Math 629: History of Mathematics
Frank Sottile

• Opening Remarks: Your book reviews are due this Wednesday.
      In your readings of Stillwell, I'd like it if you could also read the exercises, as well as the text. Stillwell puts some useful historical and mathematical gems in his exercises, and that we all have something to learn from thinking about them. Several good texts/reference works employ this trick of including additional material in the exercises.
      The historical notes at the end of the chapter provide some interesting color about the lives of the protagonists; they are likewise worth reading.
      This week's readings and exercises are all about solving equations, which has been one of the main uses of mathematics since antiquity. Of particular note was the early Renaissance explosion in what is now Northern Italy that led to solutions to the cubic and quartic equations in the space of just few years–this was the first advance in this topic in nearly a millenium, as well as the last for about 250 years. It also led to the modern understanding of and comfort with the exotic negative and complex numbers. It is worth keeping in mind that, at the start of this story, including Scipione del Ferro and Tartaglia, mathematicians avoided using subtraction or negative numbers. By then end, Rafael Bombelli was comfortable using complex numbers. When you hand your homwork in, I will be in Bologna at the oldest University in Europe. It figures into this story.
      It is in part because of these (and many other) advances in our ability to give/find/approximate/compute solutions to systems of equations, and the appearence of this in the mathematics cirriculum that a common view of a mathematician is "someone who solves equations all day". Of course, this is not the main task of mathematicians, neither academic nor those working in the private sector nor those teaching. For me, however, solving equations is a significant part of my research. I sometimes start a professional talk with the comment that I am the only person in the room who does what their mother thinks they do. Silliness aside, solving equations is important and serious business; we have seen a lot of this so far in our study of the history of mathematics; many surviving pieces of mathematics are aids to calculation, or devoted to methods to solve equations and obtain numerical answers.
• Reading:
  • Stillwell Chapter 6. As mentioned above, please read the exercises, as well as the historical notes at the end of the chapter.
  • Read the St. Andrew's page on Quadratic, cubic and quartic equations.
  • Read the St. Andrew's page on Tartaglia vs. Cardano. This is some of their correspondence. An interesting read. You may also skim stories of some of the other protagonists.
  • I have a short text that is a transcription of a math circle that I have given on solving the cubic. If you have any comments on this text, I would be happy to hear them, as I am interested in improving it.
  • Look up the recent discussion of a different way to teach how to solve a quadratic equation. (This is a Google exercise.) It is interesting to note that many people figured this out and use it---Po-Shen Loh's 'new' method is embedded in my notes on solving equations, for example. I think there is significant merit in teaching this method, as it uses one of the fundamental and visually obvious facts about parabolas. Also a method/algorithm that relies on such structure is, I think, more likely to be followed correctly than a formula, and it bolsters understanding.
• Assignment: Due Monday, February 17. (HW 6) To hand in: We are using E-campus for homework submission.
  1. Exercises 6.5.2 and 6.5.3. For 6.5.3, find the other two solutions (this will require reading my notes on cubics.)
  2. Here is another cubic to solve completely: x³=7x+6, and another: x³+2x+4i=0, where i is the imaginary unit, a square root of -1. You will find the example on the bottom of page 3 of my notes, as well as the formula (-i)³=i, useful for the second cubic.
  3. Do the exercises 6.7.1, 6.7.2, and 6.7.3 in Stillwell. These should be relatively short.
  4. Cardano led an interesting life. Look up some material on the web (two sources besides our book) and write something about his life, both mathematical and otherwise. Probably two paragraphs, and try to include some story that is not in Stillwell's potted biography.
  5. Bonus on Piazza: If anyone finds information about the story of the cubic and quartic that is interesting, yet omitted from the notes that I have, please write a short paragraph on Piazza about that.