Number Theory: An Introduction to Mathematics

6 I The Expanding Universe of Numbers We now show that there exists such a mapφ.LetC be the collection of all subsetsCofN×Asuch ...

1 Natural Numbers 7 We define thesumofmandnto be m+n=sm(n). It is not difficult to deduce from this definition and the axioms(N1 ...

8 I The Expanding Universe of Numbers We show next how a relation of order may be defined on the setN.Forany m,n∈N, we say thatm ...

1 Natural Numbers 9 Proposition 3Any nonempty subset M ofNhas a least element. Proof Assume that some nonempty subsetMofNdoes no ...

10 I The Expanding Universe of Numbers Let f:IS(m)→Inbe a map such thatf(IS(m))=Inwhich is not injective. Then there existp,q∈IS ...

2 Integers and Rational Numbers 11 However, two other natural numbersm′,n′may have the same difference asm,n,and anyway what doe ...

12 I The Expanding Universe of Numbers It follows at once from the corresponding properties of natural numbers that, also in Z, ...

2 Integers and Rational Numbers 13 Proposition 10For every a∈Z,a· 0 = 0. Proof We h av e a· 0 =a·( 0 + 0 )=a· 0 +a· 0. Adding−(a ...

14 I The Expanding Universe of Numbers We may write(P2)and(P3)symbolically in the form P+P⊆P, P·P⊆P. We now show that there are ...

2 Integers and Rational Numbers 15 Proposition 14For any integers a,b with a> 0 , there exist unique integers q,rsuch that b= ...

16 I The Expanding Universe of Numbers Addition of rational numbers is defined by (a,b)+(c,d)=(ad+cb,bd), wherebd=0sinceb=0and ...

3 Real Numbers 17 −a,wherea∈P. From the corresponding result forZ, it follows that(P1)continues to hold inQ. We will show that(P ...

18 I The Expanding Universe of Numbers But it was proved, probably by a disciple of Pythagoras, that there is no rational number ...

3 Real Numbers 19 It is obvious from the definition by set inclusion that at most one holds. Now suppose that neitherA<BnorA= ...

20 I The Expanding Universe of Numbers Proposition 16For any cuts A,B, there exists a cut C such that A+C=B if and only if A< ...

3 Real Numbers 21 It remains to show thata 1 b+a 2 c∈A(B+C)ifa 1 ,a 2 ∈A,b∈Bandc∈C.But a 1 b+a 2 c≤a 2 (b+c) ifa 1 ≤a 2 , and a ...

22 I The Expanding Universe of Numbers This completes the first stage of Dedekind’s construction. In the second stage we pass fr ...

3 Real Numbers 23 Proof LetSbe the set of all positive real numbersxsuch thatx^2 ≤a.ThesetSis not empty, since it containsaifa≤1 ...

24 I The Expanding Universe of Numbers The notion of convergence can be defined in any totally ordered set. A sequence {an}is sa ...

3 Real Numbers 25 Thenx 1 >1andx 12 >a,since(a− 1 )^2 >0. Define the sequence{xn}recursively by xn+ 1 =(xn+a/xn)/ 2 (n≥ ...