```Date: Jul 25, 2013 6:09 PM
Author: clicliclic@freenet.de
Subject: Re: An independent integration test suite

Waldek Hebisch schrieb:> > Nasser M. Abbasi <nma@12000.org> wrote:> >> > I remember reading that some math people, 100 or 200 years ago,> > decided that sqrt (of non-negative, non-complex values) was single> > valued function and its value is the non-negative root (ie.> > principal square root).> > Beware that mathematicians have Humpty Dumpty attitude:> : When I use a word. it means just what I choose it to> :  mean -- neither more nor less.> Another mathematician may use the same word with different> meaning (of course, in good math text meaning is explained).> > Before complex numbers were invented only nonnegative> square root was in use.> Euler in his "Vollständige Anleitung zur Algebra" (Complete guide toalgebra) aimed at laypeople and published in 1770 in St. Petersburg intwo volumes,  <http://math.dartmouth.edu/~euler/pages/E387.html>  <http://math.dartmouth.edu/~euler/pages/E388.html>defines the square root as double-valued (§120): "es können daher vonjedem Quadrat zwei Quadratwurzeln angegeben werden, deren eine positive,die andere negativ ist." He admits negative radicands as well (§147):SQRT(-4) = SQRT(4)*SQRT(-1) = 2*SQRT(-1), etc., and these square rootsare taken to be double-valued too (§150). On the other hand, heintroduces cube roots as single-valued: CBRT(8) = 2, CBRT(27) = 3 in§164 and CBRT(-8) = -2, CBRT(-27) = -3 in §167, and then remarks (§167):"Also werden wir hier nicht zu unmöglichen oder eingebildeten Zahlengeführt, wie bei den Quadratwurzeln der Negativzahlen geschehen.".Later on, however, in the context of solving cubic equations, he treatsall solutions (§151): any cubic equation x^3 = a has the three solutionsx = CBRT(a), x = (-1 + SQRT(-3))/2*CBRT(a), x = (-1 - SQRT(-3))/2*CBRT(a), and he adds: "woraus erhellt, daß jede Cubikwurzel dreierleiWerthe hat, von denen der erste möglich, die beiden anderen aberunmöglich sind."Euler always writes SQRT(-1) for the imaginary unit, and (§143) callssquare roots of negative numbers "unmögliche" (impossible) or"eingebildete" (imagined) numbers. These designations are justified asfollows (\$144): "Von diesen behauptet man also mit vollem Recht, daß sieweder größer noch kleiner sind als Nichts; und auch nicht einmal Nichtsselbst, weswegen sie für unmöglich gehalten werden müssen." On the otherhand (§145): "Dennoch stellen Sie unserem Verstande sich vor, und findenin unserer Einbildung Platz; daher sie auch blos eingebildete Zahlengenannt werden." And even though they are "impossible" we know how tocalculate with them: SQRT(-4)*SQRT(-4) = -4, etc.But why do it? Euler's reason sounds rather weak to me (\$151): "Endlichmuß noch das Bedenken gehoben werden, daß die Lehre von den unmöglichenZahlen als nutzlose Grille angesehen werden könne. Allein dies Bedenkenist unbegründet; diese Lehre ist in der That von der größtenWichtigkeit, indem oft Aufgaben vorkommen, von welchen man nicht sofortwissen kann, ob sie Mögliches oder Unmögliches verlangen. Wenn nun dieAuflösung derselben zu solchen unmöglichen Zahlen führt, so hat man einsicheres Zeichen, daß die Aufgabe Unmögliches verlangt." We do itbecause the occurrence of "impossible" numbers safely indicates that agiven problem has no solution.Obviously, the invention of complex numbers was a gradual process.Martin.PS: I am quoting from my undated (around 1900) Reclam version of Euler's"Algebra" in one volume, for which grammar and spelling were somewhatmodernized. Interestingly, the original German edition was preceded bythe publication in 1768/69 of a Russian edition prepared by two studentsof Euler under the title "Universal'naya arifmetika" (Universalarithmetic), volume I of which can also be found on the internet. Theoriginal German edition sells for something like Euro 7500,- nowadays:<http://www.sophiararebooks.com/pages/books/3056/leonhard-euler/vollstandige-anleitung-zur-algebra>Euler, Leonhard. Vollständige Anleitung zur Algebra. St. Petersburg: Kaiserliche Academie der Wissenschaften, 1770. Bound in 2 volumes, 188 x 121 mm, polished calf with raised bands andrichly gilt spine.Euler composed his famous 'Algebra' in German in 1765-6, soon after hisreturn to St. Petersburg from Berlin. He was by then partially blind,and dictated the work to a young valet. Publication of the originalGerman version [...] was, however, delayed until 1770 and thus came tobe preceded by a Russian translation by his students Peter Inokhodtsevand Ivan Yudin which was issued in 1768-9. [...]"Euler's Vollständige Anleitung zur Algebra is not only the most populartextbook on elementary algebra, with the exception of Euclid's Elementsit is the most widely printed book on mathematics." (Truesdell). It "wastranslated into Russian (1768-9), Dutch (1773), French (1774), Latin(1790), English (1797, 1822) and Greek (1800). The popular Germanedition from Reclam Verlag sold no less than 108,000 copies between 1883and 1943 (Reich, 1992). After 240 years this algebra textbook is stillin print today, available in several languages and editions" (Heeffer)."In 1770 Euler published his 'Vollständige Anleitung zur Algebra' ...The first volume treats of determinate algebra. This contains one of theearliest attempts to place the fundamental processes on a scientificbasis: the same subject had attracted D'Alembert's attention. This workalso includes the proof of the binomial theorem for an unrestricted realindex which is still known by Euler's name ... The second volume of thealgebra treats of indeterminate or Diophantine algebra. This containsthe solutions of some of the problems proposed by Fermat, and which hadhitherto remained unsolved." (Ball)."Euler proved the impossibility of solving x^3+y^3=z^3 in which x, y,and z are integers, x*y*z /= 0 (a particular case of Fermat's lasttheorem); his demonstration, based on the method of infinite descent andusing complex numbers of the form a+b*(-3)^(1/2), is thoroughlydescribed in his Vollständige Anleitung zur Algebra, the second volumeof which has a large section devoted to Diophantine analysis." (DSB).Truesdell, Leonard Euler: Supreme Geometer, 1972; Albrecht Heeffer, The Rhetoric of Problems in Algebra Textbooks fromPacioli to Euler, pp.4-5; Ball, A Short Account of the History of Mathematics, p.397;DSB presumably refers to: Charles Coulston Gillespie (ed.), Dictionaryof Scientific Biography, 1970-80.
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