SELECTED CHAPTERS FROM ALGEBRA I. R. Shafarevich Preface

Selected chapters from algebra 13As before, we see thatand soa k+1 =10 k+1 + + a n =10 n 6 1=10 k n 6 m +(a m+1 =10 m+1 + +(a k +1)=10 k ):Since a k 6= 9, the digit a k + 1 is one of the digits 1, 2, . . . , 9. Putc = a m+1 =10 + +(a k +1)=10 k;m :We can repeat our reasoning once more and obtain that c m. From these relationsit follows that 0 m < m + 110 m , i.e., 0 m ; m < 110 m . But this contradicts thefact that m and 0 m are distinct rational numbers having the same denominator10 m . Hence, 0 m = m for all m. But numbers a m are uniquely determined by thenumbers m ,since m ; m;1 = a m =10 m . Thus, they coincide in both fractions,too.We pass now to the second question: does every real number correspond tosome innite decimal fraction? As well as the answer, the method of proof is alreadyknown to us. We justwant toconvince ourselves that the reasoning can be basedon the axioms we formulated.First of all, let us remark that each real number is situated between twoconsecutive integers, i.e., there exists an integer A, suchthatA 6