1ST TERM MATHS PROJECT

1. 1ST TERM MATHS PROJECT DONE BY:- SOUMEE SENGUPTA 9A 9107

2. CONTENTS • LIFE OF MATHEMATICIAN G.CANTOR • LIFE OF MATHEMATICIAN R.DEDEKIND • LIFE OF MATHEMATICIAN ARCHIMEDES • LIFE OF MATHEMATICIAN RENE DESCARTES • LIFE OF MATHEMATICIAN THALES

3. Georg Ferdinand Ludwig Philipp Cantor ( March 3, 1845, St. Petersburg, Russia – January 6, 1918, Halle, Germany) was a German mathematician who is best known as the creator of set theory. He was born between 1809 and 1814 in Copenhagen, Denmark, and brought up in a Lutheran German mission in St. Petersburg. Georg Cantor's father was a Danish man of Lutheran religion. His mother, Maria Anna Böhm, was born in St. Petersburg and came from an Austrian Roman Catholic family. In 1860, Cantor graduated with distinction from the Realschule in Darmstadt; his exceptional skills in mathematics, trigonometry in particular, were noted. In 1862, following his father's wishes, Cantor entered the Federal Polytechnic Institute in Zurich, today the ETH Zurich and began studying mathematics. In 1867, Berlin granted him the Ph.D. for a thesis on number theory, De aequationibus secundi gradus indeterminatis. After teaching one year in a Berlin girls' school, Cantor took up a position at the University of Halle, where he spent his entire career. He was awarded the requisite habilitation for his thesis on number theory. LIFE OF MATHEMATICIAN G.CANTOR CHILDHOOD AND EDUCATION

4. ADULTHOOD AND HIS FINAL YEARS • Cantor was promoted to Extraordinary Professor in 1872, and made full Professor in 1879. To attain the latter rank at the age of 34 was a notable accomplishment, but Cantor very much desired a chair at a more prestigious university, in particular at Berlin, then the leading German university. • . However, Kronecker, who headed mathematics at Berlin until his death in 1891, and his colleague Hermann Schwarz were not agreeable to having Cantor as a colleague. • In 1890, Cantor was instrumental in founding the Deutsche Mathematiker-Vereinigung, chaired its first meeting in Halle in 1891, and was elected its first president. • In 1911, Cantor was one of the distinguished foreign scholars invited to attend the 500th anniversary of the founding of the University of St. Andrews in Scotland. • Cantor retired in 1913, and suffered from poverty, even hunger, during World War I. The public celebration of his 70th birthday was cancelled because of the war. He died in the sanatorium where he had spent the final year of his life.

5. HIS WORKS • He was the first to see that infinite sets come in different sizes, as follows. He first showed that given any set A, the set of all possible subsets of A, called the power set of A, exists. He then proved that the power set of an infinite set A has a size greater than the size of A (this fact is now known as Cantor's theorem). Thus there is an infinite hierarchy of sizes of infinite sets, from which springs the transfinite cardinal and ordinal numbers, and their peculiar arithmetic. His notation for the cardinal numbers was the Hebrew letter aleph with a natural number subscript; for the ordinals he employed the Greek letter omega. • Cantor was the first to appreciate the value of one-to-one correspondences (hereinafter denoted "1-to-1") for set theory. He defined finite and infinite sets, breaking down the latter into denumerable and nondenumerable setsThere exists a 1-to-1 correspondence between any denumerable set and the set of all natural numbers; all other infinite sets are nondenumerable. He proved that the set of all rational numbers is denumerable, but that the set of all real numbers is not and hence is strictly bigger. The cardinality of the natural numbers is aleph-null; that of the reals is larger, and is at least aleph-one (the latter being the next smallest cardinal after aleph-null). • Cantor was the first to show that there was more than one kind of infinity. In doing so, he became the first to invoke the notion of a 1-to-1 correspondence, albeit not calling it such. • Between 1879 and 1884, Cantor published a series of six articles in Mathematische Annalen that together formed an introduction to his set theory. • Cantor was the first to show that there was more than one kind of infinity. In doing so, he became the first to invoke the notion of a 1-to-1 correspondence. • Cantor's work did attract favorable notice beyond Hilbert's celebrated encomium. In public lectures delivered at the first International Congress of Mathematicians, held in Zurich in 1897, Hurwitz and Hadamard both expressed their admiration for Cantor's set theory.

6. HIS PICTURE AND HIS FULL NAME Georg Ferdinand Ludwig Philipp Cantor • Born: 3 March 1845 in St Petersburg, Russia • Died: 6 Jan 1918 in Halle, Germany

7. LIFE OF MATHEMATICIAN R.DEDEKIND CHILDHOOD AND EDUCATION • Richard Dedekind's father was a professor at the Collegium Carolinum in Brunswick. His mother was the daughter of a professor who also worked at the Collegium Carolinum . • He attended school in Brunswick from the age of seven . • The Collegium Carolinum was an educational institution between a high school and a university and he entered it in 1848 at the age of 16. • In the autumn term of 1850, Dedekind attended his first course given by Gauss. It was a course on least squares and [1]:- • In 1854 both Riemann and Dedekind were awarded their habilitation degrees within a few weeks of each other. Dedekind was then qualified as a university teacher and he began teaching at Göttingen giving courses on probability and geometry.

8. ADULTHOOD AND HIS WORKS • The idea that came to him on 24 November 1858 was that every real number r divides the rational numbers into two subsets, namely those greater than r and those less than r. Dedekind's brilliant idea was to represent the real numbers by such divisions of the rationals. • The Collegium Carolinum in Brunswick had been upgraded to the Brunswick Polytechnikum by the 1860s, and Dedekind was appointed to the Polytechnikum in 1862 • As well as his analysis of the nature of number, his work on mathematical induction, including the definition of finite and infinite sets, and his work in number theory, particularly in algebraic number fields, is of major importance. • It was in the third and fourth editions of Vorlesungen über Zahlentheorie, published in 1879 and 1894, that Dedekind wrote supplements in which he introduced the notion of an ideal which is fundamental to ring theory. Dedekind formulated his theory in the ring of integers of an algebraic number field. • He presented a logical theory of number and of complete induction, presented his principal conception of the essence of arithmetic, and dealt with the role of the complete system of real numbers in geometry in the problem of the continuity of space. Among other things, he provides a definition independent of the concept of number for the infiniteness or finiteness of a set by using the concept of mappingand treating the recursive definition, which is so important to the theory of ordinal numbers. • Dedekind's brilliance consisted not only of the theorems and concepts that he studied but, because of his ability to formulate and express his ideas so clearly, he introduced a new style of mathematics that been a major influence on mathematicians ever since.

9. HIS PICTURE AND HIS FULL NAME Julius Wilhelm Richard Dedekind • Born: 6 Oct 1831 in Braunschweig,Germany • Died: 12 Feb 1916 in Braunschweig,Germany

10. LIFE OF MATHEMATICIAN ARCHIMEDES CHILDHOOD AND HIS MATURED LIFE • Archimedes of Syracuse (Greek: Ἀρχιμήδης) (c. 287 BC – c. 212 BC) was a Greek mathematician, physicist, engineer, inventor, and astronomer. • Archimedes is considered to be one of the greatest mathematicians of all time.[2] He used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave a remarkably accurate approximation of Pi.[3] He also defined the spiral bearing his name, formulas for the volumes of surfaces of revolution and an ingenious system for expressing very large numbers • Archimedes was born c. 287 BC in the seaport city of Syracuse, Sicily, at that time a colony of Magna Graecia • A biography of Archimedes was written by his friend Heracleides but this work has been lost, leaving the details of his life obscure.[8] It is unknown, for instance, whether he ever married or had children • Archimedes died c. 212 BC during the Second Punic War, when Roman forces under General Marcus Claudius Marcellus captured the city of Syracuse after a two-year-long siege. According to the popular account given by Plutarch, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, saying that he had to finish working on the problem. The soldier was enraged by this, and killed Archimedes with his sword.

11. HIS MATHEMATICAL ACHIEVEMENTS • While he is often regarded as a designer of mechanical devices, Archimedes also made contributions to the field of mathematics. • Archimedes was able to use infinitesimals in a way that is similar to modern integral calculus. By assuming a proposition to be true and showing that this would lead to a contradiction, he could give answers to problems to an arbitrary degree of accuracy, while specifying the limits within which the answer lay. This technique is known as the method of exhaustion, and he employed it to approximate the value of π (Pi ) • In The Measurement of a Circle, Archimedes gives the value of the square root of 3 as being more than 265/153 (approximately 1.7320261) and less than 1351/780 (approximately 1.7320512). The actual value is approximately 1.7320508, making this a very accurate estimate. • In The Quadrature of the Parabola, Archimedes proved that the area enclosed by a parabola and a straight line is 4/3 multiplied by the area of a triangle with equal base and height. He expressed the solution to the problem as a geometric series that summed to infinity with the ratio 1/4: • If the first term in this series is the area of the triangle, then the second is the sum of the areas of two triangles whose bases are the two smaller secant lines, and so on. This proof is a variation of the infinite series1/4 + 1/16 + 1/64 + 1/256 + · · · which sums to 1/3. • Archimedes devised a system of counting based on the myriad. The word is from the Greek μυριάς murias, for the number 10,000. He proposed a number system using powers of a myriad of myriads (100 million) and concluded that the number of grains of sand required to fill the universe would be 8×1063, which can also be expressed as eight vigintillion. [35]

12. HIS SURVIVING WORKS • Surviving works • On plane equilibriums, Quadrature of the parabola, On the sphere and cylinder, On spirals, On conoids and spheroids, On floating bodies, Measurement of a circle, The Sandreckoner, On the method of mechanical problems. • Place inHistory • Generally regarded as the greatest mathematician and scientist of antiquity and one of the three greatest mathematicians of all time (together with Isaac Newton (English 1643-1727) and Carl Friedrich Gauss (German 1777-1855)).

13. HIS NAME AND HIS PICTURES Archimedes of Syracuse • Born: 287 BC in Syracuse, Sicily • Died: 212 BC in Syracuse, Sicily

14. LIFE OF MATHEMATICIAN RENE DESCARTES CHILDHOOD AND HIS MATURED LIFE • René Descartes (French IPA: [ʁəne de'kaʁt]Latin:Renatus Cartesius) (March 31, 1596 – February 11, 1650), also known as Renatus Cartesius (latinized form), was a highly influential French philosopher, mathematician, scientist, and writer. He has been dubbed the "Father of Modern Philosophy," and much of subsequent Western philosophy is a response to his writings, which continue to be studied closely. His influence in mathematics is also apparent, the Cartesian coordinate system that is used in plane geometry and algebra being named for him, and he was one of the key figures in the Scientific Revolution. • Biography • Descartes was born in La Haye en Touraine (now Descartes), Indre-et-Loire, France. When he was one year old, his mother Jeanne Brochard died of tuberculosis. His father Joachim was a judge in the High Court of Justice. At the age of eight, he entered the JesuitCollège Royal Henry-Le-Grand at La Flèche. After graduation, he studied at the University of Poitiers, earning a Baccalauréat and License in law in 1616, in accordance with his father's wishes that he should become a lawyer. • In 1622 he returned to France, and during the next few years spent time in Paris and other parts of Europe. • Despite these frequent moves he wrote all his major work during his 20 plus years in the Netherlands, where he managed to revolutionize mathematics and philosophy. • Descartes continued to publish works concerning both mathematics and philosophy for the rest of his life. In 1643, Cartesian philosophy was condemned at the University of Utrecht, and Descartes began his long correspondence with Princess Elizabeth of Bohemia. In 1647, he was awarded a pension by the King of France. Descartes was interviewed by Frans Burman at Egmond-Binnen in 1648. • René Descartes died on February 11, 1650 in Stockholm, Sweden, where he had been invited as a teacher for Queen Christina of Sweden. The cause of death was said to be pneumonia.

15. HIS ACHIEVEMENTS IN MATHEMATICS • Descartes' theory provided the basis for the calculus of Newton and Leibniz, by applying infinitesimal calculus to the tangent line problem, thus permitting the evolution of that branch of modern mathematics.Descartes created analytic geometry, and discovered an early form of the law of conservation of momentum. He outlined his views on the universe in his Principles of Philosophy • One of Descartes most enduring legacies was his development of Cartesian geometry which uses algebra to describe geometry. He also invented the notation which uses superscripts to indicate powers or exponents, for example the 2 used in x² to indicate squaring. • . By 1619, under the influence of the Dutch mathematician and scientist Beeckman, Descartes began his exceptionally fertile mathematical studies of natural phenomena. • The scientific and technical studies of these years resulted in the three texts on optics, meteorology and geometry, which were only published in 1637, and 'The World' which was published posthumously. Nevertheless, Descartes was establishing quite a reputation as a formidable mathematician. Descartes made a number of important contributions to mathematics and physics, among the most enduring of which was his foundation (with Galileo) of what is now known as analytic geometry. That is, broadly speaking, the use of geometrical analysis to solve complex algebraic problems, and vice versa. It is difficult to overestimate the importance for the history of mathematical physics of this bringing together of the sciences of geometry and algebra. • Descartes is considered a revolutionary figure, especially for his attempts to change the relationship between philosophy and theology, and integrate philosophy with the new forms of science. He is respected for his attempts to create a form of philosophical argument akin to science or mathematics.

16. HIS NAME AND HIS PICTURES Rene Descartes • Born : March 31,1596 in La Haye,France • Died : February 11,1650 in Stockholm,Sweden

17. LIFE OF MATHEMATICIAN THALES WORKS OF THE GREAT MATHEMATICIAN • Thales was a great astronomer and mathematician. • Thales of Miletus was the first known Greek philosopher, scientist and mathematician. Some consider him to be the teacher of of Pythagoras, though it may be only that he advised Pythagoras to travel to Egypt and Chaldea. • From Eudemus of Rhodes (fl ca. 320 B.C) we know that he studied in Egypt and brought these teachings to Greece. He is unanimously ascribed the introduction of mathematical and astronomical sciences into Greece. • As a mathematician, Thales is famous for his theorems, three of which are attributed to him by Proclus: circle bisected by diameter; angles at base of isosceles triangle are equal vertically opposed angles are equal. • He figured out a way to measure the height of one of the Egyptian pyramids. He waited until a time of day when his own shadow was the same height that he was, and then he measured the shadow of the pyramid. • He is also credited by tradition with having made the first proof of a geometric theorem. He is said to have demonstrated that an angle inscribed in a semi-circle is a right angle, which is known as the Theorem of Thales.

18. HIS ACHIEVEMENTS • None of his writing survives; this makes it is difficult to determine his philosophy and to be certain about his mathematical discoveries. • There is, of course, the story of his successful speculation in oil presses -- as testament to his practical business acumen. • It is reported that he predicted an eclipse of the Sun on May 28, 585 BC, startling all of Ionia. • He is credited with five theorems of elementary geometry. • None of his writing survives; this makes it is difficult to determine his philosophy and to be certain about his mathematical discoveries. • There is, of course, the story of his successful speculation in oil presses -- as testament to his practical business acumen. • It is reported that he predicted an eclipse of the Sun on May 28, 585 BC, startling all of Ionia. • He is credited with five theorems of elementary geometry. • Five basic propositions with proofs of plane geometry are attributed to Thales. • Proposition. A circle is bisected by any diameter. • Proposition. The base angles of an isosceles triangle are equal. • Proposition. The angles between two intersecting straight lines are equal. • Proposition. Two triangles are congruent if they have two angles and the included side equal. • Proposition. An angle in a semicircle is a right angle. • Thales bridged the worlds of myth and reason with his belief that to understand the world, one must know its nature ('physis', hence the modern 'physics'). He believed that all phenomena could be explained in natural terms, contrary to the popular belief at the time that supernatural forces determined almost everything. Thales professed it was "not what we know, but how we know it" (the scientific method). His contributions elevated measurements from practical to philosophical logic.

19. Thales of Miletus Born:630 B.C. in Miletus, Turkey Died:543 B.C. in Miletus, Turkey

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