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Then x − 1 ≥ y ≥ n + 1 andx ≥ n + 2. Hence,x ∈ {x ∈ N | n + 1 < x} ⇒ (x ≥ n + 2)and consequently, min{x ∈ N | n + 1 < x} = n + 2, that is, (n + 1) ∈ E.By the principle of induction E = N, and 30 is now proved.As immediate corollaries of 20 and 30 above, we obtain the following propertiesand 60 ) of the natural numbers.(40 , 50 ,40 . (m ∈ N) ∧ (n ∈ N) ∧ (n < m) ⇒ (n + 1 ≤ m).50 .
The number (n + 1) ∈ N is the immediate successor of the number n ∈ N; thatis, if n ∈ N, there are no natural numbers x satisfying n < x < n + 1.60 . If n ∈ N and n = 1, then (n − 1) ∈ N and (n − 1) is the immediate predecessorof n in N; that is, if n ∈ N, there are no natural numbers x satisfying n − 1 <x < n.We now prove one more property of the set of natural numbers.70 . In any nonempty subset of the set of natural numbers there is a minimal element.Proof Let M ⊂ N.
If 1 ∈ M, then min M = 1, since ∀n ∈ N (1 ≤ n).Now suppose 1 ∈/ M, that is, 1 ∈ E = N\M. The set E must contain a natural number n such that all natural numbers not larger than n belong to E, but2.2 Classes of Real Numbers and Computations49(n + 1) ∈ M. If there were no such n, the set E ⊂ N, which contains 1, wouldcontain along with each of its elements n, the number (n + 1) also; by the principle of induction, it would therefore equal N. But the latter is impossible, sinceN\E = M = ∅.The number (n + 1) so found must be the smallest element of M, since there areno natural numbers between n and n + 1, as we have seen.2.2.2 Rational and Irrational Numbersa. The IntegersDefinition 3 The union of the set of natural numbers, the set of negatives of naturalnumbers, and zero is called the set of integers and is denoted Z.Since, as has already been proved, addition and multiplication of natural numbersdo not take us outside N, it follows that these same operations on integers do notlead outside of Z.Proof Indeed, if m, n ∈ Z, either one of these numbers is zero, and then the summ + n equals the other number, so that (m + n) ∈ Z and m · n = 0 ∈ Z, or bothnumbers are non-zero.
In the latter case, either m, n ∈ N and then (m + n) ∈ N ⊂ Zand (m · n) ∈ N ⊂ Z, or (−m), (−n) ∈ N and then m · n = ((−1)m)((−1)n) ∈ N or(−m), n ∈ N and then (−m · n) ∈ N, that is, m · n ∈ Z, or, finally, m, −n ∈ N andthen (−m · n) ∈ N and once again m · n ∈ Z.Thus Z is an Abelian group with respect to addition. With respect to multiplication Z is not a group, nor is Z\0, since the reciprocals of the integers are not in Z(except the reciprocals of 1 and −1).Proof Indeed, if m ∈ Z and m = 0, 1, then assuming first that m ∈ N, we have0 < 1 < m, and, since m · m−1 = 1 > 0, we must have 0 < m−1 < 1 (see the con/ Z. The casesequences of the order axioms in the previous subsection). Thus m−1 ∈when m is a negative integer different from −1 reduces immediately to the onealready considered.When k = m · n−1 ∈ Z for two integers m, n ∈ Z, that is, when m = k · n for somek ∈ Z, we say that m is divisible by n or a multiple of n, or that n is a divisor of m.The divisibility of integers reduces immediately via suitable sign changes, that is,through multiplication by −1 when necessary, to the divisibility of the corresponding natural numbers.
In this context it is studied in number theory.We recall without proof the so-called fundamental theorem of arithmetic, whichwe shall use in studying certain examples.A number p ∈ N, p = 1, is prime if it has no divisors in N except 1 and p.502The Real NumbersThe fundamental theorem of arithmetic Each natural number admits a representation as a productn = p 1 · · · pk ,where p1 , . .
. , pk are prime numbers. This representation is unique except for theorder of the factors.Numbers m, n ∈ Z are said to be relatively prime if they have no common divisors except 1 and –1.It follows in particular from this theorem that if the product m · n of relativelyprime numbers m and n is divisible by a prime p, then one of the two numbers isalso divisible by p.b. The Rational NumbersDefinition 4 Numbers of the form m · n−1 , where m, n ∈ Z, are called rational.We denote the set of rational numbers by Q.Thus, the ordered pair (m, n) of integers defines the rational number q = m · n−1if n = 0.The number q = m · n−1 can also be written as a quotient2 of m and n, that is, asa so-called rational fraction mn.The rules you learned in school for operating with rational numbers in terms oftheir representation as fractions follow immediately from the definition of a rationalnumber and the axioms for real numbers.
In particular, “the value of a fraction isunchanged when both numerator and denominator are multiplied by the same nonmzero integer”, that is, the fractions mknk and n represent the same rational number. In−1−1fact, since (nk)(k n ) = 1, that is (n · k)−1 = k −1 · n−1 , we have (mk)(nk)−1 =(mk)(k −1 n−1 ) = m · n−1 .Thus the different ordered pairs (m, n) and (mk, nk) define the same rationalnumber. Consequently, after suitable reductions, any rational number can be presented as an ordered pair of relatively prime integers.On the other hand, if the pairs (m1 , n1 ) and (m2 , n2 ) define the same rational−1number, that is, m1 · n−11 = m2 · n2 , then m1 n2 = m2 n1 , and if, for example, m1and n1 are relatively prime, it follows from the corollary of the fundamental theorem−1of arithmetic mentioned above that n2 · n−11 = m2 · m1 = k ∈ Z.We have thus demonstrated that two ordered pairs (m1 , n1 ) and (m2 , n2 ) definethe same rational number if and only if they are proportional.
That is, there existsan integer k ∈ Z such that, for example, m2 = km1 and n2 = kn1 .2 The notation Q comes from the first letter of the English word quotient, which in turn comes fromthe Latin quota, meaning the unit part of something, and quat, meaning how many.2.2 Classes of Real Numbers and Computations51c. The Irrational NumbersDefinition 5 The real numbers that are not rational are called irrational.√The classical example of an irrational real number is 2, that is, the numbers ∈ R such that s > 0 and s 2 = 2.
By the Pythagorean theorem, the irrationality of√2 is equivalent to the assertion that the diagonal and side of a square are incommensurable.Thus we begin by verifying that there exists a real number s ∈ R whose squareequals 2, and then that s ∈/ Q.Proof Let X and Y be the sets of positive real numbers such that ∀x ∈ X (x 2 < 2),∀y ∈ Y (2 < y 2 ). Since 1 ∈ X and 2 ∈ Y , it follows that X and Y are nonempty sets.Further, since (x < y) ⇔ (x 2 < y 2 ) for positive numbers x and y, every elementof X is less than every element of Y .
By the completeness axiom there exists s ∈ Rsuch that x ≤ s ≤ y for all x ∈ X and all y ∈ Y .We shall show that s 2 = 2.2If s 2 < 2, then, for example, the number s + 2−s3s , which is larger than s, wouldhave a square less than 2. Indeed, we know that 1 ∈ X, so that 12 ≤ s 2 < 2, and0 < Δ := 2 − s 2 ≤ 1.
It follows that 2Δ 2ΔΔΔΔs++< s2 + 3 ·= s 2 + Δ = 2.= s2 + 2 ·< s2 + 3 ·3s3s3s3s3sΔ) ∈ X, which is inconsistent with the inequality x ≤ s for allConsequently, (s + 3sx ∈ X.2If 2 < s 2 , then the number s − s 3s−2 , which is smaller than s, would have asquare larger than 2. Indeed, we know that 2 ∈ Y , so that 2 < s 2 ≤ 22 or 0 < Δ :=s 2 − 2 < 3 and 0 < Δ3 < 1.
Hence,Δs−3s2 2ΔΔΔ+= s 2 − Δ = 2,=s −2·> s2 − 3 ·3s3s3s2and we have now contradicted the fact that s is a lower bound of Y .Thus the only remaining possibility is that s 2 = 2.Let us show, finally, that s ∈/ Q. Assume that s ∈ Q and let mn be an irreducible2representation of s. Then m = 2 · n2 , so that m2 is divisible by 2 and therefore malso is divisible by 2. But, if m = 2k, then 2k 2 = n2 , and for the same reason, n mustbe divisible by 2.
But this contradicts the assumed irreducibility of the fraction mn.We have worked hard just now to prove that there exist irrational numbers. Weshall soon see that in a certain sense nearly all real numbers are irrational. It will beshown that the cardinality of the set of irrational numbers is larger than that of theset of rational numbers and that in fact the former equals the cardinality of the setof real numbers.522The Real NumbersAmong the irrational numbers we make a further distinction between the socalled algebraic irrational numbers and the transcendental numbers.A real number is called algebraic if it is the root of an algebraic equationa0 x n + · · · + an−1 x + an = 0with rational (or equivalently, integer) coefficients.Otherwise the number is called transcendental.We shall see that the cardinality of the set of algebraic numbers is the same asthat of the set of rational numbers, while the cardinality of the set of transcendentalnumbers is the same as that of the set of real numbers.
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