Pythagorean Theorem

Fields of Study: Fields of Study: Communication; Connections; Geometry.

Summary: The Pythagorean theorem is a fundamental theorem of mathematics and has numerous applications in number theory and geometry.

The Pythagorean theorem stands as one of the great theorems of mathematics. Ancient peoples appear to have used the Pythagorean theorem to calculate the duration of lunar eclipses or to create right angles in their pyramids or buildings. Archeological evidence suggests that the truth of the result was known in Babylon more than 1000 years before Pythagoras, approximately 1900–1600 BCE Mathematicians and historians continue to debate the early history of the theorem and whether it was discovered independently in such places as Mesopotamia, India, China, and Greece. For instance, some theorize that Pythagoras may have learned the theorem during a visit to India, which in turn may have been influenced by Mesopotamia. The theorem is the culminating proposition of the first book of Euclid’s Elements. While Euclid (c. 350 BCE) did not mention Pythagoras, later writers such as Cicero and Plutarch referred to it as his discovery. As phrased in the twenty-first century, the theorem states the following:

In any right triangle, the square of the hypotenuse c is equal to the sum of the squares of the legs a and b. That is, a²+b²=c².

The theorem has inspired countless generations, and it is useful in a wide variety of contexts and applications, such as in chemistry cell-packing and music.

In Pythagoras’s day, humankind had not yet invented algebra. As such, this theorem was not viewed with algebraic perspective but rather in a distinctly geometric way. Visually, as shown in Figure 1, on the right triangle with legs a and b and hypotenuse c, the sum of the areas of the darker gray squares is equal to the area of the lightest gray square.

Proofs

Among the many remarkable features of the Pythagorean theorem, one of the most prominent is that the result admits so many different proofs, including one by former US President James Garfield in 1876. Some of the shortest representations of the Pythagorean theorem are geometric figures called “dissections.” For example, Indian mathematician Bhaskara’s dissection figure was accompanied by the word “Behold.” The Chinese also presented dissection figures that are now called “Pythagorean,” and some theorize that these may have led to the development of tangram puzzles. Complete Pythagorean proofs based on dissection figures often combine algebra and geometry.

Given a right triangle with legs of length a and b, construct a square of side length a+b. Then, along each side, mark a point that lies a units along the side. If consecutive pairs of these points are connected with line segments, four identical (congruent) copies of the original triangle have been constructed inside the large square (see Figure 2).

In addition, these four line segments have generated a quadrilateral (a four-sided polygon) in the interior of the large square. This quadrilateral’s sides each have length c, which is the hypotenuse of the given right triangle. Furthermore, a straightforward argument involving angle measurements in the triangles shows that each of the four angles in the interior quadrilateral measures 90 degrees. Hence, the inside quadrilateral is in fact a square.

Consider the area of Figure 2 in two different ways. First, the area A of the entire outside square, which has sides of length a+b, must therefore be A=(a+b)². At the same time, one can view the area of the outside square as having been subdivided into five parts. Four of those pieces are congruent right triangles whose area is each ab/2. The fifth part is the interior square, whose area is c². Thus, the area A of the outer square also satisfies the relationship that

Equating the two different expressions for A, one finds

Expanding the left side and simplifying the right, it follows that a²+2ab+b²=2ab +c².

Finally, subtracting 2ab from both sides, the conclusion of the Pythagorean Theorem follows: a²+b²=c².

Applications

Furthermore, the Pythagorean theorem is rightly viewed as one of the most central results in Euclidean geometry. Its statement is equivalent to Euclid’s parallel postulate, and therefore is directly tied to the truth of a large number of other key results.

In addition to the geometric ideas the Pythagorean theorem evokes, it generates key new ideas and questions about numbers. For instance, if one takes the legs of a right triangle to each have length 1, then it follows that the hypotenuse c is a number such that c²=2. There is no rational number (that is, no ratio of whole numbers) whose square is 2. This situation forced Greek mathematicians to reconsider their original conviction that all numbers were “commensurable”: that any possible number must be able to be expressed as the ratio of whole numbers. Remarkably, it took mathematicians another 2000 years to put the so-called real numbers, the set of numbers on which calculus is based, on solid footing.

Another Pythagorean idea that has generated a remarkable amount of mathematics is the notion of a “Pythagorean Triple,” which is an ordered triple of whole numbers like (3, 4, 5) that represents a solution to the Pythagorean theorem, since 3²+4²=5². A Babylonian clay tablet, named the “Plimpton 322 Tablet,” contains many Pythagorean triples. Some suggest that these were a set of teaching exercises, though historians and mathematicians continue to debate their role. Euclid is credited with the development of a formula that will generate a Pythagorean triple, given any two natural numbers. Indeed, there are even infinitely many “primitive” Pythagorean triples, triples in which a, b, and c share no common divisor. Algebraic extensions include investigating solutions to Pythagorean-like equations with other powers, such as a³+b³=c³.Remarkably, no three positive numbers satisfy such equations; Pierre de Fermat, a French lawyer in the seventeenth century, wrote this note (as translated by historians) in the margins of Diophantus of Alexandria’s Arithmetica:

I have discovered a truly marvelous proof that it is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. This margin is too narrow to contain it.

No one ever discovered Fermat’s proof, yet Fermat’s Last Theorem stimulated the development of algebraic number theory in the nineteenth century, and many results in mathematics were shown to be true if Fermat’s Last Theorem was true. Andrew Wiles finally proved it to be true near the end of the twentieth century.

There are many other extensions of the Pythagorean theorem. Pappus of Alexandria generalized the theorem to parallelograms. In the 1939 film The Wizard of Oz, the Scarecrow recites a version using square roots instead of squares. The Scarecrow’s theorem is false in planar geometry, but it can hold in spherical geometry. However, the Pythagorean theorem does not hold on a perfectly round planet. In this case, a²+b²>c².Writers for the animated television show Futurama named this the Greenwaldian theorem, after mathematician Sarah Greenwald. In the twenty-first century, physicists and mathematicians investigate whether the Pythagorean theorem holds in our universe.

The Pythagorean theorem is also a fundamental idea in several other areas of mathematics and applications. Essentially all of plane trigonometry rests on the Pythagorean Theorem as its starting point, and the modern notion of “orthogonality” in linear algebra is an extension and generalization of the work of Pythagoras. Both trigonometry and orthogonality lead to a wide range of interesting and important applications, including the theory of wavelets and Fourier analysis, mathematics that enables prominent image compression algorithms to help the Internet function.

Its own inherent beauty, the multitude of possible proofs, the rich mathematical ideas it spawns, and the applications that follow all contribute to making the Pythagorean theorem one of the genuine masterpieces in all of mathematics.

Bibliography

MacTutor History of Mathematics Archive. “Pythagoras’s Theorem in Babylonian Mathematics.” http://www-history.mcs.st-andrews.ac.uk/HistTopics/Babylonian‗Pythagoras.html.

Maor, Eli. The Pythagorean Theorem: A 4000-Year History. Princeton, NJ: Princeton University Press, 2007.

Posamentie, Alfred. The Pythagorean Theorem: The Story of Its Power and Beauty. Amherst, NY: Prometheus Books, 2010.