数学专业英语.doc

Mathematical EnglishDr. Xiaomin ZhangEmail: §2.4 Integers, Rational Numbers and Real numbers TEXT A Integers and rational numbersThere exist certain subsets of R which are distinguished because they have special properties not shared by all real numbers. In this section we shall discuss two such subsets, the integers and the rational numbers.To introduce the positive integers we begin with the number 1, whose existence is guaranteed by Axiom 4. The number 1+1 is denoted by 2, the number 2+1 by 3, and so on. The numbers 1, 2, 3, , obtained in this way by repeated addition of 1 are all positive, and they are called the positive integers. Strictly speaking, this description of the positive integers is not entirely complete because we have not explained in details what we mean by the expressions “and so on, or “repeated addition of 1. Although the intuitive meaning of expressions may seem clear, in a careful treatment of the real-number system it is necessary to give a more precise definition of the positive integers. There are many ways to do this. One convenient method is to introduce first the notion of an inductive set.DEFINITION OF AN INDUCTIVE SET A set of real numbers is called an inductive set if it has the following two properties:(a) The number 1 is in the set.(b) For every x in the set, the number x+1 is also in the set.For example, R is an inductive set. So is the set R+. Now we shall define the positive integers to be those real numbers which belong to every inductive set.DEFINITION OF POSITIVE INTEGERS A real number is called a positive integer if it belongs to every inductive set.Let P denote the set of all positive integers. Then P is itself an inductive set because (a) it contains 1, and (b) it contains x+1 whenever it contains x. Since the members of P belong to every inductive set, we refer to P as the smallest inductive set. This property of the set P forms the logical basis for a type of reasoning that mathematicians call proof by induction, a detailed discussion of which is given in Part 4 of this Introduction.The negatives of the positive integers are called the negative integers. The positive integers, together with the negative integers and 0 (zero), form a set Z which we call simply the set of integers.In a thorough treatment of the real-number system, it would be necessary at this stage to prove certain theorems about integers. For example, the sum, difference, or product of two integers is an integer, but the quotient of two integers need not be an integer. However, we shall not enter into the details of such proofs.Quotients of integers a/b (where b¹0) are called rational number. The set of rational numbers, denoted by Q, contains Z as a subset. The reader should realize that all the field axioms and the order axioms are satisfied by Q. For this reason, we say that the set of rational numbers is an ordered field. Real numbers that are not in Q are called irrational.NotationsField axioms A field is any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra, where division algebra, also called a "division ring" or "skew field," means a ring in which every nonzero element has a multiplicative inverse, but multiplication is not necessarily commutative. Order axioms A total order (or "totally ordered set," or "linearly ordered set") is a set plus a relation on the set (called a total order) that satisfies the conditions for a partial order plus an additional condition known as the comparability condition. A relation £ is a total order on a set S ("£ totally orders S") if the following properties hold. 1. Reflexivity: a£a for all aÎS. 2. Antisymmetry: a£b and b£a implies a=b. 3. Transitivity: a£b and b£c implies a£c. 4. Comparability (trichotomy law): For any a, bÎS, either a£b or b£a. The first three are the axioms of a partial order, while addition of the trichotomy law defines a total order. TEXT B Geometric interpretation of real numbers as points on a lineThe reader is undoubtedly familiar with the geometric representation of real numbers by means of points on a straight line. A point is selected to represent 0 and another, to the right of 0, to represent 1, as illustrated in Figure 2-4-1. This choice determines the scale. If one adopts an appropriate set of axioms for Euclidean geometry, then each real number corresponds to exactly one point on this line and, conversely, each point on the line corresponds to one and only one real number. For this reason the line is often called the real line or the real axis, and it is customary to use the words real number and point interchangeably. Thus we often speak of the point x rather than the point corresponding to the real numbers.The ordering relation among the real numbers has a simple geometric interpretation. If x<y, the point x lies to the left of the point y as shown in Figure 2-4-1. Positive numbers lie to the right of 0 and negative numbers to the left of 0. If a<b, a point x satisfies the inequalities a<x<b if and only if x is between a and b.This device for representing real numbers geometrically is a very worthwhile aid that helps us to discover and understand better certain properties of real numbers. However, the reader should realize that all properties of real numbers that are to be accepted as theorems must be deducible from the axioms without any reference to geometry. This does not mean that one should not make use of geometry in studying properties of real numbers. On the contrary, the geometry often suggests the method of proof of a particular theorem, and sometimes a geometric argument is more illuminating than a purely analytic proof (one depending entirely on the axioms for the real numbers). In this book, geometric arguments are used to a large extent to help motivate or clarify a particular discuss. Nevertheless, the proofs of all the important theorems are presented is analytic form.SUPPLEMENT Prime NumberA prime number (or prime integer, often simply called a "prime" for short) is a positive integer p>1 that has no positive integer divisors other than 1 and p itself. For example, the only divisors of 13 are 1 and 13, making 13 a prime number, while the number 24 has divisors 1, 2, 3, 4, 6, 8, 12, and 24 (corresponding to the factorization 24=23×3), making 24 not a prime number. Positive integers other than 1 which are not primes are called composite numbers. Prime numbers are therefore numbers that cannot be factored or, more precisely, are numbers n whose divisors are trivial and given by exactly 1 and n. The number 1 is a special case which is considered neither prime nor composite. With 1 excluded, the smallest prime is therefore 2 and since 2 is the only even prime, it is also somewhat special. Note also that while 2 is considered a prime today, at one time it was not. The nth prime number is commonly denoted pn, so p1=2, p2=3, and so on, and may be computed in Mathematica as Primen. The set of primes is sometimes denoted P, represented in Mathematica as Primes. Euler commented "Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate". In a 1975 lecture, D. Zagier commented "There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that, despite their simple definition and role as the building blocks of the natural numbers, the prime numbers grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behavior, and that they obey these laws with almost military precision". Large primes include the large Mersenne primes, Ferrier's prime. The largest known prime as of Feb. 2005 is the Mersenne prime 2-1 (Weisstein 2005). Prime numbers can be generated by sieving processes (such as the sieve of Eratosthenes), and lucky numbers, which are also generated by sieving, appear to share some interesting asymptotic properties with the primes. Prime numbers satisfy many strange and wonderful properties. The function that gives the number of primes less than or equal to a number n is denoted p(n) and is called the prime counting function. The theorem giving an asymptotic form for p(n) is called the prime number theorem.The fundamental theorem of arithmetic states that any positive integer can be represented in exactly one way as a product of primes. Euclid's second theorem demonstrated that there are an infinite number of primes. However, it is not known if there are an infinite number of primes of the form n2+1, whether there are an infinite number of twin primes (the twin prime conjecture), or if a prime can always be found between n2 and (n+1)2. The latter two of these are two of Landau's problems. Primes consisting of consecutive digits (counting 0 as coming after 9) include 2, 3, 5, 7, 23, 67, 89, 4567, 78901, . Primes consisting of digits that are themselves primes include 23, 37, 53, 73, 223, 227, 233, 257, 277, 337, 353, 373, 523, 557, ., which is one of the Smarandache sequences. Because a prime number p has the trivial factors 1 and p, Bill Gates accidentally referred to a trivial operation when he stated "Because both the system's privacy and the security of digital money depend on encryption, a breakthrough in mathematics or computer science that defeats the cryptographic system could be a disaster. The obvious mathematical breakthrough would be the development of an easy way to factor large prime numbers emphasis added" (Gates 1995, p. 265). NotationsMersenne prime A Mersenne prime is a Mersenne number, i.e., a number of the form Mn=2n-1 that is prime. In order for Mn to be prime, n must itself be prime. This is true since for composite n with factors r and s, n=rs. Therefore, 2n-1 can be written as 2rs-1, which is a binomial number that always has a factor 2r-1. The first few Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, . corresponding to indices n=1, 3, 5, 7, 13, 17, 19, 31, 61, 89, . Mersenne primes were first studied because of the remarkable properties that every Mersenne prime corresponds to exactly one perfect number. L. Welsh maintains an extensive bibliography and history of Mersenne numbers. It has been conjectured that there exist an infinite number of Mersenne primes. However, finding Mersenne primes is computationally very challenging. For example, the 1963 discovery that M11213 is prime was heralded by a special postal meter design, illustrated above, issued in Urbana, Illinois. G. Woltman has organized a distributed search program via the Internet known as GIMPS (Great Internet Mersenne Prime Search) in which hundreds of volunteers use their personal computers to perform pieces of the search. On November 17, 2003, a GIMPS volunteer reported discovery of the 40th Mersenne prime, a discovery that was subsequently confirmed. Almost exactly six months later, discovery of the 41st known Mersenne prime by a GIMPS volunteer was announced. The 42nd known Mersenne prime was announced on Feb. 18, 2005 and its exponent was released on Feb. 26. The efforts of GIMPS volunteers make this distributed computing project the discoverer of all eight of the largest known Mersenne primes. In fact, as of Feb. 2005, GIMPS participants have tested and double-checked all exponents below 9889900 and tested all exponents below 15130000 at least once (GIMPS). The table below gives the index p of known Mersenne primes Mp, together with the number of digits, discovery years, and discoverer. A similar table has been compiled by C. Caldwell. Note that the region after the 38th known Mersenne prime has not been completely searched, so identification of "the" 40th Mersenne prime is tentative (GIMPS). #pdigitsyeardiscoverer (reference)121antiquity 231antiquity 352antiquity 473antiquity 51341461Reguis (1536), Cataldi (1603)61761588Cataldi (1603)71961588Cataldi (1603)831101750Euler (1772)961191883Pervouchine (1883), Seelhoff (1886)1089271911Powers (1911)11107331913Powers (1914)12127391876Lucas (1876)13521157Jan. 30, 1952Robinson14607183Jan. 30, 1952Robinson151279386Jan. 30, 1952Robinson162203664Jan. 30, 1952Robinson172281687Jan. 30, 1952Robinson183217969Sep. 8, 1957Riesel1942531281Nov. 3, 1961Hurwitz2044231332Nov. 3, 1961Hurwitz2196892917May 11, 1963Gillies (1964)2299412993May 16, 1963Gillies (1964)23112133376Jun. 2, 1963Gillies (1964)24199376002Mar. 4, 1971Tuckerman (1971)25217016533Oct. 30, 1978Noll and Nickel (1980)26232096987Feb. 9, 1979Noll (Noll and Nickel 1980)274449713395Apr. 8, 1979Nelson and Slowinski (Slowinski 1978-79)288624325962Sep. 25, 1982Slowinski2911050333265Jan. 28, 1988Colquitt and Welsh (1991)3013204939751Sep. 20, 1983Slowinski3121609165050Sep. 6, 1985Slowinski32756839227832Feb. 19, 1992Slowinski and Gage33859433258716Jan. 10, 1994Slowinski and Gage34378632Sep. 3, 1996Slowinski and Gage35420921Nov. 12, 1996Joel Armengaud/GIMPS36895832Aug. 24, 1997Gordon Spence/GIMPS (Devlin 1997)37909526Jan. 27, 1998Roland Clarkson/GIMPS38Jun. 1, 1999Nayan Hajratwala/GIMPS39Nov. 14, 2001Michael Cameron/GIMPS (Whitehouse 2001, Weisstein 2001)40Nov. 17, 2003Michael Shafer/GIMPS (Weisstein 2003)41May 15, 2004Josh Findley/GIMPS (Weisstein 2004)42Feb. 18, 2005Martin Nowak/GIMPS (Weisstein 2005)Ferrier's prime According to Hardy and Wright (1979), the 44-digit Ferrier's prime is(2148+1)/17=24593863921determined to be prime using only a mechanical calculator, is the largest prime found before the days of electronic computers. Mathematica can verify primality of this number in a (small) fraction of a second, showing how far the art of numerical computation has advanced in the intervening years. lucky numbers Write out all the odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, . The first odd number greater than 1 is 3, so strike out every third number from the list: 1, 3, 7, 9, 13, 15, 19, . The first odd number greater than 3 in the list is 7, so strike out every seventh number: 1, 3, 7, 9, 13, 15, 21, 25, 31, . Numbers remaining after this procedure has been carried out completely are ca