Tuesday, Feb. 18

Read: Burton, Section 2.2, The Greatest Common Divisor
Dunham, Chapter 2, Euclid's Proof of the Pythagorean Theorem, at least through Proposition 1.1 and its proof (pages 27-37)

Do: Burton, Exercises 2cd, 3, 5, 13b, 18, 20bf from Section 2.2

Next quiz will focus on number theory (Burton) with just a bit from Dunham.

Thursday, Feb. 20

Read: Burton, Section 2.3, The Euclidian Algorithm
Dunham, Chapter 2, Euclid's Proof of the Pythagorean Theorem

Do: Burton, Exercises 1, 2ab, 3, 8, 10, 12 from Section 2.3

Get acquainted with computer resources exercise: Visit your assigned Website (see reverse). Report (in one or two pages) (1) an interesting piece of mathematics history you found there, and (2) what resources are there that you and your classmates might find useful in future mathematics history research.

Next quiz will include questions about both number theory (Burton) and geometry (Dunham).

Tuesday, Feb. 25

Read: Burton, Section 3.1, Fundamental Theorem of Arithmetic
Re-read: Dunham, Chapter 2, Euclid's Proof of the Pythagorean Theorem

Do: Burton, Exercises 2, 3abc, 4, 5, 12 from Section 3.1
Extra credit: Exercises 10, 19 from Section 3.1

Hentschke Seminar, today, 4 p.m., Hentschke 204
Speaker: Dr. Beverly West, Cornell University / Topic: Differential Equations
Refreshments served beforehand!

Next quiz, 12-14 points worth of number theory (Burton), 6-8 points worth of geometry (Dunham, Chapter 2), for a total of 20 points.

Thursday, Feb. 27

Read: Burton, Section 3.2, Sieve of Eratosthenes (and infinitude of primes)
Dunham, Chapter 3, Euclid and the Infinitude of Primes

Do: Burton, Exercises 2, 3, 5, 9, 12 from Section 3.2

Get acquainted with a mathematician and with library resources exercise: Write a brief biographical report on a mathematician of your choice.

Next quiz: Guess which theorem and proof will appear on the quiz!

Tuesday, March 4

Read: Burton, Section 3.3, The Goldbach Conjecture
Re-read: Dunham, Chapter 3, Euclid and the Infinitude of Primes
Begin reading: Dunham, Chapter 4, Archimedes' Determination of Circular Area

Do: Burton, Exercises 3, 6, 9, 10, 13, 19, 26a from Section 3.3
Extra credit: Exercises 25, 27 from Section 3.3

Hentschke Seminar, today, 4 p.m., Hentschke 204
Speaker: Dr. Lisette de Pillis, Harvey Mudd College
Title: Picturing Algorithm Efficiency: How Fast Is My Black Box?
Refreshments served beforehand!

Next quiz covers recent number theory, and a bit more from Dunham, Chapter 3.

Thursday, March 6

Read: Burton, Section 2.4, The Diophantine Equation ax + by = c
Dunham, Chapter 4, Archimedes' Determination of Circular Area
Archimedes, Quadrature of the Parabola, Proposition 23

Do: Burton, Exercises 1, 2a, 9ab from Section 2.4

Archimedes Exercise: Give a modern interpretation of the statement and proof of Proposition 23 (from Archimedes' Quadrature of the Parabola handout).

Extra credit: At right (sorry, WWW readers!) is the title page of one of the mathematics textbooks Lincoln may have used in school, Nicolas Pike's Arithmetic. A copy of this text (not Lincoln's own copy, unfortunately) resides in the Heritage Room of the Smiley Public Library, 125 W. Vine St., Redlands, where it is part of the "Books Lincoln Read" collection. Go to the Heritage Room of the Smiley Library (call first to make sure the Heritage Room is open: 798-7632) and ask to see Nicolas Pike's Arithmetic from the "Books Lincoln Read" collection (it's not on the shelves). Handle the book very carefully: make sure that your hands are very clean and that you do not crack the spine (do not force the book open all the way if it doesn't want to go!), smudge or tear the pages, etc. Take notes with pencil only (no pens!). I think you'll be surprised to see how much algebra and combinatorics the text contains, despite its title. Pick a favorite topic from the text and write a short report on how Pike's presentation of the topic is similar to and/or differs from how the topic usually is presented today. You may complete this project at any time during the semester.

Next quiz: Really big quiz (some might call it an examination)!

Tuesday, March 11

Re-read: Dunham, Chapter 4, Archimedes' Determination of Circular Area

Do: Burton, Section 3.2, Exercises 3, 5, 9, 12 (again)
Burton, Section 3.3, Exercise 6 (again)

Next quiz covers Dunham, Chapter 4, Archimedes' Determination of Circular Area.

Thursday, March 13

Read: Burton, Sections 4.1, 4.2, Carl Friedrich Gauss, Basic Properties of Congruence
Dunham, Chapter 5, Heron's Formula for Triangular Area

Do: Burton, Section 4.2, Exercises 1, 2, 3, 4a, 13

Next quiz covers Gauss, congruences, and Heron's formula---mainly congruences.

Tuesday, March 18

Read: Burton, Section 4.4, Linear Congruences
Re-read: Dunham, Chapter 5, Heron's Formula for Triangular Area

Do: Burton, Section 4.4, Exercises 1bcd, 5, 14, 18

Hentschke Seminar, 4 p.m., Hentschke 201
Speaker: Dr. David Barsky, California State University, San Marcos
Title: Introduction to Percolation Theory

Next quiz covers congruences (Burton, Chapter 4) and a bit on Dunham, Chapter 5.

Thursday, March 20

Read: Burton, Sections 5.1, 5.2, Pierre de Fermat, Fermat's Factorization Method
Al-Khwarizmi, from The Book of Algebra and Almucabola

Do: Burton, Section 5.2, Exercises 1b, 3, 4
Al-Khwarizmi Exercises: See below.

Next quiz covers linear congruences and Fermat factorization.

Al-Khwarizmi Exercises

(1) For each of Chapters I, II, III, and IV, give the general form of the algebraic equation al-Khwarizmi is solving.

(2) What specific equation does al-Khwarizmi solve in Chapter VI? How exactly does he
solve this equation? Answer this question by translating al-Khwarizmi's work into algebraic equations and labeling the three diagrams he provides. What do we call this technique today?

(3) In Chapter VI, al-Khwarizmi uses a set of diagrams in which, in the middle diagram,
four corners are missing. Draw a different set of diagrams illustrating the solution technique in which, in the middle diagram, just one corner is missing.

(4) Correct the formula in Struik's Note #5, page 203.

Tuesday, March 25

Read: Burton, Section 5.3, The Little Theorem
Begin reading: Dunham, Chapter 6, Cardano and the Solution of the Cubic
Levi ben Gerson, from the Maasei Hoshev (Art of the Calculator)

Do: Burton, Section 5.3, Exercises 1, 3, 2a, 4d, 6, 7, 11

Next quiz covers Fermat's Little Theorem with a bit from al-Khwarizmi.

Thursday, March 27

Read: Burton, Section 5.4, Wilson's Theorem
Levi ben Gerson, from the Maasei Hoshev (Art of the Calculator)
Katz, Mathematics Around the World
Dunham, Chapter 6, Cardano and the Solution of the Cubic

Do: Burton, Section 5.4, Exercises 1, 3, 5b, 10a, 11, and
Levi ben Gerson Exercises, below.

Next quiz covers ben Gerson, Katz, and Cardano.

Levi ben Gerson Exercises

(1) Levi ben Gerson states five theorems, numbered 41, 42, 63, 64 and 65. Re-state each result using modern notation---that is, using notation!

(2) Does Levi ben Gerson prove his theorems? Justify your answer. Does Levi ben Gerson prove his theorems using the Principle of Mathematical Induction? Justify your answer.


Tuesday, April 8

Read: Burton, Section 7.1, Leonhard Euler; Section 7.2, Euler's Phi-Function
Begin reading: Dunham, Chapter 7, A Gem from Isaac Newton

Do: Burton, Section 7.2, Exercises 1, 4ab, 8, 15
Extra credit: Exercises 13, 14

Hentschke Seminar, 4 p.m., Hentschke 201
Speaker: Dr. Bela Bajnok, Gettysburg College
Title: The Boolean Lattice Model of Communication Networks

Next quiz covers Euler's phi-function and Dunham, Chapter 7, through page 165.

Great Theorem paper topic due Thursday, start of class. Details elsewhere.

Thursday, April 10

Read: Burton, Section 7.3, Euler's Theorem

Do: Burton, Section 5.3, Exercises 1, 3, 11 (again!)
Burton, Section 7.3, Exercises 2, 7, 12ab
Extra credit: Exercise 5 from Section 7.3

Tuesday, April 15

Read: Burton, Section 7.4, Some Properties of the Phi-Function (through page 137 only)
Dunham, Chapter 7, A Gem from Isaac Newton

Do: Burton, Section 5.3, Exercises 2a, 4d, 6b, 7 (again!)

Newton Exercise: Use Newton's Binomial Theorem to expand (1+x)^4, (1+x)^(-2), and (1+x)^(1/2). Give the first 5 terms of the latter two expansions.

Next quiz covers Dunham, Chapter 7, A Gem from Isaac Newton.

Thursday, April 17

Read: Burton, Section 7.5, An Application to Cryptography (at least through page 145)
Dunham, Chapter 8, The Bernoullis and the Harmonic Series

Do: Burton, Section 7.5, Exercises 1-8

Next quiz covers cryptography, the Bernoullis, and a bit on the phi-function.

Great Theorem paper first preliminary report due tomorrow

Tuesday, April 22

Re-read: Burton, Section 7.5, An Application to Cryptography
Dunham, Chapter 8, The Bernoullis and the Harmonic Series

Do: Burton, Section 7.5, Exercises 1-3, 5-7, 9-11, 13-15
Extra credit: Exercises 4, 8, 12, rest of 7

Next quiz: Dunham, Chapter 8, The Bernoullis and the Harmonic Series, and cryptography.

Next class: Bring your first or second preliminary report (rewritten and/or reprinted, if you like) for peer review.

Thursday, April 24

Read: Burton, Section 8.1, The Order of an Integer Modulo n
Dunham, Chapter 9, The Extraordinary Sums of Leonhard Euler

Do: Burton, Section 8.1, Exercises 1b, 2b, 4 (first part only), 10-12
Extra credit: Exercises 5, 6a, 7 (4k+1 only), second part of 4

Next quiz covers Dunham, Chapter 9, The Extraordinary Sums of Leonhard Euler.

Great Theorem paper second preliminary report due tomorrow


Tuesday, April 29

Re-read: Burton, Section 8.1, The Order of an Integer Modulo n
Dunham, Chapter 9, The Extraordinary Sums of Leonhard Euler

Last Hentschke Seminar, today, 4 p.m., Hentschke 201
Speaker: Dr. Robert Stein, California State University, San Bernardino
Title: Interpolation: Wallis and Newton

Next quiz covers Dunham, Chapter 9, The Extraordinary Sums of Leonhard Euler, and also will include one computation of the order of an integer modulo n.


Thursday, May 1

Read: Burton, Section 8.2, Primitive Roots for Primes
Dunham, Chapter 10, A Sampler of Euler's Number Theory

Do: Burton, Section 8.2, Exercises 1, 2, 3, 10

Next quiz covers Dunham, Chapter 10, A Sampler of Euler's Number Theory, and also will include computation of the primitive roots of a prime p.

Great Theorem paper first draft due tomorrow

Tuesday, May 6

Great Theorem presentations

Thursday, May 8

Really Big Quiz covering Burton, Chapters 7 and 8; Dunham, Chapters 7-10; and any related material discussed or assigned in class

Great Theorem paper final draft due Monday

Tuesday, May 13

Great Theorem presentations


Thursday, May 15

Great Theorem presentations

Saturday, May 24, 3 p.m.

Really, Really Big Quiz (Final Examination), our classroom



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