Project Sa Advance Cal 1 to-3

7/31/2019 Project Sa Advance Cal 1 to-3

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Submitted by: Mary CherilynCabigao BSM 4D/3D

Submitted to: Ms.Yolanda Roberto


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Dame Mary Lucy Cartwright

December 17, 1900 - April 3, 1998

Mary Cartwright was born on December 17, 1900 in Aynho, Northamptonshire, England.

She graduated from the University of Oxford in 1923, having attained a First inmathematics in only the second year that women were allowed to take Final degrees atOxford. After teaching mathematics in the schools for four years, she returned to Oxford in 1928 for herD.Phil in mathematics under the supervision of G. H. Hardy and E. C. Titchmarsh, receiving the degree in1930. Her thesis was on "The Zeros of Integral Functions of Special Types." This was published in twoparts in the Quarterly Journal of Mathematics, Vol. 1 (1930) [Abstract] and Vol. 2 (1931) [Abstract].After finishing at Oxford, Cartwright obtained a Yarrow Research Fellowship at Girton College,Cambridge University, where she continued her work on the theory of functions. In 1935 she wasappointed a lecturer in mathematics at Cambridge. She held the position of University Lecturerfrom 1935 until 1959, and then Reader in Theory of Functions from 1959 until herretirement in 1968. During this time she was on the staff of Girton College, serving as Director ofStudies in Mathematics and then as Mistress of Girton College from 1949 to 1968.

During the 1940's Mary Cartwright worked with John Littlewood on the solutions of the Van der Polequation and discovered many of the phenomena that later became known as "chaos". In his review of IanStewart's book,Nature's Numbers, Dyson writes about this work:

"Cartwright had been working with Littlewood on the solutions of the [ Van der Pol] equation, whichdescribe the output of a nonlinear radio amplifier when the input is a pure sine-wave. The wholedevelopment of radio in World War Two depended on high power amplifiers, and it was a matter of lifeand death to have amplifiers that did what they were supposed to do. The soldiers were plagued withamplifiers that misbehaved, and blamed the manufacturers for their erratic behavior. Cartwright andLittlewood discovered that the manufacturers were not to blame. The equation itself was to blame. They

discovered that as you raise the gain of the amplifier, the solutions of the equation become more and moreirregular. At low power the solution has the same period as the input, but as the power increases you seesolutions with double the period, and finally you have solutions that are not periodic at all."

Cartwright had a distinguished career in analytic function theory and university administration, publishingover 100 papers on classical analysis, differential equations and related topological problems. In 1947Cartwright became the first woman mathematician to be elected as a Fellow of the Royal Society ofEngland. She was elected President of the London Mathematical Society in 1951, receivedthe Sylvester Medal of the Royal Society in 1964, the De Morgan Medal of the LondonMathematical Society in 1968, and in 1969 became Dame Mary Cartwright (the femaleequivalent of a knighthood).

After her retirement Cartwright held visiting professorships at universities in England, America, andPoland. She died in Cambridge on April 3, 1998.
http://www.agnesscott.edu/lriddle/women/abstracts/cartwright_abstract.htmhttp://www.agnesscott.edu/lriddle/women/abstracts/cartwright_abstract2.htmhttp://www.agnesscott.edu/lriddle/women/abstracts/cartwright_abstract2.htmhttp://www.agnesscott.edu/lriddle/women/abstracts/cartwright_abstract.htm


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Edith Hirsch Luchins

December 21, 1921 - November 18, 2002

Edith Hirsch was born in Brzeziny, Poland, in 1921, the oldest of four children. Her familyemigrated to New York City when she was six. Edith attended the New York City public schoolswhere her interest in mathematics developed in elementary school. While in high school she tookevery mathematics course the school offered. She was a member of the school's math club andhelped tutor other students. She went on to study mathematics at Brooklyn College where sheearned her B.A. degree in 1942.

While still in high school, Edith took a course in psychology taught by Abraham Luchins, then agraduate student in educational psychology at New York University. Thus began not only

her lifelong interest in cognitive psychology as it applied to mathematics education, but also herlifelong collaboration with her future husband whom she married several months after graduatingfrom college. After a year working in industry in support of the U.S. involvement in World WarII, Luchins entered the graduate program in mathematics at New York University while herhusband joined the army. She earned her M.S. degree in 1944 and became a doctoral student ofRichard Courant. At the same time she also began teaching at Brooklyn College as an instructorin math.

Luchins' graduate work in mathematics was interrupted, however, by the birth of her first twochildren in 1946 and 1948. She completed her coursework at NYU but never took thecomprehensive exams or wrote her thesis. She did manage to teach an evening math course at

Brooklyn College for awhile. In 1949 the family moved to Montreal, Canada, when AbrahamLuchins began teaching at McGill University. They remained in Montreal for five years. Duringthat time Edith worked with her husband in the area of psychological issues in mathematicseducation.

In 1954, Abraham Luchins obtained a position at the University of Oregon. While raisingfour children, Edith resumed her mathematics studies at Oregon, receiving her Ph.D. in1957 with a thesis entitled "On Some Properties of Certain Banach Algebras", written underthe supervision of Bertram Yood. Her fifth child was born a year later. Parts of her dissertationwere published in the articles "On strictly semi-simple Banach algebras" in the PacificJournal of Mathematics, Vol. 9, No. 2 (1959), 551-554 [Abstract], and "On radicals and

continuity of homomorphisms into Banach algebras", Pacific Journal of Mathematics, Vol.9, No. 3 (1959), 755-758 [Abstract]. Both papers were prepared while Luchins held the NewYork State Fellowship, 1957-58, of the American Association of University Women.

Edith Luchins taught mathematics at the Rensselaer Polytechnic Institute from 1962 until herretirement in 1992. She was the first woman to be appointed a full professor at Rensselaer,and among the first tenured women full professors of mathematics at a major engineeringschool. During 1991-1992 she was a distinguished visiting professor of mathematics at the
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United States Military Academy in West Point. Even after retirement she continued her workwith the Rensselaer mathematics department. In 1994 she was appointed an adjunct professor ofcognitive sciences at Renssalaer. A well-loved teacher and colleague, Luchins received theRensselaer Distinguished Teaching Award, the Darrin Counseling Award, the Martin LutherKing Jr. Award, and the Rensselaer Alumni Association Outstanding Faculty Award. She also

was awarded the Award for Distinguished Public Service at West Point. In 1998 she was madean honorary member of the international Society for Gestalt Theory and Its Applications. TheRenssalaer Campus News obituary described her professional career as follows: "Luchins'research focused on mathematics and psychology. She had worked on mathematical models oforder effects in information processing; on gender differences in cognitive processes and theirimplications for teaching and learning mathematics; and on the roles of heuristics and algorithmsin mathematical problem solving, with and without the use of computers. She was also interestedin the history of mathematics, and, in particular, the history of women in mathematics."

Luchins once directed an NSF study on why there are so few women in mathematics.Luchins and her Ph.D student, Mary Ann McLoughlin, studied and wrote about the

mathematical life ofOlga Taussky Todd. But she was also interested in many other areas ofmathematics, such as number theory, cryptography, and geometry. For example, she received anNSF grant and a Rensselaer Teaching Fellowship to integrate geometry and calculus throughcomputer graphics. Luchins wrote 12 books or monographs and over 70 articles, many with herhusband.

Edith Luchins was an active member of Congregation Beth Abraham Jacob in Albany, NewYork, where she did pioneering work in Jewish communal life. She was the first woman selectedto serve on the Board of Directors of the Orthodox Union.
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Indian Mathematician

RAMANUJAN

Dec 22 1887-april 26 1920

He was born on 22na of December 1887 in a small village of Tanjore district, Madras. Hefailed in English in Intermediate, so his formal studies were stopped but his self-study ofmathematics continued.

He sent a set of 120 theorems to Professor Hardy of Cambridge. As a result he invitedRamanujan to England.

Ramanujan showed that any big number can be written as sum of not more than four primenumbers.

He showed that how to divide the number into two or more squares or cubes.

when Mr Litlewood came to see Ramanujan in taxi number 1729, Ramanujan said that 1729 isthe smallest number which can be written in the form of sum of cubes of two numbers in twoways, i.e. 1729 = 93 + 103 = 13 + 123since then the number 1729 is called Ramanujans number.

In the third century B.C, Archimedes noted that the ratio of circumference of a circle to itsdiameter is constant. The ratio is now called pi ( ) (the 16th letter in the Greek alphabetseries)

The largest numbers the Greeks and the Romans used were 106 whereas Hindus used numbers asbig as 1053 with specific names as early as 5000 B.C. during the Vedic period.


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ArchimedesArchimedes was bornc. 287 BC in the seaport city ofSyracuse, Sicily, at that time a self-governingcolonyinMagna Graecia. The date of birth is based on a statement by theByzantine

GreekhistorianJohn Tzetzesthat Archimedes lived for 75 years.[7]

InThe Sand Reckoner,Archimedes gives his father's name as Phidias, anastronomerabout whom nothing isknown.Plutarchwrote in hisParallel Livesthat Archimedes was related to KingHiero II, theruler of Syracuse.[8]A biography of Archimedes was written by his friend Heracleides but thiswork has been lost, leaving the details of his life obscure.[9]It is unknown, for instance, whetherhe ever married or had children. During his youth, Archimedes may have studiedinAlexandria,Egypt, whereConon of SamosandEratosthenes of Cyrenewere contemporaries.He referred to Conon of Samos as his friend, while two of his works (The Method ofMechanical Theoremsand theCattle Problem) have introductions addressed to Eratosthenes.[a]

Archimedes diedc. 212 BC during theSecond Punic War, when Roman forces underGeneralMarcus Claudius Marcelluscaptured the city of Syracuse after a two-year-longsiege.According to the popular account given byPlutarch, Archimedes was contemplatingamathematical diagramwhen the city was captured. A Roman soldier commanded him tocome and meet General Marcellus but he declined, saying that he had to finish working on theproblem. The soldier was enraged by this, and killed Archimedes with his sword. Plutarch alsogives a lesser-known account of the death of Archimedes which suggests that he may have beenkilled while attempting to surrender to a Roman soldier. According to this story, Archimedes wascarrying mathematical instruments, and was killed because the soldier thought that they werevaluable items. General Marcellus was reportedly angered by the death of Archimedes, as heconsidered him a valuable scientific asset and had ordered that he not be harmed.[10]

The last words attributed to Archimedes are "Do not disturb my circles" (Greek: ), a reference to the circles in the mathematical drawing that he was supposedlystudying when disturbed by the Roman soldier. This quote is often given in Latinas "Noliturbare circulos meos," but there is no reliable evidence that Archimedes uttered these words andthey do not appear in the account given by Plutarch.[10]

The tomb of Archimedes carried a sculpture illustrating his favorite mathematical proof,consisting of asphereand acylinderof the same height and diameter. Archimedes had proventhat the volume and surface area of the sphere are two thirds that of the cylinder includingits bases. In 75 BC, 137 years after his death, the RomanoratorCicerowas serving

asquaestorinSicily. He had heard stories about the tomb of Archimedes, but none of the localswas able to give him the location. Eventually he found the tomb near the Agrigentine gate inSyracuse, in a neglected condition and overgrown with bushes. Cicero had the tomb cleaned up,and was able to see the carving and read some of the verses that had been added as aninscription.[11]A tomb discovered in a hotel courtyard in Syracuse in the early 1960s wasclaimed to be that of Archimedes, but its location today is unknown.[12]

The standard versions of the life of Archimedes were written long after his death by thehistorians of Ancient Rome. The account of the siege of Syracuse given byPolybiusin
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his Universal History was written around seventy years after Archimedes' death, and was usedsubsequently as a source by Plutarch andLivy. It sheds little light on Archimedes as a person, andfocuses on the war machines that he is said to have built in order to defend the city.[13]

While he is often regarded as a designer of mechanical devices, Archimedes also made contributions to

the field of mathematics.Plutarchwrote: "He placed his whole affection and ambition in those purer

speculations where there can be no reference to the vulgar needs of life."[43]

Archimedes used themethod of exhaustionto approximate the value ofpi.

Archimedes was able to useinfinitesimalsin a way that is similar to modernintegralcalculus. Through proof by contradiction (reductio ad absurdum), he could give answers toproblems to an arbitrary degree of accuracy, while specifying the limits within which the answerlay. This technique is known as themethod of exhaustion, and he employed it to approximate thevalue of . He did this by drawing a largerpolygonoutside acircleand a smaller polygon insidethe circle. As the number of sides of the polygon increases, it becomes a more accurateapproximation of a circle. When the polygons had 96 sides each, he calculated the lengths oftheir sides and showed that the value of lay between 3

17 (approximately 3.1429) and

31071 (approximately 3.1408), consistent with its actual value of approximately 3.1416. He alsoproved that theareaof a circle was equal to multiplied by thesquareof theradiusof the circle(r

2). InOn the Sphere and Cylinder, Archimedes postulates that any magnitude when added toitself enough times will exceed any given magnitude. This is theArchimedean propertyof realnumbers.[44]

InMeasurement of a Circle, Archimedes gives the value of thesquare rootof 3 as lying

between 265153 (approximately 1.7320261) and 1351780 (approximately 1.7320512). The actualvalue is approximately 1.7320508, making this a very accurate estimate. He introduced thisresult without offering any explanation of the method used to obtain it. This aspect of the workof Archimedes causedJohn Wallisto remark that he was: "as it were of set purpose to havecovered up the traces of his investigation as if he had grudged posterity the secret of his methodof inquiry while he wished to extort from them assent to his results."[45]

As proven by Archimedes, the area of theparabolicsegment in the upper figure is equal to 4/3that of the inscribed triangle in the lower figure.

InThe Quadrature of the Parabola, Archimedes proved that the area enclosed by aparabolaanda straight line is 43 times the area of a corresponding inscribedtriangleas shown in the figure atright. He expressed the solution to the problem as aninfinitegeometric serieswith thecommonratio14:
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ntegralhttp://en.wikipedia.org/wiki/Infinitesimalhttp://en.wikipedia.org/wiki/Pihttp://en.wikipedia.org/wiki/Method_of_exhaustionhttp://en.wikipedia.org/wiki/Archimedes#cite_note-42http://en.wikipedia.org/wiki/Plutarchhttp://en.wikipedia.org/wiki/Archimedes#cite_note-12http://en.wikipedia.org/wiki/Livy


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If the first term in this series is the area of the triangle, then the second is the sum of the areasof two triangles whose bases are the two smallersecant lines, and so on. This proof uses a

variation of the series1/4 + 1/16 + 1/64 + 1/256 + which sums to

1

3.InThe Sand Reckoner, Archimedes set out to calculate the number of grains of sandthat the universe could contain. In doing so, he challenged the notion that the numberof grains of sand was too large to be counted. He wrote: "There are some, King Gelo(Gelo II, son ofHiero II), who think that the number of the sand is infinite inmultitude; and I mean by the sand not only that which exists about Syracuse and therest of Sicily but also that which is found in every region whether inhabited oruninhabited." To solve the problem, Archimedes devised a system of counting based onthemyriad. The word is from the Greekmurias, for the number 10,000. He proposeda number system using powers of a myriad of myriads (100 million) and concluded that thenumber of grains of sand required to fill the universe would be 8vigintillion, or 81063.[46]
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EuclidBorn: Alexandria, Egypt 365 B.C.. Died: Athens, Greece 300 B.C

Euclid or more famously known as the Father of Geometry is a Greek Mathematician. Itis deduced that he was born ca.365BC and died ca.300BC . However there are no writtenrecords or facts relating to Euclids birth or death and as there is no perfect or official Euclidbiography, through the meager references found in the pages of history, one can only concludethat this famous persona flourished during 300BC. Euclid in Greek means good glory and is

aptly named so, as his written work on Elements, is one of the most important and influentialwork in the field of Mathematics which gave him immense fame and glory. Once Euclids

elements were out the whole structure and pattern of mathematics, especially geometry, waschanged and within seconds, this genius was recognized and became the Father of Geometry.Many famous artists have painted his portrait, but sadly no one actually knows how he looks and

all the kudos go to the artists imagination.From the few historical references of 5th and 4thcentury, found in the works of Porclus and Papus of Alexandria, we can conclude that he wasactive during the reign of Ptolemy I. In fact, it is said that when King Ptolemy asked Euclid toshow an easier way to learn geometry rather than using his elements, Euclid humbly replied thattheres no royal road to geometry. Presumably a prodigy of the famous philosopher and scholarPlato and the mentor of Archemedes, Euclid is a gem of a mathematician and his works areincluded in students texts to teach mathematics and geometry.

Euclid Mathematician - Founder of Elements and Logics

Proclus the famous Greek philosopher in his work mentions Euclid Mathematician and his

treatise in mathematics especially geometry. Euclid as a mathematician is best known for hiseverlasting work the Stoicheia or Elements. Based on previous works of Eudoxus, Pythagoras,Thales, Plato etc, Euclid devised the Elements, the best ever book on Geometry which till the20th century was used as the standard textbook for studying this fascinating subject. TheElements start with definitions, Euclids postulates and common opinions or assumptions,followed by rigorous geometric proof. Euclid brought in the method of exhaustion and reductionad absurdum and revolutionized the technique of geometric proof.Apart from his famous work ingeometry, Euclid is also widely known for his second theorem on prime numbers. In a clear andconcise manner, exemplifying his intellectual superiority, Euclid proved that the number ofprime numbers is infinite and till date mathematicians are trying to prove him wrong, but in vain.Euclid is also credited for his algorithm on calculating the greatest common divisor of 2 numbersand the proof of Pythagoras Theorem. The various works written on division, properties,perspective and mathematical astronomy give a glimps of this genius and proves that EuclidMathematician is one of the greatest ever found in the history.


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Father Geometry - The Alpha of Precise Geometry

The term Father Geometrywas coined for Euclid after reviewing his work Elements. Therewere many mathematicians and geometricians before and after Euclid but none of them had theprecision, logic and ability to derive perfect conclusions through systematic proofs like him.

Euclids Element was unchallengeable and indisputable and the only standard text available forstudy of geometry till recent years. Although a new geometric principle and proof is available forstudies, Euclids geometry is still under keen review and unquestionably the base for any andevery modern theorem.Divided into 13 books, the Elements exhaustively deal with PlaneGeometry, Ratios and Proportions and Spatial Geometry. With four major axioms and n numberof geometric definitions to his credit, Euclid is aptly named Father Geometry as his works havepaved the way for the modern science and logic, indirectly helping in the study of mathematics,astronomy, engineering etc.


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Gauss, Carl Friedrich17771855

Gauss, Carl Friedrich (krlfr'drikhgous) [key], born Johann Friederich Carl Gauss,17771855, German mathematician, physicist, and astronomer. Gauss was educated at theCaroline College, Brunswick, and the Univ. of Gttingen, his education and early researchbeing financed by the Duke of Brunswick. Following the death of the duke in 1806, Gaussbecame director (1807) of the astronomical observatory at Gttingen, a post he held until hisdeath. Considered the greatest mathematician of his time and as the equal of Archimedes andNewton, Gauss showed his genius early and made many of his important discoveries before hewas twenty. His greatest work was done in the area of higher arithmetic and numbertheory; hisDisquisitionesArithmeticae (completed in 1798 but not published until 1801) isone of the masterpieces of mathematical literature.

Gauss was extremely careful and rigorous in all his work, insisting on a complete proof of anyresult before he would publish it. As a consequence, he made many discoveries that were notcredited to him and had to be remade by others later; for example, he anticipated Bolyai andLobachevsky in non-Euclidean geometry, Jacobi in the double periodicity of elliptic functions,Cauchy in the theory of functions of a complex variable, and Hamilton in quaternions. However,his published works were enough to establish his reputation as one of the greatestmathematicians of all time. Gauss early discovered the law of quadratic reciprocity and,independently of Legendre, the method of least squares. He showed that a regular polygon ofnsides can be constructed using only compass and straight edge only ifn is of the form2p(2q+1)(2r+1) , where 2q + 1, 2r+ 1, are prime numbers.

In 1801, following the discovery of the asteroid Ceres by Piazzi, Gauss calculated its orbit on thebasis of very few accurate observations, and it was rediscovered the following year in the preciselocation he had predicted for it. He tested his method again successfully on the orbits of otherasteroids discovered over the next few years and finally presented in hisTheoriamotuscorporumcelestium (1809) a complete treatment of the calculation of the orbitsof planets and comets from observational data. From 1821, Gauss was engaged by thegovernments of Hanover and Denmark in connection with geodetic survey work. This led to hisextensive investigations in the theory of space curves and surfaces and his importantcontributions todifferential geometry as well as to such practical results as his invention of theheliotrope, a device used to measure distances by means of reflected sunlight.

Gauss was also interested in electric and magnetic phenomena and after about 1830 was involvedin research in collaboration with Wilhelm Weber. In 1833 he invented the electric telegraph. Healso made studies of terrestrial magnetism and electromagnetic theory. During the last years ofhis life Gauss was concerned with topics now falling under the general heading of topology,which had not yet been developed at that time, and he correctly predicted that this subject wouldbecome of great importance in mathematics.
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John von Neumann(1903-1957)

John von Neumann, who was born in Budapest, Hungary, in 1903, was primarilyamathematician, and wrote numerous papers on both pure and applied math. He also madeimportant contributions to a number of other fields of inquiry, including quantum physics,economics and computer science.

Von Neumann studied mathematics, physics and chemistry at German and Swissuniversities for several years, finally receiving a Ph.D. in mathematics from the Universityof Budapest in 1926. He taught at Berlin and Hamburg from 1927to 1930 and thenemigrated to the United States to join the faculty at Princeton University. Three years later hetook a position at the Institute of Advanced Studies at Princeton.

Until the outbreak of World War II, Von Neumann mostly did work in pure math,makingimportant contributions to the fields of mathematical logic, set theory and operator theory.However, his work in operator theory had powerful applications in theoretical physics, andhe published a book on quantum physics,The Mathematical Foundations of QuantumMechanics in 1932. This work remainsa standard text on the subject. During World War II,when the U.S. government called on a great many scientists to help out with the development ofnew technologies demanded by the war effort, Von Neumann took on numerous positions as aconsultant. He was engaged in many different research projects and proved his ability as anadministrator as well as a brilliant scientist. Among other consulting positions, he was involvedwith the development of the atomicbomb at the Los Alamos Scientific Laboratory. At about thesame time, he caused a revolution in the social sciences with his work on game theory, Theoryof

Games and Economic Behavior, written with the economist Oskar Morgensternand published in1944. In those years, he also became a principal player in the development of high-speed digitalcomputers and the stored programs used in virtually all contemporary computer applications.

While at Los Alamos, Von Neumann became impressed with the need to develop computationalequipment technology that could carry out the enormously complexmathematical calculationswhich the scientists then had to carry out by hand.In 1944, Von Neumann became involved withefforts to develop computers, mostnotably ENIAC (Electronic Numerical Integrator andCalculator), which was then the most powerful device under construction. ENIAC could beprogrammed todo different tasks, but this required a partial rewiring of the machine. Oneof thescientists working on ENIAC, J. Presper Eckert, came up with the ideaof a stored program,which would make it possible to load a computer programinto computer memory from disk. Thecomputer could then run the program without being manually reprogrammed. The idea was notused in the design of ENIAC,but a follow-up project, called EDVAC, which Von Neumann wasclosely associated with, did incorporate the stored program. A paper Von Neumann wrote in1944, entitled "First Draft of a Report on EDVAC," explained the revolutionaryideas that were togovern the development of computers for the next two decades. Von Neumann proposed aseparation of storage, arithmetic and control functions; random-access memory (RAM); storedprograms; arithmetic modification of instructions; conditional branching; a choice between


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binary number and decimal number representation; and a choice between serial and paralleloperation. Basically, he introduced new procedures in their logical organization, the"codes" bywhich a fixed system of wiring could solve a great variety of problems. Particularly the idea of astored program and the solutions for realizing the equipment that could deal with storedprograms were revolutionary, promising great gains in speed and productivity.

In summary, Von Neumann rethought the basic design of the computer into the separatecomponents of arithmetic function, central control (now known as thecentral processing unit[CPU]), memory (the hard drive) and the input and output devices. Under Von Neumann'ssupervision, a computer with these capabilities was developed at the Institute of AdvancedStudies from 1946 to 1951. Although the machine quickly became a dinosaur, it was the first trueforerunnerof the contemporary high-speed digital computer.


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Napier, John15504 April 1617

John Napier of Merchiston (15504 April 1617)also signed as Neper, NepairnamedMarvellous Merchiston, was aScottish mathematician, physicist, astronomer & astrologer,and also the 8th Laird ofMerchistoun. He was the son of Sir Archibald Napier of Merchiston.John Napier is most renowned as the discoverer of thelogarithm. Napier is the inventor of theso-called "Napier's bones". Napier also made common the use of thedecimal point inarithmetic and mathematics. Napier's birthplace, the Merchiston Tower in Edinburgh,Scotland, is now part of the facilities ofEdinburgh Napier University. After his death fromthe effects ofgout, Napier's remains were buried in St Cuthbert's Church, Edinburgh.

Napier's father was Sir Archibald Napier of Merchiston Castle, and his mother was JanetBothwell, daughter of a member of the Estates of Parliament, and a sister of the clergyman Adam

Bothwell, who became the Bishop of Orkney. Archibald Napier was 16 years old when JohnNapier was born.

As was the common practice for members of the nobility at that time, John Napier did not enterschools until he was 13. He did not stay in school very long, however. It is believed that hedropped out of school in Scotland and perhaps travelled in mainland Europe to better continuehis studies. Little is known about those years, where, when, or with whom he might have studied,although his uncle Adam Bothwell wrote a letter to John's father on 5 December 1560, saying "Ipray you, sir, to send John to the schools either to France orFlanders, for he can learn no good

at home", and it is believed that this advice was followed.

In 1571 Napier, aged 21, returned to Scotland. In 1572 he married Elizabeth Stirling, daughter ofJames Stirling, the 4th Laird ofKeir and ofCadder. Napier bought a castle at Gartness in 1574.They had two children before Elizabeth died in 1579. John Napier later married AgnesChisholm, with whom he had ten more children. Upon the death of his father in 1608, Napier andhis family moved into Merchiston Castle in Edinburgh, where he resided the remainder of hislife.

His work,Mirifici Logarithmorum Canonis Descriptio (1614) contained fifty-seven pages ofexplanatory matter and ninety pages of tables of numbers related to natural logarithms.The book also has an excellent discussion of theorems in spherical trigonometry, usually knownas Napier's Rules of Circular Parts. Modern English translations of both Napier's books onlogarithms, and their description can be found on the web, as well as a discussion of Napier'sBones (see below) and Promptuary (another early calculating device).[1]His invention oflogarithms was quickly taken up at Gresham College, and prominent English mathematicianHenry Briggs visited Napier in 1615. Among the matters they discussed was a re-scaling ofNapier's logarithms, in which the presence of the mathematical constant e(more accurately, theinteger part ofe times a large power of 10) was a practical difficulty. Napier delegated to Briggsthe computation of a revised table. The computational advance available via logarithms, theconverse of powered numbers or exponential notation, was such that it made calculations by
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hand much quicker.[2]The way was opened to later scientific advances, in astronomy, dynamics,physics; and also in astrology.

Napier made further contributions. He improved Simon Stevin's decimal notation. Arab latticemultiplication, used by Fibonacci, was made more convenient by his introduction ofNapier's

bones, a multiplication tool using a set of numbered rods. He may have worked largely inisolation, but he had contact with Tycho Brahe who corresponded with his friend John Craig.Craig certainly announced the discovery of logarithms to Brahe in the 1590s (the name itselfcame later); there is a story from Anthony Wood, perhaps not well substantiated, that Napierhad a hint from Craig that Longomontanus, a follower of Brahe, was working in a similardirection.
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Maria GaetanaAgnesiMay 16, 1718 and died on January 9, 1799

Was one of the most important Western female mathematicians and philosophers inhistory.

Early Life

Agnesi was the eldest child of a wealthy silk merchant who provided her with the best tutorsavailable. She was an extremely precocious child who mastered Latin, Greek, Hebrew, andseveral modern languages at an early age, and her father liked to host gatherings where she coulddisplay her knowledge.Propositionesphilosophicae(Propositions of Philosophy), a series ofessays on natural philosophy and history based on her discussions before such gatherings, waspublished in 1738.

Mathematics

Agnesi's best-known work,Instituzionianaliticheadusodellagioventitaliana (1748;Analytical Institutions for the Use of Italian Youth), in two huge volumes, provided aremarkably comprehensive and systematic treatment of algebra and analysis, including suchrelatively new developments as integral and differential calculus. In this text is found adiscussion of the Agnesi curve, a cubic curve known in Italian as versiera, which was confusedwith versicra(witch) and translated into English as the Witch of Agnesi. The FrenchAcademy of Sciences, in its review of theInstituzioni, stated that: We regard it as the mostcomplete and best made treatise.Pope Benedict XIV was similarly impressed and appointed

Agnesi professor of mathematics at the University of Bologna in 1750.

Later Life

However, Agnesi had turned increasingly to religion and never journeyed to Bologna. After thedeath of her father in 1752, she devoted herself almost exclusively to charitable work andreligious studies. She established various hospices and died in one of the poorhouses that she hadonce directed.
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Jacques Alexandre Csar Charlesborn Nov. 12, 1746, Beaugency, Fr.died April 7, 1823, Paris

Acques Alexandre Csar Charles was a French mathematician, physicist, and inventor who,along with Nicolas Robert, was the first to launch a hydrogen balloon. From his flights -some of which rose more than a mile, he developed Charles' law regarding thermal gasexpansion. His experiments with electricity led to several inventions, including a hydrometer,and an improved Fahrenheit aerometer.

Profile

French mathematician, physicist, and inventor who, with Nicolas Robert, was the first toascend in a hydrogen balloon (1783). About 1787 he developed Charles's law (q.v.) concerningthe thermal expansion of gases.

From clerking in the finance ministry Charles turned to science and experimented withelectricity. He developed several inventions, including a hydrometer and reflecting goniometer,and improved the Gravesand heliostat and Fahrenheit's aerometer. With the Robert brothers,Nicolas and Anne-Jean, he built one of the first hydrogen balloons (1783). In several flights herose more than a mile in altitude. He was elected (1795) to the Acadmie des Sciences andsubsequently became a professor of physics. His published papers deal mainly with mathematics.


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Robert Metcalfe(born April 7, 1946 inBrooklyn,New York)

Robert Metcalfe,engineer, technology executive and venture capitalist, invented the Ethernet.He attended MIT and Harvard University, studying mathematics and computer science,before joining Xerox's Palo Alto Research center. It was there that he was inspired to invent theEthernet in 1973, which allowed computers to send packets of information and avoid collisionswith incoming packets.

Early Life

Born in Brooklyn, Metcalfe grew up on Long Island, New York, the son of an engineeringtechnician. As an electrical engineering student at the Massachusetts Institute of Technology

(M.I.T.), Metcalfe showed little inclination to sleep. He paid for college by working nights as acomputer programmer, from midnight to 8:00 a.m. Then he'd hit the tennis courts for severalhours a day, as captain of the tennis team. After graduating from M.I.T in 1967, he enrolledas a graduate student in applied mathematics. However, this solid middle-class student foundthe elitist attitude at the school grating, and he spent his free hours working in a computer lab atM.I.T. There, he was assigned the task of building an interface allowing early Internet servers totalk to each other.
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John VennJohn VennFRS(4 August 18344 April 1923), was aBritishlogicianandphilosopher.

He is famous for introducing theVenn diagram, which is used in many fields, includingset

theory,probability,logic,statistics, andcomputer science.In 1964, Metcalfe graduated from Bay Shore Public High School. He graduated fromMITin

1969 with twoB.S.degrees, one inElectrical Engineeringand the other in Industrial Managementfrom theMIT Sloan School of Management. He then went toHarvardfor graduate school, earninghisM.S.in 1970

While pursuing a doctorate in computer science, Metcalfe took a job with MIT'sProject MACafter

Harvard refused to let him be responsible for connecting the school to the brand-newARPAnet. At

MIT'sProject MAC, Metcalfe was responsible for building some of the hardware that would link MIT's

minicomputers with the ARPAnet. Metcalfe was so enamored with ARPAnet, he made it the topic of his

doctoral dissertation. However, Harvard flunked him. His inspiration for a new dissertation came while

working atXerox PARCwhere he read a paper about theALOHA networkat theUniversity of Hawaii.

He identified and fixed some of the bugs in the AlohaNet model and made his analysis part of a revised

thesis, which finally earned him his HarvardPhDin 1973.[3]

Metcalfe was working atXerox PARCin 1973 when he andDavid BoggsinventedEthernet, a

standard for connectingcomputersover short distances. Metcalfe identifies the day Ethernet was

born as May 22, 1973, the day he circulated a memo titled "Alto Ethernet" which contained a rough

schematic of how itwould work. "That is the first time Ethernet appears as a word, as does the idea of

using coax as ether, where the participating stations, like in AlohaNet or ARPAnet, would inject their

packets of data, they'd travel around at megabits per second, there would be collisions, and

retransmissions, and back-off," Metcalfe explained. Boggs identifies another date as the birth of Ethernet:

November 11, 1973, the first day the system actually functioned.[4]

In 1979, Metcalfe departed PARC and founded3Com, a manufacturer ofcomputer

networkingequipment. In 1980 he received theAssociation for Computing MachineryGrace

Murray Hopper Awardfor his contributions to the development of local networks, specificallyEthernet. In

1990 Metcalfe lost a boardroom skirmish at3Comin the contest to succeed Bill Krause asCEO. The

board of directors chose Eric Benhamou to run the networking company Metcalfe had founded in hisPalo

Altoapartment in 1979. Metcalfe left3Comand began a 10 year stint as a publisher and pundit, writing

anInternetcolumn forInfoWorld. He became aventure capitalistin 2001 and is now a General Partner

atPolaris Venture Partners. He is a director ofPop!Tech, an executive technology conference he

cofounded in 1997. He has recently been working with Polaris-funded startup Ember to work on a new

type of energy grid, Enernet.In November 2010 Metcalfe was selected to lead innovation initiatives at The University of Texas at

Austin Cockrell School of Engineering. He began his appointment in January 2011.[5]

Metcalfe was awarded theIEEE Medal of Honorin 1996 for "exemplary and sustained leadership

in the development, standardization, and commercialization ofEthernet."[6]
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Fourier, (Jean Baptiste) Joseph, Baron(1768-1830)

Background:Fourier, (Jean Baptiste) Joseph, Baron (Pronounced: Fooryay) (1768-1830) was born inAuxerre, France. He's considered a French mathematician and physicist. He was the son ofa tailor and born into a prominent family. Although initially trained for the priesthood, heturned to mathematics and became a teacher by the age of 16. Fourier was professor between1795 and 1798) at the colepolytechnique in Paris. He accompanied Napoleon I to Egypt and in1808 he was made a baron. He died in Paris in 1830 - Fourier never married.

Contributions:

Fourier had a verydistinguised career and is famous for showing how the conduction ofheat in solid bodies could be analyzed in terms of infinite mathematical series which iscalled the 'Fourier Series'. Fourier was the first correct theory on heat diffusion. His scientificwritings are contained in two volumes. Published in 1822, the Fourier Series shows how amathematical series of sine and cosine terms can be used to analyze heat conduction in solidbodies.

Fourier came upon his idea in connection with the problem of the flow of heat in solid bodies,including the earth. The formula x/2 = sin x - (sin 2x)/2 + (sin 3x)/3 + was published byLeonhard Euler (1707-1783) before Fouriers work began, so you might like to ponder thequestion why Euler did not receive the credit for Fouriers series. The Fourier Transform and Its

Application, by Ronald N.


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Pierre de Fermat French Number Theorist

August of 1601 and died in 1665.

Pierre de Fermat (pronounced Fair-mah) was born in Beaumont-de-Lomagne, Francein August of 1601 and died in 1665. He is considered to be one of the greatest

mathematicians of the seventeenth century. Fermat's father was a leather merchant

and his mother's family was in the legal profession. Fermat attended a Franciscanmonastery before moving on to obtain a Bachelor's Degree in civil law from the University ofOrleans in 1631. He married, had five children and practiced law. For the most part, Math

was a hobby for Fermat.Fermat was a busy lawyer and did not let his love of mathcompletely take over his time. It's been said that Fermat never wanted anything to bepublished as he considered math to be his hobby. The only one thing he did publish - he did

so anonymously. He sent many of his papers by mail to some of the best mathematicians inFrance. It was his link with Marin Mersenne that gave Fermat his international reputation.Fermat loved to dabble in math and rarely provide his proofs (evidence or procedures for

reaching conclusions), he would state theorems but neglected the proofs! In fact, his most

Famous work 'Fermat's Last Theorem' remained without a proof until 1993 when Andrew J.Wiles provided the first proof. During Fermat's lifetime, he received very little recognition asa mathematician, if not for the fact that others saved his papers and letters, he may not be

the legacy that he is seen as today.

Contributions:

Fermat is considered to be one ofthe 'fathers' of analytic geometry. (Along with

Rene' Descartes.) Fermat along with Blaise Pascal is also considered to be one of the founders of

probabilitytheory. Fermat also made contributions in the field of optics and provided a law on light

travel and made wrote a few papers about calculus well before Isaac Newton andGottfried Leibniz were actually born. The Most Famous Question In Math History for 350 Years!

Fermat's most important work was done in the development of modern numbertheory which was one of his favorite areas in math. He is best remembered forhis number theory, in particular for Fermat's Last Theorem. This theoremstates that: xn + yn = zn has no non-zero integer solutions for x, y and z

when n is greater than 2.
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Andrew Wilesborn 11 April 1953

Sir Andrew John Wiles,KBE, FRS(born 11 April 1953)[1]

is a British mathematician and aRoyal Society Research Professor at Oxford University, specializing in number theory. He ismost famous for proving Fermat's Last Theorem.

Wiles is the son ofMaurice Frank Wiles (19232005), the Regius Professor of Divinity at theUniversity of Oxford[2]and Patricia Wiles (ne Mowll). His father worked as the Chaplain atRidley Hall, Cambridge, for the years 195255. Wiles was born in Cambridge, England, in1953, and he attended King's College School, Cambridge, andThe Leys School, Cambridge.

Wiles discovered Fermat's Last Theorem on his way home from school when he was 10 yearsold. He stopped by his local library where he found a book about the theorem.[3]Puzzled by the

fact that the statement of the theorem was so easy that he, a ten-year old, could understand it, hedecided to be the first person to prove it. However, he soon realized that his knowledge ofmathematics was too small, so he abandoned his childhood dream, until 1986, when he heardthat Ribet had proved Serre's-conjecture and therefore established a link betweenFermat's Last Theorem and the Taniyama-Shimura conjecture.

Wiles earned his bachelor's degree in mathematics in 1974 after his study at MertonCollege, Oxford, and a Ph.D. in 1980, after his research at Clare College, Cambridge.

After a stay at the Institute for Advanced Study in New Jersey in 1981, Wiles became aprofessor at Princeton University. In 198586, Wiles was a Guggenheim Fellow at the

Institut des Hautes tudes Scientifiques near Paris and at the cole Normale Suprieure.From 1988 to 1990, Wiles was a Royal Society Research Professor at Oxford University, andthen he returned to Princeton.

In October 2009 it was announced that Wiles would again become a Royal Society ResearchProfessor at Oxford in 2011.[4]

Wiles's graduate research was guided by John Coates beginning in the summer of 1975.Together these colleagues worked on the arithmetic ofelliptic curves with complexmultiplication by the methods ofIwasawa theory. He further worked withBarry Mazuronthe main conjecture of Iwasawa theory over the rational numbers, and soon afterward, he

generalized this result to totally real fields.
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e-WhosWho-0http://en.wikipedia.org/wiki/Fellow_of_the_Royal_Societyhttp://en.wikipedia.org/wiki/Order_of_the_British_Empire


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23

G. F. Bernhard Riemann(1826-1866)

Bernhard Riemann, born to a poor family in 1826, would rise to become one of the worlds

prominent mathematicians in the 19th Century. The list of contributions to geometry arelarge, and he has a wide range of theorems bearing his name. To name just a few:Riemannian Geometry, Riemannian Surfaces and the Riemann Integral. However, he is perhapsmost famous (or infamous) for his legendarily difficult Riemann Hypothesis; an extremelycomplex problem on the matter of the distributions of prime numbers. Largely ignored for thefirst 50 years following its appearance, due to few other mathematicians actually understandinghis work at the time, it has quickly risen to become one of the greatest open questions in modernscience, baffling and confounding even the greatest mathematicians. Although progress has beenmade, its has been incredibly slow. However, a prize of $1 million has been offered from theClay Maths Institute for a proof, and one would almost undoubtedly receive a Fields medalif under 40 (The Nobel prize of mathematics). The fallout from such a proof is hypothesized to

be large: Major encryption systems are thought to be breakable with such a proof, and all thatrely on them would collapse. As well as this, a proof of the hypothesi s is expected to use newmathematics. It would seem that, even in death, Riemanns work may still pave the way for newcontributions to the field, just as he did in life.


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Carl Friedrich Gauss(1777-1855)

Johann Carl Friedrich Gauss ( /as/; German:Gau listen (helpinfo), Latin:CarolusFridericus Gauss) (30 April 177723 February 1855) was a German mathematician andphysical scientist who contributed significantly to many fields, including number theory,statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy andoptics.

Sometimes referred to as thePrinceps mathematicorum[1](Latin, "the Prince ofMathematicians" or "the foremost of mathematicians") and "greatest mathematician sinceantiquity", Gauss had a remarkable influence in many fields of mathematics and science and isranked as one of history's most influential mathematicians.[2]He referred to mathematics as "thequeen of sciences".[3]

Carl Friedrich Gauss was born on 30 April 1777 in Braunschweig, in the duchy ofBraunschweig-Wolfenbttel, now part ofLower Saxony, Germany, as the son of poorworking-class parents.[4]Indeed, his mother was illiterate and never recorded the date of hisbirth, remembering only that he had been born on a Wednesday, eight days before the Feast ofthe Ascension, which itself occurs 40 days after Easter. Gauss would later solve this puzzle forhis birthdate in the context offinding the date of Easter,deriving methods to compute the date

in both past and future years.[5]

He was christened and confirmed in a church near the school heattended as a child.[6]

Gauss was a child prodigy. There are many anecdotes pertaining to his precocity while a toddler,and he made his first ground-breaking mathematical discoveries while still a teenager. HecompletedDisquisitiones Arithmeticae, his magnum opus, in 1798 at the age of 21, though itwas not published until 1801. This work was fundamental in consolidating number theory as adiscipline and has shaped the field to the present day.

Gauss's intellectual abilities attracted the attention of the Duke of Braunschweig,[2]whosent him to the Collegium Carolinum (now Technische Universitt Braunschweig), whichhe attended from 1792 to 1795, and to the University of Gttingen from 1795 to 1798. While inuniversity, Gauss independently rediscovered several important theorems;[citation needed] hisbreakthrough occurred in 1796 when he was able to show that any regular polygon with anumber of sides which is a Fermat prime (and, consequently, those polygons with any number of
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