Math 629: History of Mathematics
Frank Sottile

• Opening Remarks:
        Please get started on your Book Review. Let me know which book you choose.
        The reading is a bit light this week; chalk that up to my giving you some space to read your book for the book review. Also at fault is my travels: (I am writing this on a train between Frankfurt and Ulm on a new computer—I had an accident with a glass of water last night—with a German keyboard, and there is no internet on the train.)
• Reading:
  • Chapters 3 and 4 of Stillwell.
• Assignment: Due Monday, February 13. (HW 5) It is OK to discuss this among yourselves. We all have something to learn from each other.
  To hand in: Email a .pdf to both Bennett Clayton bgclayton@math.tamu.edu and Frank Sottile. fjsteachmath@gmail.com
  1. I've always been a fan of perfect numbers. Give the defining property of a perfect number. Prove that if 2ⁿ-1 is a prime number, p, then 2^n-1p is a perfect number. Write down four perfect numbers. Bonus: How many perfect numbers are there?
  2. Continued fractions are somewhat of a lost art. It is known, for example, that the truncations of a continued fraction for a real number α are the best rational approximations to α. Do all four exercises in Section 3.4 of Stillwell.
  3. Consider Archimedes' quote from The Method: "It is of course easier to supply the proof when we have previously acquired some knowledge of the questions by the method, than it is to find it without any previous knowledge." What was "the method" he is referring to? What does his quote say about the role of experimentation or studying examples in Mathematics?
  4. Archimedes' use of exhaustion to determine areas predated the Calculus by over 1800 years. Try your hand at this method by doing the exercises in Stillwell's Section 4.4 to prove the formula for the logarithm of product of rational numbers. Make sure to use exhaustion and not calculus tricks. Why did I restrict to your showing the formula for rational numbers?