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1、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 subs
2、ets, 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,
3、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 cle
4、ar, 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 a
5、n 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
6、 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 belo
7、ng 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 integer
8、s 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 exam
9、ple, 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 b0) are called rational number. The set of rational numbers, denoted by Q, contain
10、s 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 s
11、atisfies 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
12、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 orde
13、rs S) if the following properties hold. 1. Reflexivity: aa for all aS. 2. Antisymmetry: ab and ba implies a=b. 3. Transitivity: ab and bc implies ac. 4. Comparability (trichotomy law): For any a, bS, either ab or ba. The first three are the axioms of a partial order, while addition of the trichotomy
14、 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
15、, 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 rea
16、son 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 inte
17、rpretation. If xy, 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 ab, a point x satisfies the inequalities ax1 that has no positive integer divisors other than 1 and p itself. For example, the onl
18、y 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=233), making 24 not a prime number. Positive integers other than 1 which are not primes are called composite numbers. Prime numbers are theref
19、ore 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
20、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. Eule
21、r 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 n
22、umbers 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 o
23、ther 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
24、military precision. Large primes include the large Mersenne primes, Ferriers 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
25、 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 givin
26、g 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. Euclids second theorem demonstrated that there are an infinite number of primes. However, it is not kn
27、own 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 Landaus problems. Primes consisting of consecutive digits (coun
28、ting 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 a
29、nd p, Bill Gates accidentally referred to a trivial operation when he stated Because both the systems 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
30、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 s
31、ince 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, . Me
32、rsenne 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. How
33、ever, 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
34、(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, di
35、scovery 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
36、 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,
37、 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)121antiquity231antiquity352antiquity473an
38、tiquity51341461Reguis (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, 1
39、952Robinson162203664Jan. 30, 1952Robinson172281687Jan. 30, 1952Robinson183217969Sep. 8, 1957Riesel1942531281Nov. 3, 1961Hurwitz2044231332Nov. 3, 1961Hurwitz2196892917May 11, 1963Gillies (1964)2299412993May 16, 1963Gillies (1964)23112133376Jun. 2, 1963Gillies (1964)24199376002Mar. 4, 1971Tuckerman (1
40、971)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
41、. 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, 1999Nay
42、an 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)Ferriers prime According to Hardy and Wright (1979), the 44-
43、digit Ferriers 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 computa
44、tion 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