true. Their idea of proofs were established algorithms to achieve correct answers and they were limited to exactly that. Greeks on the other hand had a backstory to much of their mathematical work. For example, in proving the pythagorean theorem to be true and correct, they defended their process by diminishing any ambiguity and uncertainty by 1 Howard Eves, An Introduction to the History of Mathematics, Sixth Edition (Saunders College Publishing, 1953), Eves, Eves, Eves, 44,

2 thoroughly explaining how they arrived at each individual step. They created theorems, axioms, and postulates to utilize in their argument which assisted diminishing any questioning as to whether the pythagorean theorem was an accurate and true mathematical statement. These pieces in building a proof can be utilized in more than one given math problem. The utilization of proofs in mathematics can be both a helpful and hindering tool and this is seen through the early and quick growth of Greek mathematics as well as its decline after mathematician Appollonius. During the birth of early mathematics such as with the Egyptians and Babylonians, math was strictly utilized for practical application. The world of mathematics changed when men began asking fundamental questions about why a mathematical equation was 5 considered to be true. This led to the emergence of proving mathematical equations and leading 6 the world into transforming mathematics into a logical method of viewing the world. Greek mathematics rapidly began to grow and quickly surpassed mathematical work done anywhere else in the world. The use of proofs was hindering because it was also the cause of the mathematical hiatus or decline in greek mathematics. The decline was due to the fact that proofs 7 limited Greeks to certain areas of mathematics. Ultimately, the limitation stunted their growth. The mathematical proof I worked on was the proof of the Pythagorean theorem as done by Euclid who was a mathematician during the emergence of the use of proofs in ancient Greek mathematics. The problem for the proof is as follows: (AC) 2 = 2 JAB=2 CAD=ADKL Proposition 47 of Euclid s Elements states that a square opposite of a right angle is congruent to the sum of the other two squares. 1. We are given a right triangle with points A, B, and C with the right angle being BAC 5 Eves, David Bramlett and Carl Drake, A History of Mathematical Proof: Ancient Greece to the Computer Age, Journal of Mathematical Sciences & Mathematics Education 8, no. 2, 1. 7 Bramlett and Drake, 6.

3 2.Proposition 46 of Euclid s elements says that on any line, we can construct a square. Therefore, we can construct a square on each line of the given triangle. 3. Proposition 31 states that with any line and any point, we can construct a parallel line through given point. So with point C and line AD, we can construct line CK which is parallel to AD. 4. According to postulate 1 between any two points, we can also construct a line so we construct lines JB and CD 5. We now go back to proposition 47 that states the square opposite of the right angle is congruent to the sum of the other two squares by proving that the two smaller squares are equal to the congruent parallelograms which were created by the division of line CK. 6.We can now begin the process of proving the given proof in relation to proving the pythagorean theorem 7. We must begin by proving triangles JAB and CAD are congruent. 8.We know that we have two squares labeled AJHC and ADEB. By definition, squares have four congruent sides and four right angles. Therefore angle DAB and JAC are right angles and all right angles are congruent so DAB = JAC. From this, we are able to obtain that angle DAB and BAC are equal to JAC and BAC. 9.Upon proving this information we can now prove that angle JAB is congruent to CAD. 10. Proposition 4 states that if two triangles share one angle and two sides, the triangles must be congruent. We can see that triangle JAB and CAD fit this proposition and therefore are congruent triangles. 11. Now, proposition 41 states that if a triangle and a parallelogram share a base and are within the same parallel lines, the parallelogram is congruent to double the triangle given. We can apply proposition 41 with square AJHC and triangle JAB. They have a base in common but we must prove that they are within the same parallel lines- those lines being HB and JA. 12. We know that AJHC is a square and by definition, opposite sides are always parallel. Therefore, line JA is parallel to HC. 13. Because AJHC is a square, we know that all angles are right angles and therefore angle ACH is a right angle. We also know that angle ACB is a right angle. The sum of these two angles is the sum of two right angles. Proposition 14 states that if a straight line has two lines drawn outward from the same endpoint making the adjacent angles congruent to the

4 sum of two right angles, then the two lines must be in a straight line with each other. Because angles ACH and ABC fit this proposition, we know that line HB is a straight line. 14. Because line HC is parallel to line JA and line CB is the same as HC, line JA and line HB must be parallel. 15. We have now proven that triangle JAB and parallelogram AJHC fit proposition 41. This now proves that parallelogram AJHC is equal to two times triangle JAB. 16. Now we can look at parallelogram ADKL and triangle CAD and notice that they have a base in common and are within the same parallel lines, therefore by proposition 41 we can prove that parallelogram ADKL is equal to two times triangle CAD. 17. Going back, we remember that triangle JAB and CAD were congruent to each other. Therefore two times either triangle proves to be equal to either parallelogram which then proves that parallelogram ADKL and AJHC (AC 2 ) are equal to each other. 18. This completes the proof that (AC) 2 = 2 JAB=2 CAD=ADKL *Similarly and by the same methods, we can prove that (BC) 2 = 2 ABF= 2 EBC= BEKL Because parallelogram AJHC is proven to equal parallelogram ADKL and parallelogram GCBF is equal to parallelogram BEKL, The sum of parallelograms AJHC and GCBF must equal ADEB proving the pythagorean theorem! The advantages to this proof are the advantages as stated above in any proof. It is step by step and leaves no room for ambiguity. When given a diagram along with the written proof, you can carefully follow each and every step visually as well. The proof also contains theorems that have already been proven to be true which also gives the problem definite truth and credibility. The proof for this problem is very descriptive and dives very deep into every aspect of the problem proving why every step is true. This may be hindering to some and therefore may be easier to solve by intuition. The problem itself is not extremely complicated but given the length and breadth of its proof can make it seem more complicated than necessary and having to prove some of the simpler steps which most can solve intuitively can confuse or complicate things. For a typical student, I feel as though this problem is better done through a proof because upon proving it myself, I gained knowledge of different theorems, postulates, and axioms that are somewhat general and can be applied in different problems. I feel as though mastering a proof of the pythagorean theorem is important because the pythagorean theorem is such a widely used equation that many students become familiar with during their middle school years and continue to use throughout their time in math courses. Understanding the proof that lies behind a simple equation is humbling and this problem is a specifically great start for creating and understanding proofs in other problems.

5 Bibliography David Bramlett and Carl Drake, A History of Mathematical Proof: Ancient Greece to the Computer Age, Journal of Mathematical Sciences & Mathematics Education 8, no. 2. Howard Eves, An Introduction to the History of Mathematics, Sixth Edition (Saunders College Publishing, 1953). J.L. Heiberg, Euclid s Elements of Geometry, Trans. Richard Fitzpatrick (2007). Euclid's Proof of Pythagoras' Theorem (I.47), accessed Oct. 12,