Pythagoras and the Pythagorean Theorem. April Armstrong

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1 Armstrong 1 Math 409 Honors Fall 2016 Texas A&M University Professor: David Larson Pythagoras and the Pythagorean Theorem April Armstrong Introduction: Pythagoras is recognized for his association with the Pythagorean theorem, which is simply stated as the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse, or in mathematical terms, a 2 + b 2 = c 2 ( Pythagorean theorem ). Although this theorem is short, it is very powerful, and as Pythagoras stated, Do not say little in many words but a great deal in a few ( Pythagoras Quotes ). He and his followers surpassed the basic theorem and ventured into the principles of mathematics and the concepts in mathematics that were applied to the actual proof of the Pythagorean theorem. Many know of the theorem but not many know who Pythagoras actually was, what his love for mathematics accomplished, or what the theorem itself inspired. Pythagoras was an ancient Greek mathematician who was born in the late 6 th century B.C. on the island of Samos. His mother and father encouraged his rare intelligence and Pythagoras traveled the world to understand everything he could. In his travels, he began working under a Greek named Thales, who was another supporter of Pythagoras travels and discoveries (Diggins). Pythagoras ended up having a group of followers called Pythagoreans who were known for their pure lives as they were vegetarians, wore their hair long, wore simple clothing, went barefoot, and even refused to eat beans. They were also known for their passion

2 Armstrong 2 for music and mathematics, as they believed the two went hand in hand as two ways of making order out of chaos. To the Pythagoreans, music is noise that makes sense, and mathematics is rules for how the world works (Carr). Pythagoras is best associated with the Pythagorean theorem, hence the name, but many believe it was discovered before Pythagoras findings. It is believed that Pythagoras was possibly the one who provided the proof that the Pythagorean theorem is always true (Smoller). Pythagoras was a very passionate philosopher who wanted to travel the world and gain as much knowledge about as many topics as possible. He ventured to Babylon and studied with the Chaldean stargazers, traveled to Egypt to study the lore of priests at Memphis and Diospolis as well as studying people who were called rope-stretchers. These were the individuals that actually built the pyramids and their infrastructural secret involved a rope being tied in a circle with twelve evenly spaced knots that, when connected, would produce a right triangle (Smoller). Pythagoras continued to travel and feed his curiosity and is actually coined with being the first person to spread the idea that the earth is round. He became one of the most educated individuals in the world at the time and he wanted to further spread his knowledge but many were not as enthused as he was, so instead, he paid a homeless child to sit and listen to his stories and share his knowledge of the mathematical disciplines (Smoller). The boy became enthralled by Pythagoras words and yearned for more and eventually ended up paying Pythagoras for lessons, much like a tutor in today s world. Pythagoras eventually moved to the Isle of Croton and became the leader of The Secret Brotherhood, a religious order that he created with initiation rights and purifications. For The Secret Brotherhood, knowledge was the ultimate power and the greatest purification in their eyes. To them, mathematics was the utmost important knowledge to have (Diggins).

3 Armstrong 3 Integers that satisfy this theorem are called Pythagorean triples (Smoller). Pythagoras was obviously the inspiration behind the name of the Pythagorean theorem and although he is accredited with the theorem and the proof, he was not the first one to notice the relationship between the three sides of a right triangle. The relationship was actually discovered on a Babylonian tablet circa B.C. (Bogomolny). There is also evidence of an ancient Chinese astronomical and mathematical article called Chou Pei Suan Ching that has a geometrical demonstration of the theorem. Ancient Indian mathematicians also recognized the relationship, as it was discussed in the context of strict requirements for the orientation, shape, and area of the altars used for religious purposes. It has also been suggested that the ancient Mayas used variations of Pythagorean triples in their Long Count calendars (Smoller). The theorem serves indubitable importance in the basic understanding of the distance between two points. There are hundreds of proofs for the Pythagorean theorem. Probably the most famous of the plethora of proofs available for the Pythagorean theorem involves the first of Euclid s two proofs. The proof will reference the following figure: Proof:

4 Armstrong 4 Let ABC be a right-angled triangle having the angle BAC right. Say that the square on BC equals the sum of the squares on BA and AC. Describe the square BDEC on BC, and the squares GB and HC on BA and AC. Draw AL through A parallel to either BD or CE, and join AD and FC. Since each of the angles BAC and BAG is right, it follows that with a straight line BA, and at the point A on it, the two straight lines AC and AG not lying on the same side make the adjacent angles equal to two right angles, therefore CA is in a straight line with AG. For the same reason BA is also in a straight line with AH. And, since the angle DBC equals the angle FBA, for each is right, add the angle ABC to each, therefore the whole angle DBA equals the whole angle FBC. And, since DB equals BC, and FB equals BA, the two sides AB and BD equal the two sides FB and BC respectively, and the angle ABD equals the angle FBC, therefore the base AD equals the base FC, and the triangle ABD equals the triangle FBC. Now the parallelogram BL is double the triangle ABD, for they have the same base BD and are in the same parallels BD and AL. And the square GB is double the triangle FBC, for they again have the same base FB and are in the same parallels FB and GC. Therefore the parallelogram BL also equals the square GB. Similarly, if AE and BK are joined, the parallelogram CL can also be proved equal to the square HC. Therefore the whole square BDEC equals the sum of the two squares GB and HC. And the square BDEC is described on BC, and the squares GB and HC on BA and AC. Therefore, the square on BC equals the sum of the squares on BA and AC. (Bogomolny). The Pythagorean theorem inspired others to further investigate the theorem with higher powers, such as proving x n + y n = z n. This ended up being Fermat s Last Theorem and it states that x n + y n = z n has no integer solutions for x, y, and z when n>2. Fermat wrote, I have discovered a truly remarkable proof which this margin is too small to contain. This statement might have been after realizing that his proof was wrong but it opened challenges to himself and

5 Armstrong 5 his fellow mathematicians (O Connor). Fermat did prove that the area of a right triangle could not be a square, so in other words, a rational triangle could not be a rational square. In symbols, this means that there do not exist integers x, y, z with x 2 + y 2 = z 2 such that xy 2 is a square (O Connor). Ultimately, Pythagoras explorations and discoveries have inspired many others to dive deeper into the wonders of the theorem and other theorems similar to it. Pythagoras contributed a lot more to the history of mathematics such as uncovering that the sum of the angles in a triangle is 180 degrees and establishing the foundations of number theory, but his discoveries were not limited to strictly mathematical observations. Pythagoras also discovered that the intervals between harmonious musical notes always have whole number ratios. He became so obsessed with numbers and their properties that he and his followers believed that the entire universe was based on numbers and that the planets and stars moved in accordance to mathematical equations and how it all connected back to music (Mastin). Pythagoras was one of the greatest Greek mathematicians and philosophers who contributed an enormous amount of pivotal information to the world. The history of mathematics and even the history of music owes a lot to his contributions.

6 Armstrong 6 Homework As mentioned in the paper, there are hundreds of ways to prove the Pythagorean theorem. Apply your knowledge of the Pythagorean theorem to complete the following homework problems. 1. Prove the converse of the Pythagorean theorem, which states: In a triangle with sides a, b, and c (see Figure 1), if a 2 + b 2 = c 2 holds, then ABC is a right triangle with a right angle at C. Note that in the figure, A, B, and C are the angles of the triangle ( Proof of the Converse of the Pythagorean Theorem ). Figure 1 2. Given a triangle that has side lengths of 6 units, 7 units, and 10 units, use the Pythagorean theorem to prove that the triangle is not a right triangle (Roberts). 3. Prove the Pythagorean theorem using algebra. Refer to Figure 2 in your proof as needed. A useful hint would be to use area (Pierce). Figure 2

7 Armstrong 7 4. Prove that the difference of squares of two sides of a triangle equals the difference of squares of their projections on the third side, that is, that AB 2 BC 2 = AH 2 CH 2. Refer to Figure 3 and Figure 4 as needed (Bogomolny). Figure 4 Figure 3 5. Prove that the Arithmetic-Geometric Means Inequality, that is, for positive a and b, a+b 2 ab, with equality if and only if a = b (Bogomolny).

8 Armstrong 8 Answers to Homework 1. Let DEF be a triangle such that EF = a, DF = b and right angled at F (see figure below). If we let DE = x, since DEF is a right triangle, by the Pythagorean Theorem a 2 + b 2 = x 2 (*). But from the supposition, a 2 + b 2 = c 2 (**). From (*) and (**), x 2 = c 2 Since x and c are both positive, we can therefore conclude that x = c. This means that the length of three corresponding pairs of sides of triangle ABC and triangle DEF are equal. Therefore, by SSS congruency, ABC DEF. Since F and C are corresponding angles, F = C = 90 degrees. And hence, we have proved that triangle ABC is right angled at C. ( Proof of the Converse of the Pythagorean Theorem ).

9 Armstrong 9 2. Let a = 6, b = 7, and c = 10. Notice that the longest length must represent the length of the longest side, which is the hypotenuse, c. Using the Pythagorean theorem, we must plug the side lengths into the equation a 2 + b 2 = c 2. Thus, by substitution, a 2 + b 2 = c 2 becomes = 10 2, which, by squaring each term, becomes = 100, which, by adding the terms on the left hand side, becomes 85 = 100. But Therefore, the Pythagorean theorem does not hold for the given triangle. Since the Pythagorean does not hold unless the triangle is a right triangle, this implies that the given triangle is not a right triangle. (Roberts). 3. Observe that the entire square, call it S 1, has four sides each having a length of a + b. This, the area of S 1 is A 1 = (a + b)(a + b) = a 2 + ab + ba + b 2 = a 2 + 2ab + b 2 since we know that ab = ba by the Associative property. The entire square S 1 can be broken into parts, as shown in Figure 2. One part is the smaller square inside of S 1, call it S 2, which has four sides each having a length of c. Thus, the area of S 2 is A 2 = (c)(c) = c 2. Observe that the rest of the area of S 1 is taken up by four triangles, each with an area of A 3 = 1 ab, thus, all four of the triangles added together have a total area of A 2 4 = 4A 3 = 4 ( 1 ab) = 2ab. Adding the areas of S 2 2 and the four triangles yields A 5 = A 2 + A 4 = c 2 + 2ab. Since we know that S 1 is broken into S 2 and the four triangles, we know that the area of S 1 is equal to the combined areas of S 2 and the four triangles. Therefore, A 1 = A 2 + A 4 = A 5. This becomes a 2 + 2ab + b 2 = c 2 + 2ab, and after subtracting 2ab from both sides, we get a 2 + b 2 = c 2. This is the Pythagorean theorem, so we are done. (Pierce).

10 Armstrong For the proof, we must use the Pythagorean theorem twice. The first will be used for AB 2 = AH 2 + BH 2 and the second will be used for BC 2 = CH 2 + BH 2. Subtracting one from the other yields AB 2 BC 2 = AH 2 + BH 2 CH 2 BH 2, which simplifies to AB 2 BC 2 = AH 2 CH 2 ( ), which matches the equation provided in the problem. From Figure 3, if triangle ABC is an obtuse triangle, the CH = AC + AH and ( ) yields BC 2 = AB 2 + AC AC AH, which satisfies the properties of obtuse triangles. From Figure 4, if triangle ABC is an acute triangle, then CH = AC AH and ( ) yields BC 2 = AB 2 + AC 2 2 AC AH, which satisfies the properties of acute triangles. (Bogomolny). 5. Assume a > b and construct a right triangle with hypotenuse a+b and one side equal to a b 2. From the Pythagorean theorem, the remaining side will equal ab. Since, in a right triangle, the hypotenuse is the largest side, the inequality has been proven in the case where a > b. Now assume a = b and substitute it into the inequality. This yields a+a 2 2 aa = 2a 2 a2 = a a and a = a. Conversely, if a+b 2 = ab, then (a+b)2 4 = ab and (a + b) 2 = 4ab, but a 2 + 2ab + b 2 = 4ab is true if and only if a = b, which will yield aa + 2aa + aa = 4aa, which is then 4aa = 4aa. Therefore, it is equal if and only if a = b. (Bogomolny).

11 Armstrong 11 Works Cited Bogomolny, Alexander. "Pythagorean Theorem." Cut The Knot. University of Chicago Press, Nov Web. 24 Mar < Carr, Karen. "Who Was Pythagoras?" Greek Mathematician Pythagoras. Portland State University, Web. 24 Mar < Diggins, Julia E. String, Straightedge, and Shadow: The Story of Geometry. New York: Viking, Print. Mastin, Luke. "Greek Mathematics - Pythagoras." The Story of Mathematics Web. 11 Apr < O'Connor, John J., and Edmund F. Robertson. "Fermat's Last Theorem." MacTutor History of Mathematics Archive. School of Mathematics and Statistics University of St Andrews, Scotland, Feb Web. 24 Mar < Pierce, Rod. Math Is Fun Citation Math Is Fun. Ed. Rod Pierce. 9 Nov <

12 Armstrong 12 "Proof of the Converse of the Pythagorean Theorem." Proofs From the Book. Math Proofs, 16 Feb Web. 27 Oct < "Pythagoras Quotes." BrainyQuote. Xplore, Web. 08 Apr < "Pythagorean theorem". Encyclopædia Britannica. Encyclopædia Britannica Online. Encyclopædia Britannica Inc., Web. 24 Mar < "Pythagorean Theorem (solutions, Examples, Answers, Worksheets, Videos)." Online Math Learning, n.d. Web. 09 Nov < Roberts, Donna. Pythagorean Theorem. Pythagorean Theorem. Oswego City School District Regents Exam Prep Center, Web. 09 Nov < Smoller, Laura. "The History of the Pythagorean Theorem." The History of the Pythagorean Theorem. UALR Department of History, May Web. 24 Mar <