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         Diophantus Of Alexandria:     more books (16)
  1. Diophantus of Alexandria; a study in the history of Greek algebra by Thomas Little Heath, Leonhard Euler, 2010-08-25
  2. Die Arithmetik Und Die Schrift Über Polygonalzahlen Des Diophantus Von Alexandria (German Edition) by Gustav Wertheim, Diophantus, 2010-04-08
  3. Die Arithmetik Und Die Schrift Über Polygonalzahlen Des Diophantus Von Alexandria (German Edition) by Gustav Wertheim, Diophantus, 2010-03-23
  4. Diophantus of Alexandria; A Study in the History of Greek Algebra (Classic Reprint) by Sir Thomas L. Heath, 2010-03-16
  5. Diophantus of Alexandria: A Study in the History of Greek Algebra by Sir Thomas l. Heath, 1964
  6. Diophantus of Alexandria: Study in the History of Greek Algebra by Sir Thomas Heath, 1965-04
  7. Diophantus Of Alexandria: A Study In The History Of Greek Algebra (1910) by Thomas L. Heath, 2010-09-10
  8. Diophantus of Alexandria: An entry from Gale's <i>Science and Its Times</i> by Judson Knight, 2001
  9. Die Arithmetik und die Schrift über Polygonalzahlen des Diophantus von Alexandria (German Edition) by Diophantus, 1890-01-01
  10. Diophantus of Alexandria: a Study in the History of Greek Algebra by T. L. (Thomas Little), Sir Heath, 1910
  11. Diophantus of Alexandria a Study in The
  12. Diophantus of Alexandria by Sir Thomas L. Heath, 1910
  13. L'arithmetiqve De Simon Stevin De Brvges (French Edition) by Stevin Simon 1548-1620, Diophantus of Alexandria, et all 2010-09-28
  14. Diophanti Alexandrini Opera omnia (Latin Edition) by of Alexandria Diophantus, 1893-01-01

61. The Greeks
we know they only produced one algebraist, diophantus of alexandria (c 250 AD). Diophantus used an abridged notation for frequently occuring operations,
http://www.maths.wlv.ac.uk/mm2217/g.htm
The Development of Algebra
The Greeks
Whereas many Greeks made decisive advances in geometry, as far as we know they only produced one algebraist, Diophantus of Alexandria (c 250 A.D.). Diophantus used an abridged notation for frequently occuring operations, and a special symbol for the unknown. Thus for the unknown he wrote , if it occured once. For our 3x, he wrote , where is the plural of the unknown and represents the coefficient 3. Addition was denoted by simply placing the summands next to each other, and subtraction was indicated by the symbol . Instead of our sign for equality, he wrote . Also terms which were not tied to the unknown were preceded by the symbol . As an example, for our: x x he would write: Besides being the first to use symbols systematically in algebra, Diophantus was also the first to give general rules for the solution of an equation. An example, in our notation,
is as follows: 8x - 11 - 2x + 5 = x - 4 + 3x + 10 rearranged in the form 8 x + 5 + 4 = x + 3x + 10 + 11 + 2x or
Then Diophantus gives the following rule: "... it will be necessary to subtract like from like on both sides, until one term is found equal to one term."

62. INBOX: From Science News Online
Nearly 2000 years ago, for instance, diophantus of alexandria observed in his book Arithmetica that 65 can be written in two different ways as the sum of
http://www.math.wisc.edu/~ono/squares.html
Science News , June 16, 2001; Vol. 159, No. 24 Surprisingly Square http://xxx.lanl.gov/abs/math.NT/0008068 . Ono, K. Preprint. Representations of integers as sums of squares. Zagier, D. 2000. A proof of the Kac-Wakimoto affine denominator formula for the strange series. Mathematical Research Letters 7(September-November):597. Further Readings: Kac, V.G., and M. Wakimoto. 1994. Integrable highest weight modules over affine superalgebras and Appell's function. In Progress in Mathematics, eds. J.-L. Brylinski, et al. Boston, Mass.: Birkhauser. Milne, S.C. 1996. New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan's tau function. Proceedings of the National Academy of Sciences 93(Dec. 24):15004. Peterson, I. 1999. Curving beyond Fermat. Science News Online. Available at http://www.sciencenews.org/sn_arc99/11_20_99/mathland.htm . . 1999. Curving beyond Fermat's last theorem. Science News 156(Oct. 2):221. Sources: George E. Andrews Department of Mathematics Pennsylvania State University University Park, PA 16802-6402 Richard Askey Department of Mathematics University of Wisconsin Madison, WI 53706 Bruce C. Berndt Department of Mathematics University of Illinois 1409 West Green Street Urbana, IL 61801 Stephen C. Milne Department of Mathematics Ohio State University Columbus, OH 43210 Web site:

63. AMU CHMA NEWSLETTER #14 (03/21/1995)
diophantus of alexandria, The six books of Arithmetis about 250) 309311. Chapter 9 Solution of systems of linear equations 319-354
http://www.math.buffalo.edu/mad/AMU/amu_chma_14.html
AMUCHMA-NEWSLETTER-14 Chairman: Paulus Gerdes (Mozambique) Secretary: Ahmed Djebbar (Algeria)
TABLE OF CONTENTS NEWSLETTER #1 Objectives of AMUCHMA Three new AMUCHMA projects Meetings, expositions, events Current research interests ... back to AMUCHMA ONLINE 2. THREE NEW AMUCHMA PROJECTS * To stimulate the creation of new national and regional mathematical associations, and to facilitate learning from the experience of existing or past mathematical associations, AMUCHMA proposes to publish a book on the history of national and regional mathematical associations in Africa * To stimulate youth in general, and girls in particular, to enter mathematical carreers, AMUCHMA proposes to publish two books with (auto)biographies of African mathematicians : the first book on those mathematicians who played a pioneering role in the history of their respective countries, and the second on female mathematicians. Proposals for contributions, and suggestions for the selection of associations and mathematicians to be included in these books, may be sent to the chairman or to the secretary of AMUCHMA.
3. MEETINGS, EXPOSITIONS, EVENTS

64. AMU CHMA NEWSLETTER #11 (09/01/1994)
here argued to be companion works by the same author, namely diophantus of alexandria. This attribution has implications for the dating of Diophantus.
http://www.math.buffalo.edu/mad/AMU/amu_chma_11.html
AMUCHMA-NEWSLETTER-11 Chairman: Paulus Gerdes (Mozambique) Secretary: Ahmed Djebbar (Algeria) TABLE OF CONTENTS NEWSLETTER #11 Objectives of AMUCHMA Meetings Current research interests Notes and queries ... back to AMUCHMA ONLINE Higher Pedagogical Institute (ISP), Maputo (Mozambique), 1.9.1993 2. MEETINGS 2.1 4th MaghrIbian Symposium on the History of Arabic Mathematics The 4th MaghrIbian Symposium on the History of Arabic Mathematics was held at the Sidi Mohamed Ben Abdallah University in Fez (Morocco), organized by Mohamed Aballagh, Mohamed Abattouy, and Mohamed Mesbahi. The following papers were presented in Arabic, English, or French dealing with the arabic mathematical sciences, their application and reception during the Middle Ages: * Ahmed Djebbar (Algiers, Algeria): Some aspects of mathematical activities in the extreme Maghreb from the XIIth to the XVIth centuries; * Mohamed Benchrifa (Rabat, Morocco): Presence of mathematics in some Andalusian literary writings; * Moustafa Mawaldi (Aleppo, Syria): Study of the manuscript 'Risala fi-l-Hisab-l-Hawa'i' of Najm ad-Din al-Katibi; * Rachid Bebbouchi (Bab-Ez-Zonar, Algeria): Redaction techniques in mathematics: Arabic heritage and actual reality in Algeria;

65. Algebra
was written in the 3rd century AD by diophantus of alexandria. In the 9th century, alKhwarizmi drew on Diophantus work and on Hindu sources to produce
http://excalc.vestris.com/docs/math-algebra.html
Software Documentation
Algebra
Algebra is a branch of mathematics in which the general properties of numbers are studied by using symbols, usually letters, to represent variables and unknown quantities. For example, the algebraic statement ( x + y ) ^ 2 = x ^ 2 + 2 x y + y ^ 2 is true for all values of x and y. An algebraic expression that has one or more variables (denoted by letters) is a polynomial equation . Algebra is used in many areas of mathematics, for example, matrix algebra and Boolean algebra (the latter is used in working out the logic for computers). In ordinary algebra the same operations are carried on as in arithmetic, but, as the symbols are capable of a more generalized and extended meaning than the figures used in arithmetic, it facilitates calculation where the numerical values are not known, or are inconveniently large or small, or where it is desirable to keep them in an analyzed form. Within an algebraic equation the separate calculations involved must be completed in a set order. Any elements in brackets should always be calculated first, followed by multiplication, division, addition, and subtraction. Algebra was originally the name given to the study of equations. In the 9th century, the Arab mathematician Muhammad ibn-Musa al-Khwarizmi used the term al-jabr for the process of adding equal quantities to both sides of an equation. When his treatise was later translated into Latin, al-jabr became

66. Diophantus Biography .ms
diophantus of alexandria by JJ O Connor and EF Robertson. deDiophant von Alexandrien nlDiophantus slDiofant svDiofantos
http://diophantus.biography.ms/
Diophantus
See also Diophantus of Alexandria (circa - circa ) was an ancient Greek mathematician Little is known about his life. He lived in Alexandria and he died when he was 84 years old. Probably Diophantus was a Hellenized Babylonian He was known for his study of equations with variables which take on rational values and these Diophantine equations are named after him. Diophantus is sometimes known as the "father of Algebra It was in the margin of a translation from Greek to Latin of Diophantus' work Aritmetike The Arithmetic ) from circa that Pierre de Fermat wrote his famous comment that is known as Fermat's last theorem Diophantus also wrote a treatise about polygonal numbers
External links
de:Diophant von Alexandrien nl:Diophantus sl:Diofant sv:Diofantos
A
B C D ... Home page

67. UNH College Of Engineering And Physical Sciences Focus Faculty
In 240 AD, along came diophantus of alexandria, who published 13 volumes on arithmetic geometry, Pfau s field. He was a completely isolated phenomenon at
http://www.ceps.unh.edu/focus/articles/faculty.html
Matthias Pfau
James Leitzel
(See story below)
TWO NEW CEPS FACULTY MEMBERS
The Poetry of the Mind

Matthias Pfau, assistant professor of mathematics
Matthias Pfau isn't what you'd call a born mathematician. He didn't even like math much as a student. "In school, you don't learn about what mathematics is," he says. "Real math is about ideas and proof, rather than simply about calculating."
Pfau's passion for the pure math he now studies began in the hospital. Bedridden for several weeks during high school with a broken jaw, Pfau began reading an old four-volume set of math textbooks he'd picked up at a secondhand bookstore. Suddenly he was hooked. The dusky green volumes that inspired him still sit on his crowded bookshelves in his UNH office.
Pfau, who recently joined the faculty as an assistant professor of mathematics, grew up in Germany and received his B.S. from the Rheinische Universitaet Bonn. After receiving his Ph.D. from the University of Michigan in 1993, he held a series of visiting scholar positions at prestigious institutes, globetrotting from Germany to Princeton, New Jersey, to Taiwan, to France. Today he and his wife Wafa Wei, also a mathematician, and their young son Jacob are settled in New Hampshire.
Pfau traces his mathematical ancestry back through the centuries the way some people trace their family trees. He sets the scene with the Pythagoreans, who translated all proportions in nature into geometric forms. The Pythagorean world was a beautifully balanced realm of geometric relationships, from the orbit of the spheres to the proportions of the human body. "Then the bad news broke," says Pfau. "Everything wasn't as neat as it looks."

68. Mathematics Archives - Topics In Mathematics - History Of Mathematics
KEYWORDS alKhwarizmi, diophantus of alexandria, Cardano, Bombelli, Viète, abu-Kamil, al-Karaji, Fibonacci, Maestro Benedetto, Bibliography, History
http://archives.math.utk.edu/topics/history.html
Topics in Mathematics History of Mathematics

69. Math Forum - Ask Dr. Math
it would help if you spelled the word Diophantine the equations are named after diophantus of alexandria, a famous Greek mathematician of antiquity.
http://mathforum.org/library/drmath/view/54241.html

Associated Topics
Dr. Math Home Search Dr. Math
Diophantine Equations
Date: 11/17/97 at 15:37:25 From: Robert Felgate Subject: Diaphantine equations Dear Dr. Math, We have been investigating Pythagoras' theorem, and our teacher has suggested we search the web for information about diaphantine equations. We have not been very successful and wonder if you would help us. Thank you. Robert and his Mum http://mathforum.org/dr.math/ Associated Topics
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70. Math Forum - Ask Dr. Math
The mathematician diophantus of alexandria (200284 AD) gave a general solution for when problems of this type are solvable.
http://mathforum.org/library/drmath/view/51595.html

Associated Topics
Dr. Math Home Search Dr. Math
Integer Solutions of ax + by = c
Date: 04/03/2001 at 19:58:44 From: Chris Armentrout Subject: integer solutions to ax+by = c Dr. Math, I am teaching a class to help prepare colleagues for the California Praxis exam. This question came up for us. While we were able to find part A all right, we could not find the solution to the general solution. A) Given the equation 5y - 3x = 1, find three points where x and y are both integers. B) Show that there will always be integer points (x,y) in ax + by = c if a, b and c are all integers. If not, show a counterexample. Thanks, I'm looking forward to your reply. Chris Armentrout http://mathforum.org/dr.math/ Associated Topics
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71. HistoryMole: Mathematics (0190-)
0200, (circa 200300) Diophantus, a 3rd century Greek mathematician, 0250, (Circa) diophantus of alexandria wrote the first book on algebra,
http://www.historymole.com/cgi-bin/main/results.pl?type=theme&theme=Mathematics

72. A Timeline Of The Ancient Egyptians
250 AD diophantus of alexandria writes the Arithmetica , a treaty on the solution of algebraic equations and on the theory of numbers
http://www.scaruffi.com/politics/egyptian.html
A time-line of the ancient Egyptians
World News Politics History Editor ...
Piero Scaruffi
4000 BC : Egyptians trace their origins to the Mount Rwenzori range in East Africa
3500 BC : Egyptians invent the sail
3000 BC : Egyptians begin to measure time through a calendar based on the three natural cycles (the solar day, the lunar month and the solar year)
3100 BC : hieroglyphic writing in Egypt
3000 BC : the Egyptians worship the sun
2920 BC : pharaoh Menes/Aha conquers the north and unites most of Egypt, and builds the capital at Hiku-Ptah (Memphis), the site of the cult of Ptah (1st dynasty)
2900 BC : king Djer is buried at Abydos, the seat of the cult of Osiris, lord of the Underworld and husband of Isis, and his "mastaba" becomes considered the grave of Osiris
2890 BC : Hetepsekhemwy founds the second dynasty in Egypt
2800 BC : Egyptians begin mining in the Sinai
2700 BC : Egyptians write on papyrus 2660 BC : pharaoh Kasekhemwy completes the union of north and south Egypt, and builds the first fortress on the Nile, at Buhen 2649 BC : Zanakht founds the 3rd dynasty 2630 BC : Zanakht dies and is succeeded by Djoser 2620 BC : Imhotep, high priest of Ptah at Memphis and founder of Medicine, erects a pyramid made of stone at Saqqara (overlooking Memphis) for pharaoh Djoser ("step pyramid")

73. ENC Online: Curriculum Resources: Agnesi To Zeno: Over 100 Vignettes From The Hi
diophantus of alexandria African number systems and symbolism Nine chapters on mathematical art Hypatia of Alexandria The concept of zero
http://www.enc.org/resources/records/contents/0,1240,006398,00.shtm
Skip Navigation You Are Here ENC Home Curriculum Resources Search the Site More Options Don't lose access to ENC's web site! Beginning in August, goENC.com will showcase the best of ENC Online combined with useful new tools to save you time. Take action todaypurchase a school subscription through goENC.com Classroom Calendar Digital Dozen ENC Focus ... Ask ENC Explore online lesson plans, student activities, and teacher learning tools. Search Browse Resource of the Day About Curriculum Resources Read articles about inquiry, equity, and other key topics for educators and parents. Create your learning plan, read the standards, and find tips for getting grants.
Agnesi to Zeno: over 100 vignettes from the history of math
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ENC-006398
Publisher: Key Curriculum Press
Date:
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The unknown origin of counting
Ancient references to Pi
The pyramids of Egypt and the Americas Chinese mathematical activities The Rhind Papyrus and the St. Ives Puzzle Early astronomy Cosmologies of the ancient world Calendars of the ancient world Thales: a man of legend The Golden age of Greece Early Greek mathematicians Ideal and irrational numbers The paradoxes of Zeno Infinitude of primes The Elements of Euclid Apollonius and conic sections Eratosthenes computation Philo, religion and mathematics

74. Fermat's Last Theorem: Diophantine Equations And The Most Famous Problem In Math
An equation for which a solution in whole numbers is sought is called a Diophantine equation, after diophantus of alexandria, who lived about 250 AD There
http://www.wolfram.com/products/explorer/topics/fermat.html
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In 1993, to the extreme shock of the mathematical community, Andrew Wiles of Princeton University announced that he had found a proof of Fermat's Last Theorem. This theorem was one of the most famous unsolved problems of mathematics; hence, Wiles has achieved certain immortality with his discovery. It is called the "last theorem" because it was the last problem posed by Pierre de Fermat (1601-1665) to remain unresolved after his death. Fermat's problem is: Show that x n + y n z n has no solution in whole numbers, where n An equation for which a solution in whole numbers is sought is called a Diophantine equation, after Diophantus of Alexandria, who lived about 250 A.D. There is no general method for solving such equations. Get a taste of the field by seeing how The Mathematical Explorer can be used to solve some interesting types of Diophantine equations. Also, discover some features of the general problem that make these equations so interesting.

75. From Euclid To Newton
diophantus of alexandria (circa 200circa 284) Arithmeticorum libri sex, et De numeris multangulis liber unus Paris Sébastien Cramoisy for Ambrose
http://www.brown.edu/Facilities/University_Library/exhibits/math/nofr.html
From Euclid to Newton:
An Exhibition in Honor of the 1999 Conference of the
Mathematical Association of America Math Exhibit Home
Exhibits Home

Euclid

(ca. 326-265 BC)
Archimedes

(ca. 287-212 BC)
Apollonius of Perga

(ca. 260-200 BC)
Nichomachus of Gerasa
(ca. 100) Claudius Ptolemy (ca. 85-165) Diophantus of Alexandria (ca. 200-284) Pappus of Alexandria (ca. 300-350) Proclus (ca. 410-485) Boethius (ca. 480-524) Thomas Bradwardine (ca. 1290-1349) Girolamo Cardano Robert Recorde Johann MŸller of Kšnigsberg called Regiomontanus Franois Vite John Napier Henry Briggs Adriaan Vlacq ... Bonaventura Cavalieri (ca. 1598-1647) Christiaan Huygens RenŽ Descartes Gottfried Wilhem Leibniz Sir Isaac Newton ... Guillaume Franois Antoine l'Hospital, Marquis de Sainte-Mesme TOP
Euclid
Brown University Library possess a copy of each sixteenth-century translation of Euclid's Elements of Geometry into a modern language. These vernacular editions, grouped around the first Latin edition of 1482, are displayed in chronological sequence, from 1533 (Greek) to 1594 (Arabic). All copies are opened at Book I, proposition 47, "Pythagoras' Theorem," which asserts: "In right-angled triangles the square of the side opposite the right angle is equal to the sum of the squares of the sides containing the right angle." Most of the translations provide proof of this equation (a

76. A Look To The Past
But the most important of the Greek algebraists was diophantus of alexandria. Diophantus can be considered to be the father of ancient algebra.
http://ued.uniandes.edu.co/servidor/em/recinf/tg18/Vizmanos/Vizmanos-2.html
Will elementary algebra disappear with the use of new graphing calculators?.
A look to the past
What do we understand elementary algebra to be? Elementary algebra is the language with which we communicate the majority of mathematics. Thanks to algebra we can work with concepts at an abstract level and then apply them. Elementary algebra begins as a generalization of arithmetic and then focuses on its own structure and greater logical coherence. From there comes the importance of the various uses of algebraic symbols. When we write A + B, we can be indicating the sum of two natural numbers, the sum of two algebraic expressions, or even the sum of two matrices. Thus there is, at first, representations and symbolism, and later the development of algorithms and procedures to work formally with algebraic expressions. But what we today understand to be algebra has been the fruit of the efforts of many generations that have been contributing their grains of sand in constructing this magnificent building. It seems that the Egyptians already knew methods for solving first degree equations. In the

77. PSIgate - Physical Sciences Information Gateway Search/Browse Results
diophantus of alexandria Born about 200 Died about 284 Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index Version
http://www.psigate.ac.uk/roads/cgi-bin/search_webcatalogue2.pl?limit=125&term1=b

78. Book Collections, History Of Science. Thomas Fisher Rare Book Library, Universit
Early editions of the Greek mathematicians diophantus of alexandria, Archimedes, Pappus of Alexandria, Apollonius of Perga, and especially of Euclid,
http://www.library.utoronto.ca/fisher/collections/science.html
History of Science
The Library has large and comprehensive collections in the history of science, in several key areas. The library’s general Science Collection, which contains approximately 8,000 items, covers many branches of theoretical and applied science. One of the principal focuses of the collection is the science of the Renaissance, prior to Galileo, whose work, and that of his contemporaries, form a separate special collection.
Mathematics
The works of early Renaissance scientists are complemented by scholarly editions of the rediscovered texts of the ancients. Early editions of the Greek mathematicians Diophantus of Alexandria, Archimedes, Pappus of Alexandria, Apollonius of Perga, and especially of Euclid, are well represented. The first Latin edition of Euclid’s Elements of Geometry , printed by Ratdolt at Venice in 1482, the Greek editio princeps (Basel, 1533), and the first printed Arabic version (Rome, 1594), are complemented by translations into modern languages, including the first Italian (1543), English (1570), and German (1562) versions. In all, the library has twenty-six editions of Euclid, published between 1482 and 1620. The Renaissance produced many important mathematical texts, particularly in Italy. Multiple editions of the works of Niccolò Tartaglia, Girolamo Cardano, Federico Commandino, Raffaelle Bombelli, Bonaventura Cavalieri, to name a few, are all among the collections. French mathematics is represented by the works of Marin Mersenne, Pierre Fermat, and François Viète. The Renaissance was also renowned for fine printing, two of the most beautiful texts in the field of mathematics being Luca Pacioli's

79. Encyclopedia: Algebra
diophantus of alexandria ?fa?t ? e?a?d?e - (circa 200/214 - circa 284/298) was an ancient Greek mathematician. Arithmetica, an ancient text on
http://www.nationmaster.com/encyclopedia/Algebra

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    Encyclopedia: Algebra
    Updated 20 hours 31 minutes ago. Other descriptions of Algebra Algebra is a branch of mathematics , which studies structure and quantity . It may be roughly characterized as a generalization and abstraction of arithmetic , in which operations are performed on symbols rather than numbers . It includes elementary algebra , taught to high school students, as well as abstract algebra which covers such structures as groups rings and fields . Along with geometry and analysis , it is one of the three principal branches of mathematics. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... The structure of a thing is how the parts of it relate to each other, how it is put together. This contrast with process, which is how the thing works; but process requires a viable structure. ...

    80. MT4501
    The number theory of diophantus of alexandria. Solving quadratic, cubic and quartic equations. The work of Cardan and Tartaglia.
    http://www.mcs.st-andrews.ac.uk/ug/hon4/MT4501.shtml
    Home Personnel Info for prospective undergraduates Research and postgraduates ... MacTutor History of Mathematics
    School of Mathematics and Statistics
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    Honours
    timetable
    2005/2006 Sem. 1 2005/2006 Sem. 2 2006/2007 Sem. 1 2006/2007 Sem. 2
    MT4501 TOPICS IN THE HISTORY OF MATHEMATICS
    As well as the lecture-based content, students have the opportunity to write a project on a historical topic
    Aims
    The overall aim of the course is to give students an insight into the historical development of mathematics. The course aims to illustrate the following: - How mathematics has been, and still is, a developing subject. - How advances in mathematics are driven by problem solving and how initial formulations often lacked rigour. - How good mathematical notation is vital to the development of the subject. - That there is a great sophistication in mathematical ideas that are considered "elementary" today.
    Objectives
    By the end of the course students are expected to be able to: - understand, describe, compare and contrast the main ideas and methods studied in the course.

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