An Exploration of the Pythagorean Theorem

An Exploration

of the Pythagorean Theorem

Sam Otten MTH 210A W04

An Exploration of the Pythagorean Theorem More than two thousand years ago, a scholar wrote the treatise that has become a foundation of mathematical thinking to this day. The time was near the beginning of the third century before the common era (B.C.E.), and Euclid, after being schooled in Athens by students of Plato, went on to become a scholar and a teacher in the city of Alexandria, Egypt.6 Euclid set to work writing his masterpiece – the Elements. The Elements is a scientific effort in thirteen volumes that contains the basis for elementary geometry, number theory, proportional representations, the theory of limits, and offers a proof and analysis of the Pythagorean Theorem (which we will discuss in greater depth throughout this paper).3 While Elements is not completely his own composition, Euclid did the work of gathering past mathematical results and presenting them in a logical and scientifically rigorous fashion.6 The Elements is organized progressively, wherein results are drawn from earlier definitions, postulates, and previously proved theorems. With this in mind, Euclid started by explicitly defining terms and operations that would be used in his writing.3 He also described five postulates that are the basis of what is now referred to as Euclidean Geometry; they are 1) A line can be drawn through two points, 2) A segment of a line can be extended indefinitely, 3) A circle can be drawn around a point given a radius, 4) All right angles are equal to one another, 5) If two lines both intersect a third line, and if the sum of the interior angles on one side of the transversal is less than 180°, then the first two lines will eventually intersect on that side.3 Some interesting points about these postulates are that (4) can be replaced by rewriting (1) to say “a unique line,” and the fifth postulate of Euclid has not been proven and is therefore assumed in Euclidean Geometry but it is not assumed in the discipline known as Non-Euclidean Geometry.3 The first of the thirteen books of Euclid’s Elements deals with the fundamentals of triangles, rectangles, and parallelograms. It offers formulas and comparisons for the areas of these basic

2 figures and the book ends with arguably the most important finding in mathematical geometry – the Pythagorean Theorem.3

Euclid was not the first to discover the Geometric Representation of Py th. Theorem relationship between the sides of a right triangle. There is evidence from ancient Babylonian culture as well as some Eastern Civilizations which indicates they were aware of the fact that the sum of the areas of the squares A built on the legs of a right triangle is equal to the area of c b a square built on the remaining side.8 The Greek mathematician Pythagoras (after whom the theorem is C a B named) receives popular credit for the finding because his school managed to write it down and offer a proof in the middle of the sixth century B.C.E.8 Today, the relationship between the legs of a right triangle (a, b) and its hypotenuse (c) is familiar to everyone who has been in a middle or high school geometry class: a 2  b 2  c 2 . There are many proofs of this theorem, from such sources as Pythagoras, Euclid, Bhaskara (a 12th Century Hindu mathematician), Leonardo Da Vinci, and even former United States President James Garfield.2,7,8 A proof will be included in this paper to help us gain insight into the Pythagorean Theorem.

Theorem: If ABC is a right triangle with legs AC and CB and hypotenuse AB, then the sum of the squares of the two legs is equal to the square of the hypotenuse.

 Proof: Let ABC have angles of , , and , making it a right triangle. Let us also 2 draw in the line segment CD that is perpendicular to the hypotenuse. We now have three right triangles: ABC, ACD, and CBD.

3 A  D C A 

  C B D C D B

By the theorem which states that the sum of the angles of a triangle equals  , we can conclude

 from ABC that      . Looking at ACD we see that two of the angles are defined, 2

 namely  and . By the theorem mentioned above we know that the sum of all three angles 2 must equal  and it can be concluded that the third angle is . Similarly, two angles from

 CBD are known to be  and while the third is determinedly . We see that the three 2 triangles have the same angle measurements which means, by the Angle-Angle-Angle Similarity Theorem from Euclidean Geometry, that they are similar triangles.

Applying the Fundamental Theorem of Similar Triangles we know that the following ratios hold:

AC  AD and CB  DB . AB AC AB CB

By cross-multiplying each ratio relationship we see that

AC  AC  AD  AB and CB CB  DB  AB.

At this point we will combine the two equations above by adding them, which gives us

AC  AC  CB CB  AD  AB  DB  AB.

We will now rewrite the left-hand side using power notation and we will factor out the AB from both terms of the right-hand side, resulting in

4 D is tanc e Form ula

2 2 2 2 . x +y AC  CB  AB AD  DB y      

x The combination of the two line segments AD and DB equals

the line segment AB, therefore A ltitudes of T riangles

2 2 2 AC  CB  AB

b2 where AC and CB are the legs of a right triangle and AB is the c 2- 4 b hypotenuse. This proves that the sum of the squares of the legs of 2 a right triangle is equal to the square of its hypotenuse.2,4,7 ٱ This result has become one of the most fruitful in the field of mathematics. The Pythagorean Theorem is the basis for the distance formula in Euclidean geometry. 8 It provides the means for determining the distance between any two points when given their horizontal and vertical components. The theorem can also be used to calculate lengths of inclines, the length of rectangular diagonals, and the altitude of triangles which then leads to triangular areas.7,8 The trigonometric identity sin 2 x  cos2 x  1 has its roots in the Pythagorean Theorem as well, applying the theorem to points on the unit circle. These applications, along with countless others, are immeasurably useful in geometry, trigonometry, graphical algebra, physics, engineering, cartography, navigation, and graphic design. Thus far our interpretation of the Pythagorean Theorem has been strictly two- dimensional. Does the Pythagorean Theorem extend into higher dimensions? This is a question that mathematicians have been contemplating for centuries. In his 1673 work Aritmetika, a French mathematician by the name of Pierre de Fermat offered the following conjecture about the integral generalization of the Pythagorean Theorem into higher dimensions:

It is impossible to partition a cube into two cubes, or a biquadrate into two biquadrates, and in general any power greater than the second into two powers with the same exponent.3

5 This is known as Fermat’s Last Theorem because the mathematician failed to prove his conjecture, claiming the “margin of my notebook is too small to contain” the proof.1 This claim by Fermat essentially says that the Pythagorean Theorem only applies to squares – the second power. Scholars realized this limitation on Pythagoras’ theorem would be a vital piece of information and the search for a proof of Fermat’s Last Theorem began. Before World War 1 there was a reward offered to anyone who could supply a valid proof of the theorem, but mathematicians still came up empty.3 Proofs existed concerning the third, fourth, and fifth powers thanks to Euler, Fermat, and Legendre respectively; but Fermat’s claim was for all powers greater than two and the few sample proofs that existed did not suffice. 1 The scholarly world became doubtful, and in 1984 the Kluwer Encyclopædia of Mathematics cited a strong possibility that a proof did not exist to be found.3 In 1993, after decades of hard work by countless minds, mathematical history was made. Andrew Wiles of Princeton University announced that he had proven Fermat’s Last Theorem to be true.1 Wiles implemented the ‘proof by contradiction’ argument form. He used two previously proven results related to the theorem and, assuming Fermat’s Last Theorem was false, showed that while one held true the other could not – a contradiction. Wiles then concluded that his original assumption must have been false and therefore Fermat’s Last Theorem was true.1 While this brief summary of Wiles’ work may not seem too extraordinary, keep in mind that it required 200 pages of rigorous mathematical manipulations and had eluded mathematicians for hundreds of years.1 What does it mean for Fermat’s Last Theorem to hold true? It can be viewed as a limitation on the generalization of the Pythagorean Theorem. Algebraically it means that there are no integral solutions for the equation a n  b n  c n when n  2. 3 Geometrically it means that an integral square (a two-dimensional object) can be divided into two smaller integral squares as Pythagoras’ Theorem illustrates, but an integral cube cannot be divided into two smaller integral cubes, nor can any similar object in any higher dimension. Notice that Fermat’s Last Theorem applies only to integral values of a, b, and c, indicating the result is essentially contained within the realm of algebraic number theory.5 Therefore, the proof of Fermat’s Last Theorem does not remove all possibilities for the generalization of Pythagoras’ Theorem. In their 2003 work, Putz and Sipka were able to generalize the Pythagorean Theorem by remaining in two dimensions and by taking a geometric rather than algebraic approach. They

6 realized that the truth of Fermat’s Last Theorem demonstrates that the exponent of the Pythagorean Theorem cannot be generalized; instead, they generalized the shape that is formed using the sides of a right triangle.5 Earlier it was discussed and illustrated that the Pythagorean Theorem can be thought of as the equality of squares built on the sides of a right triangle. Putz and Sipka take this further and show that “the area of a polygon constructed on the hypotenuse of a right triangle is the sum of the areas of similar polygons constructed…on the other two sides.” (They were, of course, not the first to make this generalization, as they point out in their writing.) To prove that the area of any polygon built on the hypotenuse of a right triangle is equal to the sum of the areas of similar polygons built on the legs of the same right triangle, Putz and Sipka incorporated the use of a lemma. Before we state the lemma, let us think of two similar right triangles with ratio k. This means if the area of the original right triangle is calculated by b  h kbkh , then the similar right triangle would have an area of or a factor of k 2 compared 2 2 to the original area. This intuitively leads us to the lemma used by Putz and Sipka, which was the fact that two polygons, P and P, which are similar by a factor of k will have the following area relationship: AP  k 2 AP. 5 They proved this using Heron’s formula to first show that the relationship holds for triangles, and then they demonstrated that any polygon can be triangulated so that its area is the sum of the areas of triangles.

Shape-Generalization of Pyth. Theorem

A(1)+A(2) = A(3)

Given a right triangle, geometrically similar figures can be drawn on each side. Assuming the original figure is the polygon built around the hypotenuse c, the polygons drawn

a b on the legs a and b are said to be similar with a length-ratio of and respectively because c c

7 a b  c  a and  c  b. Letting AR represent the area of the region R, and letting P be the c c 3 polygon built on the hypotenuse and P1 , P2 be similar polygons built on a and b, Putz and Sipka used their lemma and were able to show that

2 2  a   b  a 2  b 2 AP1  AP2     AP3    AP3   AP3 .  c   c  c 2

a 2  b 2 It has already been proven in this paper that a 2  b 2  c 2 , so it holds true that is 1. c 2

Simplification then results in

AP1  AP2   AP3  and it becomes clear that the area of a polygon with one side the hypotenuse of a right triangle is equal to the sum of the areas of similar polygons built on the other two sides of the same right triangle.5 Further examination also shows that areas of circles maintain the Pythagorean relationship when built around the legs of a right triangle because circles can be viewed as “a limiting process using polygonal approximations.”5 Therefore, while the truth of Fermat’s Last Theorem limits the generalization of the exponent in Pythagoras’ Theorem, Putz and Sipka were successful in their generalization of the area equality between non-square similar figures constructed on the sides of a two-dimensional right triangle. We return now to where we started. More than two thousand years ago a brilliant man is writing the treatise of his field. The theorems that Euclid offered in his Elements, the logical approach with which he developed them, and the expansiveness of the topics that he covered has been and will continue to be a foundation for the scientific field of mathematics. The Pythagorean Theorem is a perfect case study for Euclid’s contribution to mathematical thinking and applications. Like much of his work, Euclid did not discover the Pythagorean Theorem, but he garnered its truth and placed it in a framework of logical definitions, postulations, and conclusions, making it even more potent. The importance of the theorem in today’s world is immeasurable as it reaches thoroughly through many disciplines. Also, like much of his work,

8 Euclid’s discussion and presentation of the Pythagorean Theorem has sparked mathematical thinking and promoted novel research to the present day. Fermat, Wiles, Putz, Sipka, and countless other mathematicians and thinkers become connected to Euclid (and further to Pythagoras) as they work to generalize and explore all implications of the Pythagorean Theorem. It is fascinating to look back and ponder the notion that mathematics not only has the power to help us better understand reality, it has the transcending power to form bridges that span millennia. It is even more fascinating to look ahead to the future for intellectual efforts that have yet to be made, to think about the findings that await us, and to imagine the genesis of ideas that may still grow out of what we already know.

Works Cited

(1) Begley, S., & Ramo, J.C. (1993, July 5). New answer for an old question. Newsweek,

p. 52-53.

(2) Bogolmony, A. (1996). Pythagorean theorem and its many proofs [On-line]. Retrieved Jan. 17, 2003 from source: http://www.cut-the-knot.org/pythagoras/index.shtml.

(3) Encyclopædia of Mathematics, 3. (1989). Dordrecht, The Netherlands: Kluwer Academic

Publishers.

(4) Greenberg, M.J. (1974). Euclidean and non-Euclidean geometries: development and history. San Francisco, CA: W. H. Freeman.

(5) Putz, J.F., & Sipka, T.A. (2003, Sept). On generalizing the pythagorean theorem. The College Mathematics Journal, 34, p. 291-295.

(6) Ryan, P.J. (1986). Euclidean and non-Euclidean geometry: an analytic approach. New York, NY: Cambridge University Press.

9 (7) Usiskin, Z., Hirschhorn, D., et al. (2002). Geometry: second edition, p. 467-468. New Jersey, NJ: Prentice Hall, Inc.

(8) West, B.H., Griesbach, E.N., et al. (1982). The prentice hall encyclopedia of mathematics, p. 461-465. Englewood Cliffs, NJ: Prentice Hall Publishing.