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Thus s = i = c. This uniquely determined number c is assigned tothe corresponding point of the line.The assignment of coordinates to points of the line just described provides avisualizable model for both the order relation in R (hence the term “linear ordering”)and for the axiom of completeness or continuity in R, which in geometric languagemeans that there are no “holes” in the line L, which would separate it into two pieceshaving no points in common. (Such a separation could only come about by use ofsome point of the line L.)We shall not go into further detail about the construction of the mapping f : L →R, since we shall invoke the geometric interpretation of the set of real numbers only562The Real Numbersfor the sake of visualizability and perhaps to bring into play the reader’s very usefulgeometric intuition.
As for the formal proofs, just as before, they will rely either onthe collection of facts we have obtained from the axioms for the real numbers ordirectly on the axioms themselves.Geometric language, however, will be used constantly.We now introduce the following notation and terminology for the number setslisted below:]a, b[ := {x ∈ R | a < x < b} is the open interval ab;[a, b] := {x ∈ R | a ≤ x ≤ b} is the closed interval ab;]a, b] := {x ∈ R | a < x ≤ b} is the half-open interval ab containing b;[a, b[ := {x ∈ R | a ≤ x < b} is the half-open interval ab containing a.Definition 6 Open, closed, and half-open intervals are called numerical intervalsor simply intervals.
The numbers determining an interval are called its endpoints.The quantity b–a is called the length of the interval ab. If I is an interval, weshall denote its length by |I |. (The origin of this notation will soon become clear.)The sets]a, +∞[ := {x ∈ R | a < x},]−∞, b[ := {x ∈ R | x < b}[a, +∞[ := {x ∈ R | a ≤ x},]−∞, b] := {x ∈ R | x ≤ b}and ]−∞, +∞[ := R are conventionally called unbounded intervals or infinite intervals.In accordance with this use of the symbols +∞ (read “plus infinity”) and −∞(read “minus infinity”) it is customary to denote the fact that the numerical set X isnot bounded above (resp.
below), by writing sup X = +∞ (inf X = −∞).Definition 7 An open interval containing the point x ∈ R will be called a neighborhood of this point.In particular, when δ > 0, the open interval ]x − δ, x + δ[ is called the δneighborhood of x. Its length is 2δ.The distance between points x, y ∈ R is measured by the length of the intervalhaving them as endpoints.So as not to have to investigate which of the points is “left” and which is “right”,that is, whether x < y or y < x and whether the length is y − x or x − y, we can usethe useful function⎧when x > 0,⎨xwhen x = 0,|x| = 0⎩−x when x < 0,which is called the modulus or absolute value of the number.2.2 Classes of Real Numbers and Computations57Definition 8 The distance between x, y ∈ R is the quantity |x − y|.The distance is nonnegative and equals zero only when the points x and y arethe same.
The distance from x to y is the same as the distance from y to x, since|x − y| = |y − x|. Finally, if z ∈ R, then |x − y| ≤ |x − z| + |z − y|. That is, theso-called triangle inequality holds.The triangle inequality follows from a property of the absolute value that is alsocalled the triangle inequality (since it can be obtained from the preceding triangleinequality by setting z = 0 and replacing y by −y). To be specific, the inequality|x + y| ≤ |x| + |y|holds for any numbers x and y, and equality holds only when the numbers x and yare both negative or both positive.Proof If 0 ≤ x and 0 ≤ y, then 0 ≤ x + y, |x + y| = x + y, |x| = x, and |y| = y, sothat equality holds in this case.If x ≤ 0 and y ≤ 0, then x + y ≤ 0, |x + y| = −(x + y) = −x − y, |x| = −x,|y| = −y, and again we have equality.Now suppose one of the numbers is negative and the other positive, for example,x < 0 < y.
Then either x < x +y ≤ 0 or 0 ≤ x +y < y. In the first case |x +y| < |x|,and in the second case |x + y| < |y|, so that in both cases |x + y| < |x| + |y|.Using the principle of induction, one can verify that|x1 + · · · + xn | ≤ |x1 | + · · · + |xn |,and equality holds if and only if the numbers x1 , . . . , xn are all nonnegative or allnonpositive.The number a+b2 is often called the midpoint or center of the interval with endpoints a and b, since it is equidistant from the endpoints of the interval.In particular, a point x ∈ R is the center of its δ-neighborhood ]x − δ, x + δ[ andall points of the δ-neighborhood lie at a distance from x less than δ.b.
Defining a Number by Successive ApproximationsIn measuring a real physical quantity, we obtain a number that, as a rule, changeswhen the measurement is repeated, especially if one changes either the method ofmaking the measurement or the instrument used. Thus the result of measurement isusually an approximate value of the quantity being sought. The quality or precisionof a measurement is characterized, for example, by the magnitude of the possiblediscrepancy between the true value of the quantity and the value obtained for itby measurement.
When this is done, it may happen that we can never exhibit theexact value of the quantity (if it exists theoretically). Taking a more constructiveposition, however, we may (or should) consider that we know the desired quantity582The Real Numberscompletely if we can measure it with any preassigned precision. Taking this positionis tantamount to identifying the number with a sequence6 of more and more preciseapproximations by numbers obtained from measurement. But every measurementis a finite set of comparisons with some standard or with a part of the standardcommensurable with it, so that the result of the measurement will necessarily beexpressed in terms of natural numbers, integers, or, more generally, rational numbers. Hence theoretically the whole set of real numbers can be described in terms ofsequences of rational numbers by constructing, after due analysis, a mathematicalcopy or, better expressed, a model of what people do with numbers who have nonotion of their axiomatic description.
The latter add and multiply the approximatevalues rather than the values being measured, which are unknown to them. (To besure, they do not always know how to say what relation the result of these operationshas to the result that would be obtained if the computations were carried out withthe exact values. We shall discuss this question below.)Having identified a number with a sequence of approximations to it, we shouldthen, for example, add the sequences of approximate values when we wish to addtwo numbers.
The new sequence thus obtained must be regarded as a new number,called the sum of the first two. But is it a number? The subtlety of the questionresides in the fact that not every randomly constructed sequence is the sequenceof arbitrarily precise approximations to some quantity. That is, one still has to learnhow to determine from the sequence itself whether it represents some number or not.Another question that arises in the attempt to make a mathematical copy of operations with approximate numbers is that different sequences may be approximatingsequences for the same quantity. The relation between sequences of approximationsdefining a number and the numbers themselves is approximately the same as thatbetween a point on a map and an arrow on the map indicating the point.
The arrowdetermines the point, but the point determines only the tip of the arrow, and does notexclude the use of a different arrow that may happen to be more convenient.A precise description of these problems was given by Cauchy,7 who carried outthe entire program of constructing a model of the real numbers, which we have onlysketched.
One may hope that after you study the theory of limits you will be able torepeat these constructions independently of Cauchy.What has been said up to now, of course, makes no claim to mathematical rigor.The purpose of this informal digression has been to direct the reader’s attention tothe theoretical possibility that more than one natural model of the real numbers mayexist. I have also tried to give a picture of the relation of numbers to the world aroundus and to clarify the fundamental role of natural and rational numbers. Finally, Iwished to show that approximate computations are both natural and necessary.6 Ifn is the number of the measurement and xn the result of that measurement, the correspondencen → xn is simply a function f : N → R of a natural-number argument, that is, by definition a sequence (in this case a sequence of numbers). Section 3.1 is devoted to a detailed study of numericalsequences.7 A.
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