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Unformatted text preview: HISTORY OF MATHEMATICS – LECTURE 5 – THURSDAY 8TH SEPTEMBER Euclid While very little (see page 143) is known about the life of Euclid (c. 325‐265 B.C.), his major work, the Elements, containing all the known geometry and number theory at that time, has become the most widely read mathematical textbook in history, surviving in print to this day. It was used as the basic text on geometry in the western world for over 2000 years, and during that period almost every educated person would have read it during their schooling. The 13 different books that make up the Elements are thought to contain the mathematics of Pythagoras, Hippocrates (not the one associated with medicine), Eudoxus (see page 118), and other Pythagorean mathematicians. While it is not known how much of the mathematics was contributed by Euclid himself, the fact that he authored several other works and founded a school in Alexandria would indicate that he deserves to be considered among the greatest mathematicians. While the majority of the Elements, and its enduring fame, is connected with geometry, this lecture will instead first focus on a topic in the 10th book, where (at least in older editions) it is proven that √2 is irrational. Irrational numbers The Pythagoreans thought it intuitive that given any two lengths, there had to be a small measurement which would allow them to exactly compute the ratio of the two lengths using integers. For example, if two lengths measure 2.25 feet and 3.25 feet, then by using 0.25 as the unit of measurement, we can form the ratio 9:13. The fact that this turns out to not always be the case posed a serious problem to Pythagorean mathematics, since it violated their assumption that number and geometry were inseparable. Theorem: √2 is irrational Proof: Approximating √ While we cannot write √2 as the ratio of two whole numbers, we cannot however obtain an arbitrarily accurate approximation. In fact the Babylonians considered this problem over 1000 years before Euclid’s time (see page 112), and their approximation of 1.414213 was correct to six decimal places. The Greek mathematician Theon (c. 70‐135 A.D.) devised the recursive method below (known as Theon’s Ladder) for obtaining increasingly accurate approximations. It should be noted that in each row To prove that the (Diophantine) equation (*) holds true Number Theory Books VII, VIII, and IX of Euclid’s Elements form the building blocks of what today is called Number Theory. Book VII begins with definitions that allow us to generate many of the famous results derived during this period. Definition: Given two integers a and b, we say a divides b if and only if b is an integer multiple of a. Symbolically we write a | b b = qa, where q If a divides b, then we also say that b is divisible by a. Examples: i) 4 | 12 ii) 4 | ‐12 iii) 1 | 728 iv) 728 | 0 Theorem (Division Algorithm): If a is any integer and b is a positive integer, then there is a unique pair of integers q (the quotient) and r (the remainder) such that a = qb + r, where 0 r < b Ex. If a = 29, and b = 6, then it follows that q = 4 and r = 5. Ex. If a = ‐17 and b = 3, then it follows that q = and r = . The division algorithm in itself is not especially useful. However we apply it when using the Euclidean Algorithm, which allows us to find the greatest common divisor of two integers. Before stating it, we need a smaller result (which we call a “lemma”), which allows us to work with smaller numbers without changing the desired result. Lemma: If a = qb + r, then gcd(a, b) = gcd(b, r). Proof: Ex. gcd(128, 12) = gcd(12, 8) Ex. gcd(1492, 1066) = gcd(1066, 426) Ex. gcd(4999, 1109) = gcd(1109, 563) In the first example, it is clear after using the lemma once that the answer is 4. However in the second and third example we need to use the lemma repeatedly before the answer becomes clear. Ex. gcd(1492, 1066) = gcd(1066, 426) Ex. gcd(4999, 1109) = gcd(1109, 563) = gcd(426, 214) = gcd(563, 546) = gcd(214, 212) = gcd(546, 17) = gcd(212, 2) = gcd(17, 2) = gcd(2, 0) = gcd(2, 1) = 2 = gcd(1, 0) = 1 This process is the Euclidean Algorithm, and it must terminate because the remainder becomes smaller at each step. Theorem: The greatest common divisor is the last non‐zero remainder upon repeated application of the lemma. In Book VII Euclid also included the definition of a prime number, and followed it by providing proofs of two famous results. Definition: A natural number is said to be prime if it has exactly two distinct natural number divisors, i.e. 1 and itself. Ex. The primes less than 50 are: Note: This definition (slightly different from the one in the text) explains partly why the number 1 is not considered prime, as it only has one divisor. The other (and more important) reason is that the Fundamental Theorem of Arithmetic (see below) would not hold if we allowed 1 to be prime. Theorem (Fundamental Theorem of Arithmetic): Every positive number greater than 1 can be expressed as a unique product of primes (up to permutation of the factors). Ex. 144 = 2 2 2 2 3 3 = 24 32 Ex. 4725 = 3 3 3 5 5 7 = 33 52 7 We can use prime factorizations to quickly find greatest common divisors. Ex. gcd (144, 4725) = 32 = 9 Theorem: There are infinitely many prime numbers. Proof: ...
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