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Euclid's Number Theory Euclid of Alexandria, II: Number Theory Euclid of Alexandria, II: Number Theory Waseda University, SILS, History of Mathematics Euclid of Alexandria, II: Number Theory Outline Introduction Euclid’s number theory The overall structure Definitions for number theory Theory of prime numbers Properties of primes Infinitude of primes Euclid of Alexandria, II: Number Theory Introduction Concepts of number § The natural numbers is the set N “ t1; 2; 3;::: u. § The whole numbers is the set W “ t0; 1; 2; 3;::: u. § The integers is the set of positive and negative whole numbers Z “ t0; 1; ´1; 2; ´2;::: u. § The rational numbers is the set Q, of numbers of the form p{q, where p; q P Z,1 but q ‰ 0. § The real numbers, R, is the set of all the values mapped to the points of the number line. (The definition is tricky.) § An irrational number is a number that belongs to the reals, but is not rational. 1The symbol P means “in the set of,” or “is an element of.” Euclid of Alexandria, II: Number Theory Introduction The Greek concept of number § Greek number theory was exclusively interested in natural numbers. § In fact, the Greek also did not regard “1” as a number, but rather considered it the unit by which other numbers are numbered (or measured). § We can define Greek natural numbers as G “ t2; 3; 4;::: u. (But we can do most Greek number theory with N, so we will generally use this set, for simplicity.) Euclid of Alexandria, II: Number Theory Introduction Number theory before Euclid § The semi-legendary Pythagorus himself and other Pythagoreans are attributed with a fascination with numbers and with the development of a certain “pebble arithmetic” which studied the mathematical properties of numbers that correspond to certain geometry shapes (figurate numbers). § Philolaus of Croton (late 5th, earth 4th BCE) is attributed with some numerological speculations related to music theory and cosmology. § Archytus of Tarentum (4th BCE) is attributed with some theorems of number theory, most of which are directly applicable to ancient music theory. Euclid of Alexandria, II: Number Theory Euclid’s number theory The overall structure Elements VII–IX As in earlier books, Euclid probably based much of his work on the discoveries of others, but the organization and presentation was his own. § Book VII: numbers and proportions, theory of divisors, theory of least common multiples § Book VIII: theory of figurate numbers, mean proportionals § Book IX: numbers and proportions, theory of prime numbers, theory of even and odd, perfect numbers2 2A perfect number is a number whose factors sum together to equal the number, ex. 6 “ 1 ` 2 ` 3, or 28 “ 1 ` 2 ` 4 ` 7 ` 14. Euclid of Alexandria, II: Number Theory Euclid’s number theory Definitions for number theory Definitions for Euclidean number theory § 1: A unit is that by virtue of which each of the things that exists is called one. § 2: A number is a multitude made up of units. § 5: The greater number is a multiple of the lesser when it is measured by 3 the lesser number. § 11: A prime number is that which is measured by a unit alone. § 12: Numbers prime to one another are those which are measured by a unit alone as common measure. § 13: A composite number is that which is measured by some number. § 22: A perfect number is that which is equal to its own parts. 3“Measured by” is an undefined concept in Euclid’s theory. It means something like divided by, with no remainder. Euclid of Alexandria, II: Number Theory Theory of prime numbers Properties of primes Theory of Primes § We start with some “problems” that show how to determine if two numbers are relatively prime (VII 1), or, if not, to find their greatest common factor (VII.2). § If p ab, then p a or p b, where p is prime (VII.30).4 § “Any composite number is divisible by some prime number.” (VII.31) [That is, given ab, there exists some p such that p ab.] A proof by cases and by contradiction... § “If a number be the least that is measured by prime numbers, it will not be measured by any prime number except those originally measuring it.” (IX.14) [That is, if x1 x2 x3 a “ p1 p2 p3 :::, then there is no pn R tp1; p2; p3; :::u, such that pn a.] A proof by contradiction. 4The expression a b means “a divides into b with no remainder,” or as the Greeks would say, “a measures b.” That is, there is some n P N, such that an “ b. Euclid of Alexandria, II: Number Theory Theory of prime numbers Infinitude of primes Elements IX.21 — The primes are infinite § “Prime numbers are more than any assigned multitude of prime numbers.” § A proof by construction that uses cases and an indirect argument. § Preliminary: If g a and g b then g pa ´ bq. That is, a “ gm and b “ gn ñ a ´ b “ gpm ´ nq. § Let tp1; p2; p3u be the greatest known set of primes. Take the number a “ p1p2p3 ` 1. (A construction.) § There are two cases: either, (1) a is a prime, or (2) a is composite and has some prime factor pn. § In case (2), we use an indirect proof to show that pn ‰ p1; p2; p3. § The arguments in Elements IX.14 and IX.21 provide a sort of proof strategy that we could apply to any other set of numbers that was assumed to be the set in question. Euclid of Alexandria, II: Number Theory Theory of prime numbers Infinitude of primes Euclid’s approach to studying the nature of numbers § We begin with some “intuitive” assumptions about numbers and some carefully designed definitions. § We do some “problems” that show us how to carry out certain operations or algorithms. § We develop a theory of the concepts of measure, divisibility, etc., which leads to a theory of primes as the fundamental building blocks of all numbers. § We see again how mathematical theories help us develop our ideas of certain key concepts..
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