Search Results

This chapter clarifies some unfamiliar concepts of Euclidean number theory and examines the bricks, constituents, and formative elements of numbers. It also considers three famous propositions from Euclid’s Elements. The best-known theorem is 9. 20, which states that there are more prime numbers than any assigned multitude of prime numbers. The last theorem, 9. 36, deals with perfect numbers – ones that equal the sum of its proper divisors. The third famous theorem, 9. 14, states that if a number is the least that is measured by (some) prime numbers, it will not be measured by any other prime number except those originally measuring it. Thus, 9. 14 is equivalent to the Fundamental Theorem of Arithmetic (FTA). When multiplication broke loose from addition and evolved its own algebra, it was fairly easy for Johan Carl Friedrich Gauss to realise that the FTA must be true, and set about to prove it along the lines that Euclid had drawn.

“Be contradictory, because of the infinitude of primes” offers encouragement and practice with the “proof-by-contradiction” method of mathematical proof. Any mathematician will tell you that the collection of prime numbers is infinite. However, readers are guided in proving this statement by contradicting it. The activity pushes readers to engage deeply with the reasons supporting the fact that there are an infinite number of primes. Mathematics students and enthusiasts are encouraged to debate with enthusiasm in their mathematical and life pursuits. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

The Pythagoreans developed many of the ideas related to numbers that have become so familiar to us, including even and odd numbers, square numbers, triangular numbers, and so on. They also discovered that some integers cannot be decomposed into factors. These are called prime numbers and they constitute the building blocks of all the other integers, called composite. This chapter deals with the prime numbers in a general non-technical way, since much of the writing about them is quite specialized. The prime numbers have remarkable properties, many of which are still resistant to being proved. Prime numbers matter deeply to mathematics, not to mention to the progress of human knowledge generally. Pythagoras believed that prime numbers were part of a secret code which, if deciphered, would allow us to unlock the mysteries of the cosmos itself.

This chapter looks at some of the hardest problems in NP. Most of the NP problems that people considered in the mid-1970s either turned out to be NP-complete or people found efficient algorithms putting them in P. However, some NP problems refused to be so nicely and quickly characterized. Some would be settled years later, and others are still not known. These NP problems include the graph isomorphism, one of the few problems whose difficulty seems somewhat harder than P but not as hard as NP-complete problems like Hamiltonian paths and max-cut. Other NP problems include prime numbers and factoring, and linear programming. The linear programming problem has good algorithms in theory and practice—they just happen to be two very different algorithms.

This chapter considers the mental pictures of thought and images in relation to algebraic symbols. According to Ludwig Wittgenstein, “We make to ourselves pictures of facts.” For Wittgenstein, the picture is a model of what we take to be real. The geneticist Francis Galton claimed that his thoughts almost never suggested words, and when those rare moments did suggest words, they were nonsense words like “the notes of a song might accompany thought.” As for words, the French mathematician Jacques Hadamard suggested that words are neither followed by thoughts, nor thoughts by words. He goes on to say that this is also the case when he is thinking about algebraic symbols. More revealing is Hadamard's presentation of his mental pictures of the steps in a proof that there are an unlimited number of prime numbers. The chapter also discusses thought without verbal language and in relation to proofs.