Japheth Wood

Author: japheth

  • Typesetting synthetic division

    I’m teaching an Algebra course that highlights the Fundamental Theorem of Algebra. So of course we’re looking closely at polynomial division, and in particular at synthetic division. My students are preparing their homework assignments using LaTeX, so this begs the question about how to typeset their computations.

    One of my students found the LaTeX package polynom, which can automatically compute and typeset polynomial long division and synthetic division. The examples in the polynom manual are:

    \polylongdiv{X^3+X^2-1}{X-1}

    which yields

    Screen Shot 2014-09-07 at 2.57.19 PM

     

     

     

     

     

    and for synthetic division, the polynom command

    \polyhornerscheme[x=1]{x^3+x^2-1}

    which yields

    Screen Shot 2014-09-07 at 2.57.28 PM

     

     

     

    But what if you want to show division by x - m using synthetic division, rather than x - 1? These commands no longer work.

    Instead, you can make fancy use of \cline and \multicolumn in the array environment:

    \begin{array}{r|rrrr}
    & 1 & 1 & 0 & -1 \\
    m & & m & (m^2 + m) & (m^3 + m^2) \\ \cline{2-5}
    \multicolumn{2}{r}{1} & (m + 1) & (m^2 + m) & (m^3 + m^2 - 1)
    \end{array}

    which yields

    Screen Shot 2014-09-07 at 3.02.21 PM

     

     

     

    Interesting! If you divide a polynomial by x - m, the remainder is just the value of that polynomial at m!

  • How to Give a Good Math Talk

    There are a lot of sites with useful advice out there. This post is intended to collect several links for future reference.

    Technically Speaking – Videos of math research students presenting their findings. I love the examples of bad style along with good style. This comes out of an NSF-funded project.

    Joe Gallian’s Advice on Giving a Good PowerPoint Presentation from his article in Math Horizons.

    Giving a Conference Talk by Mike Dahlin.

    Oral Presentation Advice by Mark D. Hill, includes “How to Give a Bad Talk” by David A. Patterson.

    Giving an Academic Talk by Jonathan Shewchuk.

    I also have several resources on good mathematical writing that I’ll share later.

  • A brief 4,000 year history of Diophantine Equations

    I filled in for a NY Math Circle class over the weekend. Since the topic was Primitive Pythagorean Triples, I had a blast. I also shared the following outline with the students. Each item is full of wonderful mathematics and anecdotes!

    Plimpton 322, a Babylonian cuneiform tablet @ Columbia University. From 1900BCE – 1600BCE, and allegedly includes the Pythagorean triple (12709, 13500, 18541).

    Pythagoras was born ca 580BCE on the island of Samos. Famous quote: “All is Number”. His proof of the theorem that bears his name involved cutting up a square of side a+b and rearranging the pieces.

    Proclus (5th century CE) credits Pythagoras with the formula (2n+1, 2n^2 + 2n, 2n^2 + 2n + 1) of (necessarily) primitive pythagorean triples where the hypotenuse is 1 more than one of the sides. Proclus also credits Plato with the formula (2n, n^2 – 1, n^2 + 1) where the hypotenuse is 2 more than one of the sides. (For what n is this primitive?)

    Euclid of Alexandria (born 365 BCE) is famous for the 13-book Elements. The Theorem of Pythagoras is Book I, Proposition 47. Analysis on Primitive Pythagorean Triples appears as Lemma 1 “To find two square numbers such that their sum is also a square” in Book X, just before Proposition 29.

    By the way, Elisha Scott Loomis, an early 20th century mathematician, published The Pythagorean Proposition, which has 370 proofs of the theorem, and not a single one used trigonometry. This was republished in 1968 by the NCTM.

    Diophantus of Alexandria (200CE – 298CE) is famous for his book Arithmetica. He sought integer (and perhaps rational too?) solutions to algebraic equations. The term Diophantine Equation typically refers to equations where we seek positive integer solutions.

    Fermat (1608CE – 1665CE) wrote in the margin of his copy of Arithmetica that there are no integral values of x, y, z so that x^n + y^n = z^n if n > 2.

    Andrew Wiles of Princeton University announced a proof in 1993 of Fermat’s Last Theorem, after working in secret for 7 years. An error was found in his proof, which was salvaged in 1994. Wiles’ proof was published in 1995.

    Jumping back to 1900, David Hilbert asked mathematicians at the International Congress of Mathematicians to devise a method to determine whether a Diophantine equation has solutions. This is known as Hilbert’s 10th Problem.

    Julia Robinson (1919-1985) was a Californian mathematician who worked on Hilbert’s 10th problem for decades, from the 1940s until final achieving a solution (in joint work with Martin Davis, Hilary Putnam, Yuri Matiyasevich and others) in 1970. In general, there is no such algorithm!

    Mirroring the Hilbert Problems of 1900, the Clay Institute of Mathematics issued the Millennium Problems in 2000.

  • Kevin’s Books


    One of the great friends that I made in my time at Vanderbilt University was Kevin Blount. Kevin knew all the graduate students and professors, and often hosted dinners and movies at his nearby apartment.
    Kevin ended up writing his Ph.D. dissertation On the Structure of Residuated Lattices with Constantine Tsinakis, and moved on to an academic position at Sacred Heart University in Connecticut.

    Kevin passed away on May 30, 2006, which surprised all of us. I had just moved back East, and so one of the first trips was attending his memorial service at Sacred Heart.

    At the end of 2008, Kevin’s wife, Xiaoyu, gave me Kevin’s math books. After some brief discussions with some mathematical colleagues, the books ended up being stored in my attic. I’m now offering these books to those who knew Kevin. I am sure that Kevin would have been happy to have his math books shared among his friends and colleagues, and I hope that this will help keep Kevin’s memory alive in the mathematical community.

    If you are interested in some of the books below, leave a comment or contact me privately. I’ll send them your way. (Pardon the typos – I’ll correct those as they are noticed).

    Kevin’s Math Books

    1. Albert, Introduction to Algebraic Theories, Matt Insall
    2. Artin, Geometric Algebra, Tracts in Mathematics Number 3 – John Raymonda
    3. Baumslag and Chandler, Group Theory
    4. Bennet, Affine and Projective Geometry
    5. Bjarni Jonsson, Topics in Universal Algebra, Lecture Notes, Vanderbilt University, 1969-70 – John Snow
    6. Bollobas, Graph Theory, GTM 63, Matt Insall
    7. Bond/Keane, An Introduction to Abstract Mathematics, Instructors
    8. Bondy and Murty, Graph Theory with Applications, Matt Insall
    9. Bonele, Non-Euclidean Geometry, Dover
    10. Bourbaki, Elements of Mathematics, General Topology, Part 1, Addison Wesley
    11. Burnside, Theory of Groups of Finite Order, Princeton
    12. Curry, Foundations of Mathematical Logic, Dover
    13. Dieudonne, Introduction to the Theory of Formal Groups, Dekker
    14. Gruenberg and Weir, Linear Geometry, Van Nostrand
    15. Hodel, An Introduction to Mathematical Logic – Jan Gałuszka
    16. Hall, The Theory of Groups, Macmillan
    17. Springer, Geometry and Analysis of Projective Spaces
    18. Sternberg, Lectures on Differential Geometry, Prentice Hall
    19. Hennie, Introduction to Computability
    20. Lang, Introduction to Algebraic Geometry, Tracts in Mathematics Number 5 – John Raymonda
    21. Hochschild, Introduction to Affine Algebraic Groups
    22. Nagaia, Local Rings, Tracts in Mathematics Number 13
    23. Spanier, Algebraic Topology
    24. Lipschitz, Discrete Mathematics, Schaum McGraw Hill
    25. Solutuion Manual for, Brooks/Cole
    26. Curtis Clark, An Approach to Graph Achievement Games: Ultimately Economical Graphs
    27. Hungerford, Algebra, Springer 73 – John Snow
    28. Mendelson, Introduction to Mathematical Logic, 3e, Wadsworth & Brooks/Cole Mathematics Series
    29. Moore, Elementary General Topology, Prentice-Hall
    30. Mitchell, Theory of Categories, Academic Press, Matheamtics XVII
    31. Herken, The Universal Turing Machine, Oxford – John Snow
    32. ??, Fundamental Concepts of Algebra, Addison Wesley
    33. Kelley, General Topology, Van NostrandO??, Theory of LIE Groups, Princeton
    34. Munkres, Elements of Algebraic Topology
    35. Sawyer, A Geometric Approach to Abstract Algebra, Freeman
    36. McCoy, Rings and Ideals – John Raymonda
    37. Smullyan, First-Order Logic, Dover
    38. Veblen and Young, Projective Geometry, Volume 1
    39. Gelfond, Transcendental & Algebraic Numbers, Dover – John Snow
    40. Freese & McKenzie, Commutator Theory for Congruence Modular Varieties – Jonathan Farley
    41. Davey & Priestley, Introduction to Lattice and Order, Cambridge – John Snow
    42. An Algebraic Introduction to Matheamtical Logic, GTM 52 – John Snow
    43. Categories for the Working Mathematician, GTM 5 – John Snow
    44. Manin, A Course in Mathematical Logic, GTM 53
    45. Burris/Sankappanavar, A Course in Universal Algebra, GTM 78 – John Snow
    46. Hirsch, Differential Topology, GTM 33
    47. Gaum, Elements of Point Set Topology, Prentice-Hall
    48. Bourbaki, Elements of Mathematics, General Topology Part 2
    49. Eisenberg, Topology, Holt Rinehart Winston
    50. Pareigis, Categories and Functors
    51. Kaplansky, Set Theory and Metric Spaces – John Raymonda
    52. McKenzie, McNulty, Taylor, Algebras, Lattices, Varieties, Volume I, Wadsworth & Brooks/Cole Matheamtics Series, – Jonathan Farley
    53. Monk, Mathematical Logic, GTM 37 – John Snow
    54. Lightstone, Symbolic Logic and the REAL NUMBER SYSTEM – John Snow
    55. The Essentials of Logic
    56. Halmos & Givant, Logic as Algebra – Jan Gałuszka
    57. Jacobson, Lectures in Abstract Algebra, I Basic Concepts, Van Nostrand – John Raymonda
    58. Husain, Introduction to Topological Groups, Matt Insall
    59. Tarski, Introduction to Logic and to the methodology of deductive sciences, Dover
    60. Church, Introduction to Mathematical Logic, Princeton – John Snow
    61. Rosenbloom, an introduction to Symbolic Logic, Dover
    62. Krantz, How to Teach Mathematics: a personal perspective
    63. Thomas Rishel, Teaching First – A Guide for New Mathematicians, MAA
    64. Hobby and McKenzie, The Structure of Finite Algebras, AMS Comm 76, – Jonathan Farley

  • AP Classes Are a Scam

    I heard about the following Atlantic article from @stevenstrogatz : October 13th, AP Classes Are a Scam which I found quite interesting.

    I thought much the same in those years when I taught a lot of freshman Calculus. My main observations were that

    1. Most students who had taken AP Calculus in High School had to take the Calculus sequence anyway, and resented that they had to essentially repeat a course.
    2. Students who had not taken AP Calculus in High School felt intimidated that they were in class with students who had, and felt completely inadequate.
    3. As a result, the AP students barely worked at all, since they had a superficial knowledge of Calculus, while the non-AP students worked very hard.
    4. Since AP Calculus is not college level Calculus, the effects were clear by midterms: the AP students had fallen too far behind, and the non-AP students were learning the material, and starting to enjoy it.
    5. I suspected that there were other, successful AP students, who weren’t in my class, and never took another math class in their lives. Thus, some of the most enthusiastic math students at the high school level were diverted out of the math major, since they saw Calculus as the final math class.

    So why did I come away with the impression that AP Calculus, presented as the highest level math class one could take in college, was essentially a terminal math class, serving to prevent bright and hard-working high school math kids from continuing in mathematics?

    I’ve come to understand this more in the contrast between acceleration and enrichment. Our educational system emphasizes acceleration, and works hard to move kids rapidly through material. There are a lot of incentives for this, like granting college credit. An alternative is to enrich the curriculum, and allow students to go deeper into the material.

    When I was in high school, I used math team to enrich my studies, as well as my own mathematical reading of fantastic authors like Martin Gardner. I don’t think I ever earned an academic credit for this enrichment, but it was profoundly enjoyable, and directed me into mathematics. I did benefit from acceleration as well, but my most memorable mathematical moments were from some inspirational math enrichment.

    So, why do we bother with AP courses? I think our students would benefit greatly if a similar amount of resources were invested into academic enrichment. I’d love to see after-school math circles, math clubs and math teams in every school, and I can imagine similar enrichment in other subjects.

  • Math Formula Poetry Slam

    Last December, I got to meet Daniel, aka Jarabe Del Sól, a poet from the Readnex Poetry Squad. He showed me in his notebook where he had written all sorts of arcane symbols and characters, perhaps from undiscovered alphabets. I got to fill up a few pages of his notebook with math symbolism off the top of my head, and then he proceeded to respond with a poetic response and free verse. This was a lot of fun, and a few days later, he sent me photos of what I had written. If you click on the photos below, you can see what Jarabe and I wrote, a math formula poetry slam! Play this track while you’re looking, for the full experience.

    11 If gravity could fly one 64

  • Singing and Dancing Mathematics

    Long Division Style

    Recently, a Gangnam Style-inspired video came out of I.S. 285 Meyer Levin in Brooklyn. In this video, students sing and dance the procedure for long division. It’s absolutely delightful.

    The Quadratic Formula Song

    Music versions of math formulas and procedures are not new. One that came up recently in the Algebra class I teach for in-service teachers is the Quadratic Formula, set to the tune Pop Goes the Weasel.

    x equals negative b
    plus or minus the square root
    of b squared minus 4ac
    all over 2a

    Please sing this to your students!

    Solving the Cubic in a Poem

    Did you know that there is something similar for the Cubic Formula? Tartaglia was one of the first mathematicians to solve (a specific form of) the cubic. His solution was secret, but he did communicate his solution to Cardano in a poem, but swore Cardano to secrecy.

    Here’s the poem, in translation (from the MAA Digital Library, a fantastic resource).

    Tartaglia’s Poem

    Tartaglia divulged to Hieronimo Cardano (1501-1576) the solution of the three cubic equations without the quadratic term on March 25, 1539 in Cardano’s house in Milano in the form of a famous poem (translated here by the author):

    01) When the cube with the cose beside it
    02) Equates itself to some other whole number, <=q>
    03) Find two other, of which it is the difference.
    04) Hereafter you will consider this customarily
    05) That their product always will be equal
    06) To the third of the cube of the cose net. 3/3, instead of (p/3)3>
    07) Its general remainder then
    08) Of their cube sides , well subtracted,
    09) Will be the value of your principal unknown. <=x>
    10) In the second of these acts,
    11) When the cube remains solo ,
    12) You will observe these other arrangements:
    13) Of the number you will quickly make two such parts,
    14) That the one times the other will produce straightforward
    15) The third of the cube of the cose in a multitude, 16) Of which then, per common precept,
    17) You will take the cube sides joined together. 
    19) The third then of these our calculations
    20) Solves itself with the second, if you look well after,
    21) That by nature they are quasi conjoined.
    22) I found these, & not with slow steps,
    23) In thousand five hundred, four and thirty
    24) With very firm and strong foundations
    25) In the city girded around by the sea.

  • Math Conference photos

    Going through some old papers, I found the following conference photos from

    • Fall 1996 Fields Institute, Algebraic Model Theory Program
    • July 1-10, 1998, XI Simposio Latinoamericano Lógica Matemática, Mérida, Venezuela
    • 1998 Szeged Conference on Lattices and Universal Algebra

    1996 Fields Institute
    The Fall 1996 Algebraic Model Theory Program at the Fields Institute
    Photo key

    XI Simposio Latinoamericano Lógica Matemática
    1998 Szeged Conference on Lattices and Universal Algebra

  • Found Math

    I sent this photo in to the MAA Found Math series, and it was published! This looks very much like a geometric dissection proof of the Theorem of Pythagoras, but in the form of a discarded foldable card table.

    Pythagoras
    A proof without words found in the trash

  • What is Algebra good for?

    Algebra represents a shift into mathematical abstraction, providing tools necessary for further study, a focus on properties and principles that deepen our understanding of number, and also reasoning and insight that transfer to other domains of human knowledge.