Famous Classical Theorems
There are many famous theorems in mathematics. Some are known for their importance, others for their depth, usefulness, or sheer beauty. In this chapter we discuss seven of the most remarkable classical theorems; in the next chapter, we discuss three othe
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Famous Classical Theorems
There are many famous theorems in mathematics. Some are known for their importance, others for their depth, usefulness, or sheer beauty. In this chapter we discuss seven of the most remarkable classical theorems; in the next chapter, we discuss three others from more recent times. Our choices for this top ten list were motivated primarily by the nature of their proofs; we apologize if we did not choose your favorite theorem. (A more representative top 40 list can be found in Appendix D at the end of the book.) Here we included theorems that are considered to have the oldest, the most well-known, the most surprising, the most elegant, and the most unsettling proofs. Some of the theorems in our list were disappointing— even angering—to mathematicians of the time, others were celebrated instantly by most. The first four of our theorems come from antiquity, and their proofs will be studied in detail. However, as we turn to more recent results, we will not be able to provide proofs—this would be far beyond the scope of this book. We start with what historians of mathematics regard as the oldest theorem in mathematics, oldest in the sense that it was the first statement for which a rigorous proof was given. This is the following theorem discovered in the sixth century BCE by the Greek mathematician and scientist Thales of Miletus: Theorem 5.1 (Thales’s Theorem). If a triangle is inscribed in a circle so that one of its sides goes through the center of the circle, then the angle of the triangle that is opposite to this side is a right angle. A proof to Thales’s Theorem, using basic properties of triangles, can be established easily—we leave this as Problem 1. Our next theorem might be the “most well-known” theorem in mathematics. While once thought to have been discovered by Pythagoras and his circle of friends at the end of the sixth century BCE, we now know that the Babylonians as well as the Chinese knew of this result about a 1,000 years earlier.
B. Bajnok, An Invitation to Abstract Mathematics, Undergraduate Texts in Mathematics, DOI 10.1007/978-1-4614-6636-9 5, © B´ela Bajnok 2013
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5 Famous Classical Theorems
Theorem 5.2 (The Pythagorean Theorem). If a and b are the lengths of the two legs of a right triangle and c is the length of its hypotenuse, then a2 C b 2 D c 2 : There are many nice proofs of this theorem (a collection of 370 proofs, published by Elisha Scott Loomis, appeared in 1927); one such proof is assigned as Problem 2. About a century and a half after Pythagoras’s time came the shocking discovery that not every number can be written as a fraction of two integers, as was believed by the Greeks of the fifth century BCE. In particular, the diagonal of a square is incommensurable with its sides: there is no unit length (no matter how small) such that both the side and the diagonal of the square have lengths that are integer multiples of this unit length. Appliedpto the square with side length 1 and using today’s terminology, we can say that 2 is not a rational number. T
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