‎Matemáticas‎

‎HEWITT, EDWIN and KENNETH A. ROSS.‎

‎Abstract Harmonic Analysis. Vol. I: Structure of Topological Groups, Integration Theory, Group Representations.‎

‎Berlin, Göttingen, Heidelberg, Springer-Verlag, 1963. Orig. full cloth. VIII,547 pp. Clean and fine.‎

Referência livreiro : 48367

‎"CHURCH, ALONZO (+) ALAN TURING (+) EMIL POST.‎

‎[Church:] A note on the Entscheidungsproblem (+) Correction to A note on the Entscheidungsproblem (+) Review of ""A. M. Turing. On Computable numbers, with an application to the Entscheidungsproblem"" (+) [Post:] Finite combinatory processes-formulation... - [LANDMARK VOLUME IN THE HISTORY OF LOGIC]‎

‎[No place], The Association for Symbolic Logic, 1936 & 1937. Royal8vo. Bound in red half cloth with gilt lettering to spine. In ""Journal of Symbolic Logic"", Volume 1 & 2 bound together. Barcode label pasted on to back board. Small library stamp to lower part of 16 pages. A very fine copy. [Church:] Pp. 40-1" Pp. 101-2. [Post:] Pp. 103-5. [Turing:] Pp. 153-163" 164. [Entire volume: (4), 218, (2), IV, 188 pp.]‎

‎First edition of this collection of seminal papers within mathematical logic, all constituting some of the most important contributions mathematical logic and computional mathematics. A NOTE ON THE ENTSCHEIDUNGSPROBLEM (+) CORRECTION TO A NOTE ON THE ENTSCHEIDUNGSPROBLEM (+) REVIEW OF ""A. M. TURING. ON COMPUTABLE NUMBERS, WITH AN APPLICATION TO THE ENTSCHEIDUNGSPROBLEM"":First publication of Church's seminal paper in which he proved the solution to David Hilbert's ""Entscheidungsproblem"" from 1928, namely that it is impossible to decide algorithmically whether statements within arithmetic are true or false. In showing that there is no general algorithm for determining whether or not a given statement is true or false, he not only solved Hilbert's ""Entscheidungsproblem"" but also laid the foundation for modern computer logic. This conclusion is now known as Church's Theorem or the Church-Turing Theorem (not to be mistaken with the Church-Turing Thesis). The present paper anticipates Turing's famous ""On Computable Numbers"" by a few months. ""Church's paper, submitted on April 15, 1936, was the first to contain a demonstration that David Hilbert's 'Entscheidungsproblem' - i.e., the question as to whether there exists in mathematics a definite method of guaranteeing the truth or falsity of any mathematical statement - was unsolvable. Church did so by devising the 'lambda-calculus', [...] Church had earlier shown the existence of an unsolvable problem of elementary number theory, but his 1936 paper was the first to put his findings into the exact form of an answer to Hilbert's 'Entscheidungsproblem'. Church's paper bears on the question of what is computable, a problem addressed more directly by Alan Turing in his paper 'On computable numbers' published a few months later. The notion of an 'effective' or 'mechanical' computation in logic and mathematics became known as the Church-Turing thesis."" (Hook & Norman: Origins of Cyberspace, 250) Church coined in his review of Turing's paper the phrase 'Turing machine'.FINITE COMBINATORY PROCESSES-FORMULATION I: The Polish-American mathematician Emil Post made notable contributions to the theory of recursive functions. In the 1930s, independently of Turing, Post came up with the concept of a logic automaton similar to a Turing machine, which he described in the present paper (received on October 7, 1936). Post's paper was intended to fill a conceptual gap in Alonzo Church's paper on 'An unsolvable problem of elementary number theory'. Church had answered in the negative Hilbert's 'Entscheidungsproblem' but failed to provide the assertion that any such definitive method could be expressed as a formula in Church's lambda-calculus. Post proposed that a definite method would be one written in the form of instructions to mind-less worker operating on an infinite line of 'boxes' (equivalent to the Turing machines 'tape'). The range of instructions proposed by Post corresponds exactly to those performed by a Turing machine, and Church, who edited the Journal of Symbolic Logic, felt it necessary to insert an editorial note referring to Turing's ""shortly forthcoming"" paper on computable numbers, and asserting that ""the present article ... although bearing a later date, was written entirely independently of Turing's"". (Hook & Norman: Origins of Cyberspace, 356).COMPUTABILITY AND LAMBDA-DEFINABILITY (+) THE Ø-FUNCTION IN LAMBDA-K-CONVERSION: The volume also contains Turing's influential ""Computability and lambda-definability"" in which he proved that computable functions ""are identical with the lambda-definable functions of Church and the general recursive functions due to Herbrand and Gödel and developed by Kleene"". (Hook & Norman: Origins of Cyberspace, 395).‎

Referência livreiro : 48376

‎"KRIPKE, SAUL A.‎

‎A Completeness Theorem in Modal Logic. - [KRIPKE MODELS FOR MODAL LOGIC]‎

‎(No place), The Association for Symbolic Logic, 1959. Lev8vo. Bound in red half cloth with gilt lettering to spine. In ""Journal of Symbolic Logic"", Volume 24. Barcode label pasted on to back board. Small library stamp to lower part of 6 pages. A very fine copy. Pp. (1) - 14. [Entire volume: VI, 374 pp.).‎

‎The seminal first printing of Kripke's debut article, which provided the basis for his logic and for the model theory for modal logic in general. The work constitutes the very beginning of Kripke Semantics (often called possible world semantics). Kripke's works in general are rare in fist editions. Many of them remain unpublished and are only known in privately circulated manuscripts.The American philosopher Saul A. Kripke (born 1940) is an exceedingly important logician and philosopher of language and one of the most powerful and influential thinkers of analytic and Anglo-American philosophy. He is considered the greatest living philosopher and perhaps the greatest since Wittgenstein. In 2001 he was awarded the Schock Prize in Logic and Philosophy, which is considered the philosopical equivalent of the Nobel Prize.Kripke, who grew up in Omaha in a religious Jewish family, was somewhat of a prodigy child. During grammar school he got intimately acquainted with and mastered to perfection algebra, geometry and calculus, and very early on he took up philosophy, which later became his career. Still a teenager, in high school, he wrote a work that was to change the face of philosophical logic forever, namely the groundbreaking paper ""A Completeness Theorem for Modal Logic"", which was printed a few years later, in 1959, in the Journal of Symbolic Logic, while he was in his first year at Harvard University. This seminal debut work proposed what later came to be known as Kripke models for modal logic. The story goes that the paper earned a letter from the department of mathematics urging Kripke to apply for a job there, to which he is said to have written an answer explaining ""My mother said that I should finish high school and go to college first.""In 1962 he graduated from Harvard University, where he remained until 1968, first as a member of the Harvard Society of Fellows and then as a lecturer. During these years he developed the logical theories founded in the ""Completeness Theorem"" further and made seminal contributions to the field of logic and semantics. Kripke Semantics is a formal semantics for non-classical logic systems that Kripke began developing in his teenage years, first published something on in 1959 (the present work) and further developed in the 60'ies and. The development of Kripke Semantics was no less than a breakthrough in the making of non-classical logics, of which no model theory existed before Kripke's. With this work, Kripke laid the foundation for proving completeness theorems for modal logic, and for identifying the weakest normal modal logic, which is now named K after him.‎

Referência livreiro : 48378

‎"TURING, M. (+) H. A. NEWMAN.‎

‎A Formal Theorem in Church's Theory of Types (+) Practical forms of type theory‎

‎(No place), The Association for Symbolic Logic, 1942, 1943 &1948. Lev8vo. Bound in two uniform red half cloth with gilt lettering to spine. In ""Journal of Symbolic Logic"", Volume 7, 8 [Bound together] & 13.. Barcode label pasted on to back board. Small library stamp to lower part of 6 pages. Minor scratches to extremities of volume 13. A fine set. Pp. 28-33" Pp. 80-94. [Entire volumes: IV, 164 pp." IV, 236 pp.).‎

‎First printing of the two important - but often overlooked - papers by Turing which provide ""information about Turing's thoughts on the logical foundations of mathematics which is not to be found elsewhere in his writings"". (Copeland, The Essential Turing, P. 206).‎

Referência livreiro : 48381

‎BIEBERBACH, LUDWIG.‎

‎Vorlesungen über Algebra. Unter Benutzung der dritten Auflage des gleichnamigen Werkes von Gustav Bauer. In vierter Vermehrte Auflage. 16 Figuren im Text und auf 1 Tafel.‎

‎Leipzig, B.G. Teubner, Orig. full buckram. X,334 pp. Name cut from lower right of titlepage, no loss of text. Internally clean and fine.‎

Referência livreiro : 48428

‎COURANT, R.‎

‎Vorlesungen über Differential- und Integralrechnung. 2 Bde. (1. Funktionen einer Veränderlichen. Zweite, verb. Aufl. - 2. Funktionen mehrerer Veränderlichen. Zweite verb.Aufl.)‎

‎Berlin, J. Springer, 1930-31. 2 orig. full cloth. Gilt. XIV,410,VIII,412 pp. textfigs. Clean and fine.‎

Referência livreiro : 48431

‎BIEBERBACH, LUDWIG.‎

‎Theorie der Differentialgleichungen. Zweite neubearbeitete Auflage. Mit 22 Abbildungen.‎

‎Berlin, Springer, 1926. Orig. full cloth. IX,345,(2) pp. Clean and fine.‎

Referência livreiro : 48434

‎GENOCCHI, ANGELO. (= GIUSEPPE PEANO).‎

‎Differentialrechnung und Grundzüge der Integralrechnung. Herausgegeben von Giuseppe Peano. Autorisierte Deutsche Übersetzung von G. Bohlmann und A. Schepp. Mit einem Vorwort von A. Mayer.‎

‎Leipzig, B.G. Teubner, 1899. Wrappers blank. Old owners name on titlepage. VII,308 pp.‎

‎First German edition of Peano's first major publication, his first book, the work that brought him international fame, and one of the most important calculus texts since the time of Euler and Cauchy.The present book, which has a somewhat strange history, is considered a constitutional work of the science of infinitesimal calculus.‎

Referência livreiro : 48466

‎BLASCHKE, WILHELM.‎

‎Analytische Geometrie. Zweite Auflage.‎

‎Basel/Stuttgart, Birkhäuser, 1954. Lex8vo. Orig. full cloth. Name on titlepage. 190 pp. Internally fine and clean.‎

Referência livreiro : 48480

‎"BOLLÉE, LÉON. - THE INVENTION OF ""THE MILLIONAIRE CALCULATOR""‎

‎Sur une nouvelle machine à calculer.‎

‎(Paris, Gauthier-Villars, 1889). 4to. No wrappers. Disbound. In: ""Comptes Rendus Hebdomadaires des Séances de L'Academie des Sciences"", Tome 109, No 20. Pp. (723-) 758 (entire issue offered). Bollée's paper: pp. 737-739. Disbound but clean.‎

‎First printing of the paper in which Bollée describes his invention of a new calculating machine, later called ""The Millionaire"", based on a multiplying mechanism which was capable of performing multiplication directly instead of using repeated addition. As it allows multiplication by any digit it was used by government agencies ans scientists, especially astronomers, well into the twentieths century.Bollée did not produced his machine comercially, ""... but in 1893 Otto Steiger of Munich patented a calculator based on Bollée's approach, which was manufactured between 1895 and 1935 by the firm Hans W. Egli of Switzerland and marketed under the name of ""Millionaire"".... Forty-six hundred ""Millionaires"" were sold, primarly in Europe."" (Hook & Norman ""Origins of Cyberspace"" : 288).‎

Referência livreiro : 48744

‎"ELFVING, GUSTAV.‎

‎Convex Sets in Statistics.‎

‎Lund, Håkan Ohlssons,1953. 8vo. In the original grey printed wrappers. Offprint from ""Comptes Rendu du Douzième Congrès des Mathématiciens Scandinaves tenu à Lund 10-15 août 1953"". A very fine clean copy, near mint. Pp. (2), 35-39.‎

‎Scare offprint of Elfving's important paper on ""Elfving sets"". Elfving are famous for having introduced optimality theory of regression theory.‎

Referência livreiro : 48791

‎"SHEWHART, W.A.‎

‎Correction of Data for Errors of Measurement. (+) Correction of Data for Errors of Averages. (2 papers).‎

‎New York, Bell Telephone Laboratories, 1926. Orig. printed wrappers to both papers. Both punched with three holes in inner margin as issued. Offprints from ""The Bell System Technical Journal"", Vol. V. The offprints separately paginated: 16 pp. and 12 pp. The first clean and fine, the second with light wear to wrappers.‎

‎First editions in the offprint issues of two of Shewhart's main papers that reveals the evolution of his thinking and methods on the way to founding the science of statistical quality control or statistical proces control.""Shewhart was concerned that statistical theory serve the needs of industry. He exhibited the restlessness of one looking for a better way. A man of science who patiently developed and tested his ideas and the ideas of others, he was an astute observer of developments in the world of science and technology. While the literature of the day discussed the stochastic nature of both biological and technical systems, and spoke of the possibility of applying statistical methodology to these systems, Shewhart actually showed how it was to be done"" in that respect, the field of quality control can claim a genuine pioneer in Shewhart. His monumental work, Economic Control of Quality of Manufactured Product, published in 1931, is regarded as a complete and thorough exposition of the basic principles of quality control"" (ASQ).‎

Referência livreiro : 48799

‎CARTAN, È.‎

‎Leçons sur la géométrie des espaces de Riemann. Deuxieme edition, revue et augmentée.‎

‎Paris, Gauthier-Villars, 1946. 8vo. Uncut. frontwrapper lacks. Sewing a bit loose. Wrinkled.‎

Referência livreiro : 48833

‎"HOTELLING, HAROLD.‎

‎Causes of Birth Rate Fluctuations.‎

‎[No place, no printer], 1931. 8vo. Without wrappers (as issued). Offprint from: ""Journal of the American Statistical Association"", June, 1931. Fine and clean. Pp. 135-149.‎

‎Scarce offprint of Hotelling's important paper on in which it for the first time is statistically demonstrated that a delayed dip in birthrates will occur with populations iwhen subjected to harvest failures or epidemics.‎

Referência livreiro : 48847

‎"TUKEY, JOHN. (+) ET AL‎

‎Compactness in General Spaces (+)Generalized ""Sandwich"" Theorems (+) Named and Faceless Values: An Initial Exploration in Memory of Prasanta C. Mahalanobis (+) Some Sèlected Quick and Easy Methods of Statistical Analysis (+) Correlation of Ranges of c... - [LARGE COLLECTION OF OFFPRINTS]‎

‎[Various places and printer] ,1945 - 1974. Collection of 24 offprint from various academic journals. All with wrappers (or as issued) and in fine condition. Contained in a black kassett.‎

‎A large collection of offprint by American physicist John Tukey known for development of the FFT algorithm and box plot. Tukey's range test, the Tukey lambda distribution, Tukey's test of additivity and Tukey's lemma all bear his name.""John Tukey's whole life was one of public service, and as the preceding quotes make clear, he had profound influence. He was a member of the President's Scientific Advisory Committee for each of Presidents Eisenhower, Kennedy, and Johnson. He was special in many ways. He merged the scientific, governmental, technological, and industrial worlds more seamlessly than, perhaps, anyone else in the 1900s. His scientific knowledge, creativity, experience, calculating skills, and energy were prodigious. He was renowned for creating statistical concepts and words. JWT's graduate work was in mathematics, but driven by World War II, he left that field to go on to revolutionize the world of the analysis of data. At the end of the war he began a joint industrial-academic career at Bell Telephone Laboratories and at Princeton University. Science and the analysis of data were ubiquitous. This split career continued until he retired in 1985. Even after retirement his technical and scientific work continued at a very high level.He is said to have introduced the terms: ""bit"", ""linear programming"", ""ANOVA"", ""Colonel Blotto"", and was first into print with ""software"". Of these efforts L. Hogben and M. Cartwright wrote, ""The introduction by Tukey of bits for binary digits has nothing but irresponsible vulgarity to commend it."" Tukey's word ""polykay"" was described as ""linguistic miscegenation"" by Kendall and Stuart because of its combining a Greek prefix with a Latin suffix. JWT did it again later with ""polyspectrum"". (Brillinger, John Wilder Tukey).‎

Referência livreiro : 48852

‎NEYMAN. JERZY (+) ET AL‎

‎Sur une famille de tests asympopiques des hypotheses statistiques composees (+) Philosophical anlysis of 'objective' probability in science (+) An optimistic model of the correlation between light and severe accidents (+) True or false contagion (+)... - [COLLECTION OF OFFPRINTS]‎

‎[Various places and printer], 1935 -1962. Collection of 17 offprint from various academic journals. All with wrappers (or as issued) and in fine condition. Contained in a black kassett.‎

‎Collection of Polish mathematician and statistician Jerzy Neyman famous for being the first to introduce the modern concept of a confidence interval into statistical hypothesis testing.‎

Referência livreiro : 48855

‎"COX, DAVID ROXBEE (+) ET AL.‎

‎Some properties of multiserver queues with appointments (+) Regression models and life-tables (+) The continuity correction (+) Recent work on the design of experiments - a bibliography and a review (+) On distribution and their transformation to nor... - [LARGE COLLECTION OF OFFPRINTS]‎

‎[Various places and printers], 1962 - 1981. Collection of 36 offprint from various academic journals. All with wrappers (or as issued) and in fine condition. A few of them with ""Cox"" written to top of front wrapper. Housed in two black kasetts.‎

‎Large collection of offprints by British statistician David Cox who made pioneering and important contributions to numerous areas of statistics and applied probability, of which the best known is perhaps the proportional hazards model, which is widely used in the analysis of survival data. An example is survival times in medical research that can be related to information about the patients such as age, diet or exposure to certain chemical substances. The Cox process was named after him.‎

Referência livreiro : 48857

‎"NEUMANN, JOHN von.‎

‎Einige Bemerkungen zur Diracschen Theorie des relativistischen Drehelektrons.‎

‎Berlin, Julius Springer, 1927. 8vo. Contemp. hcloth. Gilt lettering to spine. In: ""Zeitschrift für Physik"" Vol. 48. VIII,891 pp. (Entire volume offered). Neumann's paper: pp. 868-881. A stamp to titlepage. Clean.‎

‎First printing of a famous paper in which Neumann extends the relativistic wave equation of the electron set forth by Dirac earlier the same year and Neumann here gives the five fundamental covariants of the Dirac theory.""Von Neumann’s most famous work in theoretical physics is his axiomatization of quantum mechanics. When he began work in that field in 1927, the methods used by its founders were hard to formulate in precise mathematical terms"" ""operator"" on ""functions"" were handled without much consideration of their domain or definition to their topological properties: and it was blithely assumed that such ""operators,"" when self-adjoint, could always be ""diagonalized"" (as in th finite dimensional case), at the expense of introducing ""Dirac functions"" as ""eigenvectors."" Von Neumann showed that mathematical rigor could be restored by taking as basic axioms the assumptions that the states of a physical system were poinds of a Hilbert space and that the measurable quantities were Hermitian (generally unbounded) operators densely efined in that space.""(DSB).‎

Referência livreiro : 48882

‎HAUGSTED, FRIDERICUS CHRISTIANUS.‎

‎Thymi in homine ac per seriem animalium descriptio anatomica-physiologicae‎

‎Hauniae [Copenhagen], 1831. 8vo. In the original printed boards. A few occassional brown spots, otherwise a fine copy. (6), 135, (1) pp + 3 folded plates.‎

‎First printing of Haugsted's early work on the thymus.‎

Referência livreiro : 48887

‎"KARHUNEN, KARI.‎

‎Über die Struktur stationärer zufälliger Funktionen.‎

‎Stockholm, Almqvist & Wiksells Boktryckeri AB, 1949. 8vo. In the original printed wrappers. Offprint from: ""Arkiv för matemativ, Utgivet av Kungl. Svenska Vetenskapsakademien"", Band 1, Nr 13. Previous owner's name to top of front wrapper. Pp. 141-160.‎

‎Scace offprint of Karhunen's paper on the structure of stationary random functions. Karhunen is famous for the Karhunen-Loève theorem and Karhunen-Loève transform.‎

Referência livreiro : 48891

‎"HOEM, JAN.‎

‎On the Optimality of Modified Minimum Chi-Square Analytical Graduation.‎

‎(Copenhagen), 1976. 8vo. Without wrappers (as issued). Offprint from ""Scand. J. Statist 3"", 1977: 1-20. A very fine and clean copy. 20 pp.‎

‎Offprint of Hoem's paper on Chi-Square analytical graduation. Hoem is best known for his work on event-history analysis - contributions that have helped shape demographic methodology.‎

Referência livreiro : 48893

‎"KRUSKAL, WILLIAM.‎

‎Helmert's Distribution.‎

‎[No place, no printer], 1946. 8vo. In the original printed wrappers. Offprint from: ""American Mathematical Monthly"", Vol. LIII, No. 8, 1946. A very fine and clean copy. Pp. 435-38.‎

‎Offprint of William Henry Kruskal first published paper. Kruskal, American mathematician and statistician, is famous for having formulated the Kruskal-Wallis one-way analysis of variance (together with W. Allen Wallis), a widely-used nonparametric statistical method.‎

Referência livreiro : 48895

‎"HUMBERT, MARIE-GEORGES.‎

‎Sur les courbes algébriques planes rectifiables (+) Sur quelques propriétés des aires sphériques (+) Sur le théorème d'Abel et quelques-unes de ses applications a la geometrie (+) Sur une classe de courbes planes, et sur une surface remarquable du qua...‎

‎Paris, Gauthier-Villars, 1888-1890 4to. Bound with the front wrappers in contemporary half cloth. In ""Journal de Mathématiques pures et appliquées"". With two paper labels to spine and with the author's name to top of each wrapper. A stamp to front free end paper and first wrapper. Otherwise fine.‎

‎First appearance of this collection of important papers which, amongst others, contains the questions to whether the involutes are algebraic.""Besides his two pedagogical works, it was through numerous memoirs (approximately 150, which have been collected) that Humbert held a major place in the mathematical discovery and production of his time. His writings were inspired by his interest in the study of algebraic curves and surfaces and were marked by the lucidity with which he related the problems biography in encountered in this area to questions of analysis and number theory.He thus enriched analysis and gave the complete solution of the two great questions of the transformation of hyperelliptic functions and of their complex multiplication. He also pointed out the resulting consequence: the existence of a group of transformations of certain surfaces into themselves constitutes an essential difference between the geometry of surfaces and that of curves. But, most important, he completed the work of Hermite by pursuing the applications to number theory throughout his life.""The present volume also contain contribution by Caspary and Hilbert.‎

Referência livreiro : 49018

‎"JACOBI, C. G. J.‎

‎Ueber die integration der partiellen differentialgleichungen erster ordnung.‎

‎Berlin, G. Reimer, 1832. 4to. As extracted from ""Journal für die reine und angewandte Mathematik, 1832, band 8"". Without backstrip. Fine and clean. Pp. 317-329 [Entire extract: Pp. 307-346 ].‎

‎First appearance of Jacobi's paper on the integration of partial differential equations of first order.""Most of Jacobi's fundamental research articles in the theory of elliptic functions, mathematical analysis, number theory, geometry, and me-chanics were published in Crelle's Journal fue die reine and angewandte Mathematik. With an average of three articles per volume, Jacobi was one of its most active contributors and quickly helped to establish its international fame. Yet his tireless occupation with research did not impair his teaching. On the contrary- never satisfied to lecture along trodden paths, Jacobi presented the substance of his own investigations to his students. He would lecture up to eight or ten hours a week on his favorite subject, the theory of elliptic functions, thus demanding the utmost from his listeners. He also inaugurated what was then a complete novelty in mathematics-research seminars-assembling the more advanced students and attracting his nearest colleagues."" (DSB).‎

Referência livreiro : 49049

‎"LIOUVILLE, JOSEPH. - THE EXISTANCE OF TRANSCENDENTAL NUMBERS PROVED.‎

‎Remarques... à des classes très-étendues de quantités dont la valeur n'est ni rationelle ni méme réductible à des irratonnelles algébriques... à un passage du livre des principes ou Newton... (Séance du Lundi 13 mai 1844). (+) Nouvelle démonstratio...‎

‎(Paris, Bachelier), 1844 4to. No wrappers. In: ""Comptes Rendus Hebdomadaires des Séances de L'Academie des Sciences"", Tome 18, No 20 a. 21. Pp. (883-) 898 a. pp. (899-) 960. (Entire issues offered). Liouville's papers: pp. 883-885 a. 910-911.‎

‎First printing of the papers in which Liouville proved the existence of transcendental numbers. ""J. Liouville..... invented a method (1844) for constructing any one of an extensive class of transcendental numbers. His numbers were the first to be proved transcendental" Hermite's proof of the transcendence of e ... followed in 1873" F. Lindemnn's for pi in 1882.""(Bell ""Development of Mathematics"", p. 275).Liouville showed that not only are these numbers irrational, they are always transcendental. Thus he was the first to prove the existence of transcendental numbers and also that there were an infinite number of them. Liouville suggested that e was transcendental, and provided the first example of a provably transcendental number, now known as Liouville’s constant.Parkinson ""Breakthroughs"", 1844 M.‎

Referência livreiro : 49077