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Cauchy (1789–1857) – French mathematician, one of the most active creators of the languageof mathematics and the machinery of classical analysis.2.2 Classes of Real Numbers and Computations59The next part of the present section is devoted to simple but important estimatesof the errors that arise in arithmetic operations on approximate quantities. Theseestimates will be used below and are of independent interest.We now give precise statements.Definition 9 If x is the exact value of a quantity and x̃ a known approximation tothe quantity, the numbersΔ(x̃) := |x − x̃|andδ(x̃) :=Δ(x̃)|x̃|are called respectively the absolute and relative error of approximation by x̃. Therelative error is not defined when x̃ = 0.Since the value x is unknown, the values of Δ(x̃) and δ(x̃) are also unknown.However, one usually knows some upper bounds Δ(x̃) < Δ and δ(x̃) < δ for thesequantities.

In this case we say that the absolute or relative error does not exceed Δ orδ respectively. In practice we need to deal only with estimates for the errors, so thatthe quantities Δ and δ themselves are often called the absolute and relative errors.But we shall not do this.The notation x = x̃ ± Δ means that x̃ − Δ ≤ x ≤ x̃ + Δ.For example,gravitational constantspeed of light in vacuoPlanck’s constantcharge of an electronrest mass of an electronG = (6.672598 ± 0.00085) × 10−11 N · m2 /kg2 ,c = 299 792 458 m/s (exactly),h = (6.6260755 ± 0.0000040) × 10−34 J · s,e = (1.60217733 ± 0.00000049) × 10−19 C,me = (9.1093897 ± 0.0000054) × 10−31 kg.The main indicator of the precision of a measurement is the relative error inapproximation, usually expressed as a percent.Thus in the examples just given the relative errors are at most (in order):13 × 10−5 ;0;6 × 10−7 ;31 × 10−8 ;6 × 10−7or, as percents of the measured values,13 × 10−3 %;0 %;6 × 10−5 %;31 × 10−6 %;6 × 10−5 %.We now estimate the errors that arise in arithmetic operations with approximatequantities.Proposition If|x − x̃| = Δ(x̃),|y − ỹ| = Δ(ỹ),602The Real NumbersthenΔ(x̃ + ỹ) := (x + y) − (x̃ + ỹ) ≤ Δ(x̃) + Δ(ỹ),Δ(x̃ · ỹ) := |x · y − x̃ · ỹ| ≤ |x̃|Δ(ỹ) + |ỹ|Δ(x̃) + Δ(x̃) · Δ(ỹ);(2.1)(2.2)if, in addition,y = 0,thenỹ = 0 and δ(ỹ) =Δ(ỹ)< 1,|ỹ| x x̃ |x̃|Δ(ỹ) + |ỹ|Δ(x̃)1x̃:= − ≤·.Δỹy ỹ1 − δ(ỹ)ỹ 2(2.3)Proof Let x = x̃ + α and y = ỹ + β.

ThenΔ(x̃ + ỹ) = (x + y) − (x̃ + ỹ) = |α + β| ≤ |α| + |β| = Δ(x̃) + Δ(ỹ),Δ(x̃ · ỹ) = |xy − x̃ · ỹ| = (x̃ + α)(ỹ + β) − x̃ · ỹ == |x̃β + ỹα + αβ| ≤ |x̃||β| + |ỹ||α| + |αβ| == |x̃|Δ(ỹ) + |ỹ|Δ(x̃) + Δ(x̃) · Δ(ỹ), x x̃ x ỹ − y x̃ x̃== − = Δỹy ỹy ỹ (x̃ + α)ỹ − (ỹ + β)x̃ |x̃||β| + |ỹ||α|11≤·= =·221 + β/ỹ1 − δ(ỹ)ỹỹ=|x̃|Δ(ỹ) + |ỹ|Δ(x̃)1.·21 − δ(ỹ)ỹThese estimates for the absolute errors imply the following estimates for therelative errors:δ(x̃ + ỹ) ≤Δ(x̃) + Δ(ỹ),|x̃ + ỹ|δ(x̃ · ỹ) ≤ δ(x̃) + δ(ỹ) + δ(ỹ) · δ(ỹ), δ(x̃) + δ(ỹ)x̃≤.δỹ1 − δ(ỹ)(2.1 )(2.2 )(2.3 )In practice, when working with sufficiently good approximations, we have Δ(x̃) ·Δ(ỹ) ≈ 0, δ(x̃) · δ(ỹ) ≈ 0, and 1 − δ(ỹ) ≈ 1, so that one can use the followingsimplified and useful, but formally incorrect, versions of formulas (2.2), (2.3), (2.2 ),and (2.3 ):Δ(x̃ · ỹ) ≤ |x̃|Δ(ỹ) + |ỹ|Δ(x̃),2.2 Classes of Real Numbers and Computations61 x̃|x̃|Δ(ỹ) + ỹΔ(x̃)Δ,≤ỹỹ 2δ(x̃ · ỹ) ≤ δ(x̃) + δ(ỹ), x̃≤ δ(x̃) + δ(ỹ).δỹFormulas (2.3) and (2.3 ) show that it is necessary to avoid dividing by a numberthat is near zero and also to avoid using rather crude approximations in which ỹ or1 − δ(ỹ) is small in absolute value.Formula (2.1 ) warns against adding approximate quantities if they are close toeach other in absolute value but opposite in sign, since then |x̃ + ỹ| is close to zero.In all these cases, the errors may increase sharply.For example, suppose your height has been measured twice by some device,and the precision of the measurement is ±0.5 cm.

Suppose a sheet of paper wasplaced under your feet before the second measurement. It may nevertheless happenthat the results of the measurement are as follows: H1 = (200 ± 0.5) cm and H2 =(199.8 ± 0.5) cm respectively.It does not make sense to try to find the thickness of the paper in the form of thedifference H2 − H1 , from which it would follow only that the thickness of the paperis not larger than 0.8 cm. That would of course be a crude reflection (if indeed onecould even call it a “reflection”) of the true situation.However, it is worthwhile to consider another more hopeful computational effectthrough which comparatively precise measurements can be carried out with crudedevices. For example, if the device just used for measuring your height was used tomeasure the thickness of 1000 sheets of the same paper, and the result was (20 ±0.5) cm, then the thickness of one sheet of paper is (0.02 ± 0.0005) cm, which is(0.2 ± 0.005) mm, as follows from formula (2.1).That is, with an absolute error not larger than 0.005 mm, the thickness of onesheet is 0.2 mm.

The relative error in this measurement is at most 0.025 or 2.5 %.This idea can be developed and has been proposed, for example, as a way ofdetecting a weak periodic signal amid the larger random static usually called whitenoise.c. The Positional Computation SystemIt was stated above that every real number can be presented as a sequence of rational approximations. We now recall a method, which is important when it comes tocomputation, for constructing in a uniform way a sequence of such rational approximations for every real number. This method leads to the positional computationsystem.Lemma If a number q > 1 is fixed, then for every positive number x ∈ R thereexists a unique integer k ∈ Z such thatq k−1 ≤ x < q k .622The Real NumbersProof We first verify that the set of numbers of the form q k , k ∈ N, is not boundedabove. If it were, it would have a least upper bound s, and by definition of the leastupper bound, there would be a natural number m ∈ N such that qs < q m ≤ s.

Butthen s < q m+1 , so that s could not be an upper bound of the set.Since 1 < q, it follows that q m < q n when m < n for all m, n ∈ Z. Hence we havealso shown that for every real number c ∈ R there exists a natural number N ∈ Nsuch that c < q n for all n > N .It follows that for any ε > 0 there exists M ∈ N such that q1m < ε for all naturalnumbers m > M.Indeed, it suffices to set c = 1ε and N = M; then 1ε < q m when m > M.Thus the set of integers m ∈ Z satisfying the inequality x < q m for x > 0 isbounded below. It therefore has a minimal element k, which obviously will be theone we are seeking, since, for this integer, q k−1 ≤ x < q k .The uniqueness of such an integer k follows from the fact that if m, n ∈ Z and,for example, m < n, then m ≤ n − 1.

Hence if q > 1, then q m ≤ q n−1 .Indeed, it can be seen from this remark that the inequalities q m−1 ≤ x < q m andq n−1 ≤ x < q n , which imply q n−1 ≤ x < q m , are incompatible if m = n.We shall use this lemma in the following construction. Fix q > 1 and take anarbitrary positive number x ∈ R. By the lemma we find a unique number p ∈ Zsuch thatq p ≤ x < q p+1 .(2.4)Definition 10 The number p satisfying (2.4) is called the order of x in the base qor (when q is fixed) simply the order of x.By the principle of Archimedes, we find a unique natural number αp ∈ N suchthatαp q p ≤ x < αp q p + q p .(2.5)Taking (2.4) into account, one can assert that αp ∈ {1, . .

. , q − 1}.All of the subsequent steps in our construction will repeat the step we are aboutto take, starting from relation (2.5).It follows from relation (2.5) and the principle of Archimedes that there exists aunique number αp−1 ∈ {0, 1, . .

. , q − 1} such thatαp q p + αp−1 q p−1 ≤ x < αp q p + αp−1 q p−1 + q p−1 .If we have made n such steps, obtaining the relationαp q p + αp−1 q p−1 + · · · + αp−n q p−n ≤≤ x < αp q p + αp−1 q p−1 + · · · + αp−n q p−n + q p−n ,(2.6)2.2 Classes of Real Numbers and Computations63then by the principle of Archimedes there exists a unique number αp−n−1 ∈{0, 1, . . . , q − 1} such thatαp q p + · · · + αp−n q p−n + αp−n−1 q p−n−1 ≤≤ x < αp q p + · · · + αp−n q p−n + αp−n−1 q p−n−1 + q p−n−1 .Thus we have exhibited an algorithm by means of which a sequence of numbersαp , αp−1 , . .

. , αp−n , . . . from the set {0, 1, . . . , q − 1} is placed in correspondencewith the positive number x. Less formally, we have constructed a sequence of rational numbers of the special formrn = αp q p + · · · + αp−n q p−n ,(2.7)and such thatrn ≤ x < rn +1.q n−p(2.8)In other words, we construct better and better approximations from belowand from above to the number x using the special sequence (2.7). The symbolαp . . .

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