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αp−n . . . is a code for the entire sequence {rn }. To recover the sequence {rn }from this symbol it is necessary to indicate the value of p, the order of x.For p ≥ 0 it is customary to place a period or comma after α0 ; for p < 0, theconvention is to place |p| zeros left of αp and a period or comma right of the leftmostzero (we recall that αp = 0).For example, when q = 10,123.45 := 1 × 102 + 2 × 101 + 3 × 100 + 4 × 10−1 + 5 × 10−2 ,0.00123 := 1 × 10−3 + 2 × 10−4 + 3 × 10−5 ;and when q = 2,1000.001 := 1 · 23 + 1 · 2−3 .Thus the value of a digit in the symbol αp . . . αp−n . . .

depends on the position itoccupies relative to the period or comma.With this convention, the symbol αp . . . α0 . . . makes it possible to recover thewhole sequence of approximations.It can be seen by inequalities (2.8) (verify this!) that different sequences {rn } and{rn }, and therefore different symbols αp . . . α0 .

. . and αp . . . α0 . . . , correspond todifferent numbers x and x .We now answer the question whether some real number x ∈ R corresponds toevery symbol αp . . . α0 . . . . The answer turns out to be negative.We remark that by virtue of the algorithm just described for obtaining the numbers αp−n ∈ {0, 1, . . . , q − 1} successively, it cannot happen that all these numbersfrom some point on are equal to q − 1.642The Real NumbersIndeed, ifrn = αp q p + · · · + αp−k q p−k + (q − 1)q p−k−1 + · · · + (q − 1)q p−nfor all n > k, that is,rn = rk +11−,q k−p q n−p(2.9)then by (2.8) we haverk +1q k−p−1q n−p≤ x < rk +1q k−p.Then for any n > k0 < rk +11− x < n−p ,q k−pqwhich, as we know from 80 above, is impossible.It is also useful to note that if at least one of the numbers αp−k−1 , . .

. , αp−n isless than q − 1, then instead of (2.9) we can writern < rk +11−q k−p q n−por, what is the samern +11< rk + k−p .q n−pq(2.10)We can now prove that any symbol αn . . . α0 . . . , composed of the numbers αk ∈{0, 1, . . . , q −1}, and in which there are numbers different from q −1 with arbitrarilylarge indices, corresponds to some number x ≥ 0.Indeed, from the symbol αp .

. . αp−n . . . let us construct the sequence {rn } of theform (2.7). By virtue of the relations r0 ≤ r1 ≤ rn ≤ · · · , taking account of (2.9) and(2.10), we haver0 ≤ r1 ≤ · · · ≤ · · · < · · · ≤ rn +1q n−p≤ · · · ≤ r1 +1q 1−p≤ r0 +1q −p.(2.11)The strict inequalities in this last relation should be understood as follows: every element of the left-hand sequence is less than every element of the right-handsequence. This follows from (2.10).If we now take x = supn∈N rn (= infn∈N (rn + q −(n−p) )), then the sequence {rn }will satisfy conditions (2.7) and (2.8), that is, the symbol αp .

. . αp−n . . . correspondsto the number x ∈ R.Thus, we have established a one-to-one correspondence between the positivenumbers x ∈ R and symbols of the form αp . . . α0 , . . . if p ≥ 0 or 0, 0 . . . 0 αp . . . |p| zeros2.2 Classes of Real Numbers and Computations65if p < 0. The symbol assigned to x is called the q-ary representation of x; the numbers that occur in the symbol are called its digits, and the position of a digit relativeto the period is called its rank.We agree to assign to a number x < 0 the symbol for the positive number −x,prefixed by a negative sign. Finally, we assign the symbol 0.0 .

. . 0 . . . to the number 0.In this way we have constructed the positional q-ary system of writing real numbers.The most useful systems are the decimal system (in common use) and for technical reasons the binary system (in electronic computers). Less common, but alsoused in some parts of computer engineering are the ternary and octal systems.Formulas (2.7) and (2.8) show that if only a finite number of digits are retainedin the q-ary expression of x (or, if we wish, we may say that the others are replacedwith zeros), then the absolute error of the resulting approximation (2.7) for x doesnot exceed one unit in the last rank retained.This observation makes it possible to use the formulas obtained in Paragraph bto estimate the errors that arise when doing arithmetic operations on numbers as aresult of replacing the exact numbers by the corresponding approximate values ofthe form (2.7).This last remark also has a certain theoretical value.

To be specific, if we identifya real number x with its q-ary expression, as was suggested in Paragraph b, oncewe have learned to perform arithmetic operations directly on the q-ary symbols, wewill have constructed a new model of the real numbers, seemingly of greater valuefrom the computational point of view.The main problems that need to be solved in this direction are the following:To two q-ary symbols it is necessary to assign a new symbol representing theirsum. It will of course be constructed one step at a time.

To be specific, by addingmore and more precise rational approximations of the original numbers, we shallobtain rational approximations corresponding to their sum. Using the remark madeabove, one can show that as the precision of the approximations of the terms increases, we shall obtain more and more q-ary digits of the sum, which will then notvary under subsequent improvements in the approximation.This same problem needs to be solved with respect to multiplication.Another, less constructive, route for passing from rational numbers to all realnumbers is due to Dedekind.Dedekind identifies a real number with a cut in the set Q of rational numbers,that is, a partition of Q into two disjoint sets A and B such that a < b for all a ∈ Aand all b ∈ B. Under this approach to real numbers our axiom of completeness(continuity) becomes a well-known theorem of Dedekind.

For that reason the axiomof completeness in the form we have given it is sometimes called Dedekind’s axiom.To summarize, in the present section we have exhibited the most importantclasses of numbers. We have shown the fundamental role played by the naturaland rational numbers. It has been shown how the basic properties of these numbers662The Real Numbersfollow from the axiom system8 we have adopted. We have given a picture of variousmodels of the set of real numbers. We have discussed the computational aspects ofthe theory of real numbers: estimates of the errors arising during arithmetical operations with approximate magnitudes, and the q-ary positional computation system.2.2.5 Problems and Exercises1. Using the principle of induction, show thata) the sum x1 + · · · + xn of real numbers is defined independently of the insertionof parentheses to specify the order of addition;b) the same is true of the product x1 · · · xn ;c) |x1 + · · · + xn | ≤ |x1 | + · · · + |xn |;d) |x1 · · · xn | = |x1 | · · · |xn |;e) ((m, n ∈ N) ∧ (m < n)) ⇒ ((n − m) ∈ N);f) (1 + x)n ≥ 1 + nx for x > −1 and n ∈ N, equality holding only when n = 1or x = 0 (Bernoulli’s inequality);n n−1n−2 b2 + · · · + n(n−1)···2 abn−1 + bn (Newa b + n(n−1)g) (a + b)n = a n + 1!2! a(n−1)!ton’s binomial formula).2.

a) Verify that Z and Q are inductive sets.b) Give examples of inductive sets different from N, Z, Q, and R.3. Show that an inductive set is not bounded above.4. a) Show that an inductive set is infinite (that is, equipotent with one of its subsetsdifferent from itself).b) The set En = {x ∈ N | x ≤ n} is finite. (We denote card En by n.)5. (The Euclidean algorithm) a) Let m, n ∈ N and m > n. Their greatest commondivisor (gcd(m, n) = d ∈ N) can be found in a finite number of steps using thefollowing algorithm of Euclid involving successive divisions with remainder.m = q1 n + r1 (r1 < n),n = q2 r1 + r2 (r2 < r1 ),r1 = q3 r2 + r3 (r3 < r2 ),...rk−1 = qk+1 rk + 0.Then d = rk .8 It was stated by Hilbert in almost the form given above at the turn of the twentieth century.

See forexample Hilbert, D. Foundations of Geometry, Chap. III, §13. (Translated from the second editionof Grundlagen der Geometrie, La Salle, Illinois: Open Court Press, 1971. This section was basedon Hilbert’s article “Über den Zahlbegriff” in Jahresbericht der deutschen Mathematikervereinigung 8 (1900).)2.2 Classes of Real Numbers and Computations67b) If d = gcd(m, n), one can choose numbers p, q ∈ Z such that pm + qn = d;in particular, if m and n are relatively prime, then pm + qn = 1.6. Try to give your own proof of the fundamental theorem of arithmetic (Paragraph a in Sect.

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