Introduction

Abstract

In this chapter we review a few different proofs of the Pythagorean Theorem. We also define Pythagorean triples, and explain the types of problems we will be interested in studying in the book.

Appendices

Exercises

• 1.1 Let a, b, c be the side lengths of a right angle triangle with c the length of the hypotenuse. Use the dissection in Figure 1.5 of a $c \times c$ square into four triangles and a square to give a proof of the Pythagorean Theorem. This proof is due to the famous 12th century Indian mathematician Bhaskara, [9, §3.3].

• 1.2 Suppose a, b, c are the side lengths of a right triangle. Use Figure 1.6 to give a proof of the Pythagorean Theorem. In the diagram, the three triangles are similar to the original triangle with scaling factors a, b, and c.

• 1.3 Here is an alternative formulation of the idea exploited in Garfield’s proof. Again, suppose a, b, c are the sides of a right triangle. Use Figure 1.7 to give one more proof of the Pythagorean Theorem.

• 1.4 Let ABC be a triangle. Show that

  $$ \ \mathrm {sgn}\ (\angle A + \angle B - \angle C) = \ \mathrm {sgn}\ (BC^2 + AC^2 - AB^2). $$

  Here $\ \mathrm {sgn}\$ is the following function:

  $$ \ \mathrm {sgn}\ (x) = {\left\{ \begin{array}{ll} +1 &{} x > 0; \\ 0 &{} x = 0; \\ -1 &{} x <0. \end{array}\right. } $$

• 1.5 ($\maltese$) List all Pythagorean triples (a, b, c), with $a \le b < c \le 100$.

• 1.6 ($\maltese$) Let N(B) be the number of Pythagorean triples (a, b, c), with $a, b, c < B$. Compute N(B) for some large values of B like 1000, 15000, 100000. Does N(B) / B approach a limit as B gets large? We will investigate this limit in Chapter 13.

Fig. 1.5

The dissection in Problem

Fig. 1.6

Figure for Problem

Fig. 1.7

The diagram for Problem

Notes

1.1 Pythagoreans

Pythagoreans certainly deserve a good deal of credit for their contributions to mathematics, if nothing else for their formalization of the concept of proof. While they may have in fact been the first people in history to have written down a formal proof of Theorem 1.1, there is no doubt that the theorem itself was known much earlier. For example, the Babylonian clay tablet Plimpton 322 described in [9, §2.6], dated between 1900 and 1600 BCE, contains fifteen pairs of fairly large natural numbers x, z, every one of which is the hypotenuse and a leg of some right triangle with integer sides. Even though the tablet does not contain a diagram showing a right triangle, it is hard to imagine these numbers would have appeared in a context other than the Pythagorean Theorem. Furthermore, given the sizes of the entries, 8161 and 18541, among others, it is only natural to assume that these numbers were not the result of random guesswork, and that the Babylonian mathematicians responsible for the content of the tablet actually had a method to produce integral solutions.

Mathematicians in Egypt too were certainly aware of the Pythagorean Theorem. The Cairo Mathematical Papyrus, described again in [9, §2.6], contains a variety of problems, some of them fairly sophisticated, dealing directly with the Pythagorean Theorem. There is also evidence to suggest that the theorem and something resembling a geometric proof of it were known to Chinese mathematicians some 300 years before Euclid, c.f. [9, §3.3]. Dickson [16, Ch. IV] reports that the Indian mathematicians, Baudhayana and Apastamba, had obtained a number of solutions to the Pythagorean Equation independently of the Greeks around 500 BCE.

At any rate, Pythagoreans were led to irrational numbers from the Pythagorean Theorem. Kline [29, Ch. 3] writes: “The discovery of incommensurable ratios [irrational numbers] is attributed to Hippasus of Metapontum (5th cent. B.C). The Pythagoreans were supposed to have been at sea at the time and to have thrown Hippasus overboard for having produced an element in the universe which denied the Pythagorean doctrine that all phenomena in the universe can be reduced to whole numbers or their ratios.”

This most likely refers to the discovery of $\sqrt{2}$. Some historians dispute the story that Hippasus was thrown overboard. The basic argument seems to be that the drowning of the discoverers sounds unlikely—which considering the fact that at the time of this writing fundamentalism in all of its shapes and forms has been eradicated in the world, the skepticism of these historians is justified. There is apparently no historical evidence that Pythagoras himself ever knew of irrational numbers—which, as little as we know of the life of the man, this is not surprising. The earliest reference to irrational numbers is in Plato’s Theaetetus [38, Page 200] where it is said of Theodorus: “was writing out for us something about roots, such as the roots of three or five, showing that they are incommensurable by the unit: he selected other examples up to seventeen—there he stopped.”

Since Theodorus skips over 2 then presumably this means that the irrationality of root 2 must have already been known. In fact there is mention of this in passing in Aristotle’s Prior Analytics [3, §23] and this appears to be the first place this is written down somewhere: “prove the initial thesis from a hypothesis, when something impossible results from the assumption of the contradictory. For example, one proves that the diagonal is incommensurable because odd numbers turn out to be equal to even ones if one assumes that it is commensurable.”

To learn more about Pythagoras and his school, we refer the reader to [9], especially Chapter 3. For the philosophical contributions of the Pythagoreans, see Russell’s fantastic book [42]. For Greek mathematics in general, see Artman [5]. To see some original writings by the Greek masters, see Thomas [51].

1.2 Pythagorean triples throughout history

Proclus, in his commentary on Euclid, states that Pythagoras had obtained the family of Pythagorean triples

$$ {\left\{ \begin{array}{ll} x = 2 \alpha + 1, \\ y = 2\alpha ^2 + 2 \alpha , \\ z = 2 \alpha ^2 + 2 \alpha + 1, \end{array}\right. } $$

for $\alpha$ a natural number, c.f. [16, §IV]. As we will see in §3.1 this family does not cover all solutions. Euclid obtained the solutions

$$ {\left\{ \begin{array}{ll} x = \alpha \beta \gamma , \\ y = \frac{1}{2} \alpha (\beta ^2 - \gamma ^2), \\ z= \frac{1}{2} \alpha (\beta ^2 + \gamma ^2). \end{array}\right. } $$

Diophantus may have been the first person to write the solutions as

$$\begin{aligned} {\left\{ \begin{array}{ll} x= m^2 - n^2, \\ y= 2mn, \\ z= m^2 + n^2. \end{array}\right. } \end{aligned}$$

(1.4)

Dickson [16, §IV] mentions an anonymous Arabic text from the tenth century where necessary and sufficient conditions are derived for the integers m, n so that the triple (1.4) is primitive. The same reference contains numerous other works by many mathematicians which provide various formulations of the solutions of the Pythagorean Equation.

Our purpose here is not to review the history of Pythagorean Equation in its entirety—the references [9, 16] do an impressive job at reviewing the history of the subject, though, see Historical References in Notes to Chapter 2. Our goal in mentioning the above isolated anecdotes is to highlight the fact that mathematics, as all other branches of human knowledge, progresses very slowly—and sometimes what in hindsight looks completely obvious, takes years, centuries, and sometimes millennia, to develop and mature. We sometimes feel smarter than our predecessors because we have learned their works, but in reality the mathematicians of the antiquity were every bit as brilliant and hardworking as the best of us.