Math 629: History of Mathematics
Frank Sottile

• Opening Remarks:     Do think about your Reflections on the History of Mathematics paper. Actually, do more than merely think this through, but start your planning, as the deadline will creep up on you.
      Our readings this week concentrate on the Calculus, continuing from last week. Chapter 10 in Stillwell's book covers many of the topics that are taught in the Calculus, as well as quite a lot about formulas involving infinite processes, including infinite sums and infinite products. Hopefully, you all recall how to derive the formula for the sum of a geometric series (both finite and infinite)–it is good to recall that before reading much of the chapter. (You can ask for help on Piazza.) The top of page 186 (my version, but in Section 10.2 on power series), where Stillwell talks about the definitions of logarithm, arcsin, and arctan in terms of integrals, is important. Once logarithm and the other functions are defined, one may consider their inverse functions, the exponential, sine (and cosine) and tangent. Furthermore, identities of the original integral in which the limits are transformed lead to famous formulas such as e^a+b = e^ae^b, and the angle addition formulas for sine and cosine and tangent.
  Section 10.4 contains examples where audaciously false (or unsubstantiated/unsubstantiatable) reasoning leads to correct results. The Numberphile video below (and Exercise 6 on the homework) explores a modern version of that with a twist.
  Notice the role of partial fractions in the formula for Fibonacci numbers, which is a bit different from how generating functions for recursive sequences are often used to get formulas (this is a common topic in combinatorics classes, which I sometimes teach).
  There is also an historical perspective on the Calculus, which you see starts before Newton, and continues until the 18th century, which we will eventually get to. For those of us who teach or will teach calculus, this additional material should be helpful in giving us, if not our students, some perspective on the material and the times of its development.
• Reading:
  • Chapter 10 in Stillwell's book.
  • J. V. Grabiner, "The Changing Concept of Change: The Derivative from Fermat to Weierstrass", Mathematics Magazine 56 (1983) 195–206.
    I met Judy Grabiner in graduate school (her husband was a mathematical friend of my then Ph.D. advisor, Dr. Graham Allan at Cambridge), and later her son was a postdoc with me at the MSRI in 1997. I have found her articles on history of mathematics to be interesting reading, and I hope that you will like this one from 41 years ago.
  • Watch the entertaining Numberphile video on 1+2+3+....
  • Look up some sources about logarithms. What were they used for? Who invented them?
    Up until the 1970s logarithms were commonly used as a shortcut to multiplication, because the logarithm turns multiplication (hard) into addition (easy). (For those who have had modern algebra, this is because the logarithm function is a group isomorphism from the multiplicative group of the positive real numbers to the additive group of the real numbers—This is a consequence of the formula that we proved about logarithms using exhaustion.)
    When I was a student in high school, they had just stopped teaching logarithms as a tool for calculation. Perhaps more interesting is that our sensory perceptions, specifically hearing and sight, are on a logarithmic scale. (As are the scales for loudness and Earthquake strength.) You might want to think about that as you look up material and do the homework problem on the distribution of data (Benford's Law). Here is an award-winning article about Benford's Law.
• Assignment: Due Monday, March 4. (HW 07)
  Here is a .pdf and a LaTeX source of the assignment.
  The group assignment is a fun project involving Benford's Law. Here is a .pdf and a LaTeX source of the group assignment. You can find a .html version of it here, and it has links to a .pdf for doodling on.
  You may also want the image files TenScale.pdf, NormalScale.pdf, and LogScale.pdf.
  
  I will have a shorter assignments the next two weeks as several people are traveling in March, (Spring Breaks).
  To hand in: We are using Gradescope for homework submission.
  1. Exercise 10.5.1 on page 192. This shows a limitation of power series, which is clear if you look at the graph of the function y=√x (  =   x^1/2 ).
  2. Everybody loves the Fibonacci sequence (from a problem of Fibonacci about rabbits; it is interesting to look up the source.) Do the three problems in Section 10.6 in Stillwell about this interesting sequence.
  3. Do Exercise 10.7.3. For this, use the formula for ζ(1-s) given at the bottom on page 196. You have to understand that Γ(n) = (n-1)!, when n is an integer, such as 2, as well as the remarkable formula for ζ(2), as discussed at the start of Section 10.4.
  4. Write a coherent couple of sentences about at least one thing that is completely wrong with the Numberphile video. (For the critical thinkers, this is easy.)
     This should not be hard mathematically, but may cause some conceptual reorientation.

Last modified: Wed Feb 28 04:05:39 PST 2024 by sottile