:: High-school knowledge
(formula for a line between two points, the Quadratic Formula,
intersecting a line with a parabola, etc.), and some
:: Sets&Logic ideas (Equivalence relations, partial orders, binary operators, induction, pigeon-hole principle, cardinality).
:: Mathematical maturity (Do you remember your basic calculus stuff? Do you remember how to sum a geometric series?)
:: Math-Greek alphabet (pdf), which we will use in class frequently.
This is a 1-semester course for folks interested in Mathematical Cryptography (major topic) and other aspects of Coding: Data Compression and (time permitting) Error-correcting codes. We may also cover one example of an Isomorphism code.
It is accessible to anyone who has Sets&Logic knowledge, whether or not you've take that course. I would like you to know:
What a prime number is, and What mathematical induction is and What an equivalence relation is. Also helpful is how to “add two numbers mod N”, and what the Euler phi function is.
Good choices [all these books are in Marston Science Library, on campus] for a bit of self-study are:
Our Teaching Page has useful information for students in all of my classes. It has my schedule, LOR guidelines, and Usually Useful Pamphlets. One of them is the Checklist (pdf) which gives pointers on what I consider to be good mathematical writing. Further information is at our class-archive URL (I email this private URL directly to students).
ALGEBRA comes from the title of a work written in Arabic about 825 by al-Khowarizmi, al-jabr w'al-muqabalah, in which al-jabr means "the reunion of broken parts." When this was translated from Arabic into Latin four centuries later, the title emerged as Ludus algebrae et almucgrabalaeque.
In 1140 Robert of Chester translated the Arabic title into Latin as Liber algebrae et almucabala.
In the 16th century it is found in English as algiebar and almachabel, and in various other forms but was finally shortened to algebra. The words meanrestoration and opposition.
In Kholâsat al-Hisâb (Essence of Arithmetic), Behâ Eddîn (c. 1600) writes,The member which is affected by a minus sign will be increased and the same added to the other member, this being algebra; the homogeneous and equal terms will then be canceled, this being al-muqâbala.
The Moors took the word al-jabr into Spain, an algebrista being a restorer, one who resets broken bones. Thus in Don Quixote (II, chap. 15), mention is made ofun algebrista who attended to the luckless Samson.At one time it was not unusual to see over the entrance to a barber shop the wordsAlgebrista y Sangrador[bonesetter and bloodletter] (Smith vol. 2, pages 389-90).
The earliest known use of the word algebra in English in its mathematical sense is by Robert Recorde in The Pathwaie to Knowledge in 1551:Also the rule of false position, with dyvers examples not onely vulgar, but some appertayning to the rule of Algeber.
The phrase an algebra is found in 1849 Trigonometry and Double Algebra by Augustus de Morgan:Ordinary langauge has methods of instantaneously assigning meaning to contadictory phrases: and thus it has stronger analogies with an algebra (if there were such a thing) in which there are preorganized rules for explaining new contradictory symbols as they arise, than with one in which a single instance of them demands an immediate revision of the whole dictionary[University of Michigan Historical Math Collection]. …
Real Analysis is no more about reality than Complex Analysis is about complexity.-P. Boyland, paraphrased
Cryptography (or cryptology; derived from Greek κρυπτός kryptós "hidden," and the verb γράφω gráfo "write") is the study of message secrecy. In modern times, it has become a branch of information theory, as the mathematical study of information...
... Steganography (i.e., hiding even the existence of a message so as to keep it confidential) was also first developed in ancient times. An early example, from Herodotus, concealed a message - a tattoo on a slave's shaved head - by regrown hair. More modern examples of steganography include the use of invisible ink, microdots, and digital watermarks to conceal information .
Ciphertexts produced by classical ciphers always reveal statistical information about the plaintext, which can often be used to break them. After the Arab discovery of frequency analysis (around the year 1000), nearly all such ciphers became more or less readily breakable by an informed attacker. ... Essentially all ciphers remained vulnerable to cryptanalysis using this technique until the invention of the polyalphabetic cipher by Leon Battista Alberti around the year 1467. Alberti's innovation was to use different ciphers (ie, substitution alphabets) for various parts of a message (often each successive plaintext letter). He also invented what was probably the first automatic cipher device, a wheel which implemented a partial realization of his invention. In the polyalphabetic Vigenère cipher, encryption uses a key word, which controls letter substitution depending on which letter of the key word is used.
Diophantus of Alexandria - (Greek: Διόφαντος ὁ Ἀλεξανδρεύς , circa 200/214 – circa 284/298) was a Greek mathematician of the Hellenistic era. Little is known of his life except that he lived in Alexandria, Egypt ...
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 thefather of Algebra. He wrote a total of thirteen books on these equations. Diophantus also wrote a treatise on polygonal numbers.
In 1637, while reviewing his translated copy of Diophantus' Arithmetica (pub. ca.250) Pierre de Fermat wrote his famousLast Theoremin the page's margins. His copy with his margin-notes survives to this day.
Although little is known about his life, some biographical information can be computed from his epitaph (see links below). He lived in Alexandria and he died when he was 84 years old. Diophantus was probably a Hellenized Babylonian.
A 5th and 6th century math puzzle involving Diophantus' age: He was a boy for one-sixth of his life. After one-twelfth more, he acquired a beard. After another one-seventh, he married. In the fifth year after his marriage his son was born. The son lived half as many as his father. Diophantus died 4 years after his son. How old was Diophantus when he died?
The answer: 84 The answer is determined from two methods: 1. Finding the common multiple of 12, 6, and 7 (which is 84). 2. Taking 14 (the age up to which would be considered a boy; one-sixth of his life) multiplied by 6, which equals 84.
we discover to our delight that.
This branch of mathematics is the only one, I believe, in which good writers frequently get results entirely erroneous. ... It may be doubted if there is a single extensive treatise on probabilities in existence which does not contain solutions absolutely indefensible.C. S. PEIRCE,
in Popular Science Monthly (1878).
I recoil with dismay and horror at this lamentable plague of functions which do not have derivatives.-Charles Hermite
There's a delta for every epsilon, It's a fact that you can always count upon. There's a delta for every epsilon And now and again, There's also an N. But one condition I must give: The epsilon must be positive A lonely life all the others live, In no theorem A delta for them. How sad, how cruel, how tragic, How pitiful, and other adjec- Tives that I might mention. The matter merits our attention. If an epsilon is a hero, Just because it is greater than zero, It must be mighty discouragin' To lie to the left of the origin. This rank discrimination is not for us, We must fight for an enlightened calculus, Where epsilons all, both minus and plus, Have deltas To call their own.Words and Music by Tom Lehrer,
American Mathematical Monthly, 81 (1974)
Linear Algebra – Linear algebra stems from the need to solve systems of linear equations. For small systems, ad hoc methods are sufficient. Larger systems require one to have more systematic methods. The modern day approach can be seen 2,000 years ago in a Chinese text, the Nine Chapters on the Mathematical Art (Traditional Chinese: 九章算術; Simplified Chinese: 九章算术; pinyin: Jiǔzhāng Suànshù).
Chinese mathematicians developed a system in which they organized linear equations in a rectangular pattern called Fāng Chéng (方程) in Chinese, involving horizontal and vertical counting rods. This rectangular representation of linear equations is the equivalent of today's matrix.
One of the key developments in linear algebra was the modern day method of solving linear systems known as Gauss-Jordan elimination, after German mathematician Carl Friedrich Gauss (1777-1855) and German engineer Wilhelm Jordan (1844-1899). Gauss called the method eliminatio, even though the Chinese were using an almost identical method nearly two millennia prior. This method stemmed really from Gauss's laziness in leaving off variable stems such as x1, x2, etc. in solving large n-tuples of linear equations while following the asteroid now known as Ceres. His method is explained in his book Theoria Motus Corporum Coelestium (1809).
NUMBER THEORY. According to Diogenes Laertius, Xenocrates of Chalcedon (396 BC - 314 BC) wrote a book titled The theory of numbers.
A letter written by Blaise Pascal to Fermat dated July 29, 1654, includes the sentence,The Chevalier de Mèré said to me that he found a falsehood in the theory of numbers for the following reason.
The term appears in 1798 in the title Essai sur la théorie des nombres by Adrien-Marie Legendre (1752-1833).
In English, theory of numbers appears in 1811 in the title An elementary investigation of the theory of numbers by Peter Barlow.
Number theory appears in 1853 in Manual of Greek literature from the earliest authentic periods to the close of the Byzantine era by Charles Anthon: "The ethics of the Pythagoreans consisted more in ascetic practice and in maxims for the restraint of the passions, especially of anger, and the cultivation of the power of endurance, than in scientific theory. What of the latter they had was, as might be expected, intimately connected with their number-theory" [University of Michigan Digital Library].
Number theory appears in 1864 in A history of philosophy in epitome by Dr. Albert Schwegler, translated from the original German by Julius H. Seelye: "Not only the old Pythagoreans, who have spoken of him, delighted in the mysterious and esoteric, but even his new-Platonistic biographers, Porphyry and Jamblichus, have treated his life as a historico-philosophical romance. We have the same uncertainty in reference to his doctrines, i. e. in reference to his share in the number-theory. Aristotle, e. g. does not ascribe this to Pythagoras himself, but only to the Pythagoreans generally, i. e. to their school" [University of Michigan Digital Library]. ...
Quantifiers ∀ and ∃ (“for all” and “there exists”) are like nitroglycerin, in that one little mis-step leads to the whole thing blowing up in your face.
There is no partial credit when it comes to Explosives and Quantifiers.-JLF King
We are sorry, but the number you have dialed is imaginary. Please rotate your phone by 90 degrees and try again.-David Grabiner
The four-year-old niece of a mathematician was playing a game in which she was the conductor on a train and her mother was a passenger.Wait a minute,said Nancy,we have to get some paper to make tickets.Oh,said her mother, who had probably had a long day,do we really need them? After all, it's only a pretend game with pretend tickets.
Oh no, Mommy, you're wrong,replied Nancy;they're pretend tickets, but it's a real game.
(recounted by David Gale. This appeared
in The Intelligencer.)
I will be running a Special Topics NT course for Fall 2007, as long as it has at least 14 students.
This course does NOT require NT 1 (MAS4203). [For those who have already taken MAS4203, I have designed this course to be a good successor.] For students who have not already taken MAS4203, I ask that (during the summer) they read about 30 pages from Shoup's free online text [linked below].
If potentially interested in taking the course, please email me at
squashATuflDOTedu (Prof. JLF King)
and I will add you to the mailing list. Most of the information about the course will be distributed by email.
Welcome! Our Teaching Page has useful information for students in all of my classes. It has my schedule, LOR guidelines, and Usually Useful Pamphlets. One of them is the Checklist (pdf) which gives pointers on what I consider to be good mathematical writing. Further information is at our class-archive URL (I email this private URL directly to students).
In all of my courses, attendance is absolutely required (excepting illness and religious holidays). In the unfortunate event that you miss a class, you are responsible to get all Notes / Announcements / TheWholeNineYards from a classmate, or several. All my classes have a substantial class-participation grade.
A syllabus (text file) is available. On the first day of class I will hand-out a paper syllabus, which has the instructions on how to use the class archive. The archive is at a private URL, only for the use of the students in our class.
Whatever you do, Don't look at Past courses with notes, exams and links!
Astonishingly, one earns genuine coin-of-the-realm CP points (CP = "class participation") by posting a solution or improving a solution.
The various Number Theory czars who help out.
|?||?||Charlye||Marshall||Jimmy-C & Dream||Cameron|
There will be not be a final exam . Rather, there will be an optional end-of-semester project, to be done individually. The project will be due, slid u n d e r my office door (Little Hall 402, Northeast corner) , no later than noon, Friday, 12Dec2008.
The final project must be carefully typed. I recommend learning the free mathematics-typesetting language . (It is the archive language of the American Mathematical Society.) It can be learned in a week. It is very good for typesetting homework.
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