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For that reason the difficultiesinvolved in exhibiting specific transcendental numbers – more precisely, provingthat a given number is transcendental – seem at first sight paradoxical and unnatural.For example, it was not proved until 1882 that the classical geometric number πis transcendental,3 and one of the famous Hilbert4 problems was to prove the transcendence of the number α β , where α is algebraic, (α√> 0) ∧ (α = 1) and β is anirrational algebraic number (for example, α = 2, β = 2).2.2.3 The Principle of ArchimedesWe now turn to the principle of Archimedes,5 which is important in both its theoretical aspect and the application of numbers in measurement and computations. Weshall prove it using the completeness axiom (more precisely, the least-upper-boundprinciple, which is equivalent to the completeness axiom).

In other axiom systemsfor the real numbers this fundamental principle is frequently included in the list ofaxioms.We remark that the propositions that we have proved up to now about the naturalnumbers and the integers have made no use at all of the completeness axiom. As3 Thenumber π equals the ratio of the circumference of a circle to its diameter in Euclidean geometry.

That is the reason this number has been conventionally denoted since the eighteenth century,following Euler by π , which is the initial letter of the Greek word περιϕ έρια – periphery (circumference). The transcendence of π was proved by the German mathematician F.

Lindemann(1852–1939). It follows in particular from the transcendence of π that it is impossible to constructa line segment of length π with compass and straightedge (the problem of rectification of the circle), and also that the ancient problem of squaring the circle cannot be solved with compass andstraightedge.4 D.Hilbert (1862–1943) – outstanding German mathematician who stated 23 problems from different areas of mathematics at the 1900 International Congress of Mathematicians in Paris.

Theseproblems came to be known as the “Hilbert problems”. The problem mentioned here (Hilbert’s seventh problem) was given an affirmative answer in 1934 by the Soviet mathematician A.O. Gel’fond(1906–1968) and the German mathematician T. Schneider (1911–1989).5 Archimedes (287–212 BCE) – brilliant Greek scholar, about whom Leibniz, one of the founders ofanalysis said, “When you study the works of Archimedes, you cease to be amazed by the achievements of modern mathematicians.”2.2 Classes of Real Numbers and Computations53will be seen below, the principle of Archimedes essentially reflects the properties ofthe natural numbers and integers connected with completeness. We begin with theseproperties.10 .

Any nonempty subset of natural numbers that is bounded from above containsa maximal element.Proof If E ⊂ N is the subset in question, then by the least-upper-bound lemma,∃! sup E = s ∈ R. By definition of the least upper bound there is a natural numbern ∈ E satisfying the condition s − 1 < n ≤ s. But then, n = max E, since a naturalnumber that is larger than n must be at least n + 1, and n + 1 > s.Corollaries 20 The set of natural numbers is not bounded above.Proof Otherwise there would exist a maximal natural number. But n < n + 1.30 . Any nonempty subset of the integers that is bounded from above contains amaximal element.Proof The proof of 10 can be repeated verbatim, replacing N with Z.40 .

Any nonempty subset of integers that is bounded below contains a minimal element.Proof One can, for example, repeat the proof of 10 , replacing N by Z and using thegreatest-lower-bound principle instead of the least-upper-bound principle.Alternatively, one can pass to the negatives of the numbers (“change signs”) anduse what has been proved in 30 .50 . The set of integers is unbounded above and unbounded below.Proof This follows from 30 and 40 , or directly from 20 .We can now state the principle of Archimedes.60 . (The principle of Archimedes).

For any fixed positive number h and any realnumber x there exists a unique integer k such that (k − 1)h ≤ x < kh.Proof Since Z is not bounded above, the set {n ∈ Z | xh < n} is a nonempty subsetof the integers that is bounded below. Then (see 40 ) it contains a minimal element k,that is (k − 1) ≤ x/ h < k.

Since h > 0, these inequalities are equivalent to thosegiven in the statement of the principle of Archimedes. The uniqueness of k ∈ Zsatisfying these two inequalities follows from the uniqueness of the minimal elementof a set of numbers (see Sect. 2.1.3).542The Real NumbersAnd now some corollaries:70 .For any positive number ε there exists a natural number n such that 0 <1n< ε.Proof By the principle of Archimedes there exists n ∈ Z such that 1 < ε · n.

Since0 < 1 and 0 < ε, we have 0 < n. Thus n ∈ N and 0 < n1 < ε.80 . If the number x ∈ R is such that 0 ≤ x and x <1nfor all n ∈ N, then x = 0.Proof The relation 0 < x is impossible by virtue of 70 .90 . For any numbers a, b ∈ R such that a < b there is a rational number r ∈ Qsuch that a < r < b.Proof Taking account of 70 , we choose n ∈ N such that 0 < n1 < b − a. By themprinciple of Archimedes we can find a number m ∈ Z such that m−1n ≤ a < n . Thenmm−1mn < b, since otherwise we would have n ≤ a < b ≤ n , from which it wouldmfollow that n1 ≥ b − a.

Thus r = mn ∈ Q and a < n < b.100 . For any number x ∈ R there exists a unique integer k ∈ Z such that k ≤ x <k + 1.Proof This follows immediately from the principle of Archimedes.The number k just mentioned is denoted [x] and is called the integer part of x.The quantity {x} := x − [x] is called the fractional part of x. Thus x = [x] + {x},and {x} ≥ 0.2.2.4 The Geometric Interpretation of the Set of Real Numbersand Computational Aspects of Operations with RealNumbersa.

The Real LineIn relation to real numbers we often use a descriptive geometric language connectedwith a fact that you know in general terms from school. By the axioms of geometrythere is a one-to-one correspondence f : L → R between the points of a line Land the set R of real numbers.

Moreover this correspondence is connected withthe rigid motions of the line. To be specific, if T is a parallel translation of theline L along itself, there exists a number t ∈ R (depending only on T ) such thatf (T (x)) = f (x) + t for each point x ∈ L.The number f (x) corresponding to a point x ∈ L is called the coordinate of x. Inview of the one-to-one nature of the mapping f : L → R, the coordinate of a point isoften called simply a point.

For example, instead of the phrase “let us take the point2.2 Classes of Real Numbers and Computations55whose coordinate is 1” we say “let us take the point 1”. Given the correspondencef : L → R, we call the line L the coordinate axis or the number axis or the real line.Because f is bijective, the set R itself is also often called the real line and its pointsare called points of the real line.As noted above, the bijective mapping f : L → R that defines coordinates on Lhas the property that under a parallel translation T the coordinates of the imagesof points of the line L differ from the coordinates of the points themselves by anumber t ∈ R, the same for every point.

For this reason f is determined completelyby specifying the point that is to have coordinate 0 and the point that is to havecoordinate 1, or more briefly, by the point 0, called the origin, and the point 1. Theclosed interval determined by these points is called the unit interval. The directiondetermined by the ray with origin at 0 containing 1 is called the positive directionand a motion in that direction (from 0 to 1) is called a motion from left to right. Inaccordance with this convention, 1 lies to the right of 0 and 0 to the left of 1.Under a parallel translation T that moves the origin x0 to the point x1 = T (x0 )with coordinate 1, the coordinates of the images of all points are one unit largerthan those of their pre-images, and therefore we locate the point x2 = T (x1 ) withcoordinate 2, the point x3 = T (x2 ) with coordinate 3, .

. . , and the point xn+1 =T (xn ) with coordinate n + 1, as well as the point x−1 = T −1 (x0 ) with coordinate−1, . . . , the point x−n−1 = T −1 (x−n ) with coordinate −n − 1. In this way we obtainall points with integer coordinates m ∈ Z.Knowing how to double, triple, . . . the unit interval, we can use Thales’ theoremto partition this interval into n congruent subintervals. By taking the subintervalhaving an endpoint at the origin, we find that the coordinate of its other end, whichwe denote by x, satisfies the equation n · x = 1, that is, x = n1 . From this we find allpoints with rational coordinates mn ∈ Q.But there still remain points of L, since we know there are intervals incommensurable with the unit interval.

Each such point, like every other point of theline, divides the line into two rays, on each of which there are points with integeror rational coordinates. (This is a consequence of the original geometric principleof Archimedes.) Thus a point produces a partition, or, as it is called, a cut of Qinto two nonempty sets X and Y corresponding to the rational points (points withrational coordinates) on the left-hand and right-hand rays. By the axiom of completeness, there is a number c that separates X and Y , that is, x ≤ c ≤ y for allx ∈ X and all y ∈ Y . Since X ∪ Y = Q, it follows that sup X = s = i = inf Y . Forotherwise, s < i and there would be a rational number between s and i lying neitherin X nor in Y .

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