Math 629: History of Mathematics
Frank Sottile

• Opening Remarks:
  I am still waiting to hear directly from five of the seven groups. One member from each of those groups should send me an email message letting me know who are the members of their group.       These deal with the Greek's forays into number theory and their treatment of infinity. It is interesting to note that some of this work that they were fascinated with turned out to be duds (perfect numbers, while cute, are not particularly important), while polygonal numbers are simply not important or interesting (except for squares). On the other hand, Euclid's algorithm not only finds the gcd of two integers (and is the foundation for elementary number theory), it also finds the gcd of two univariate polynomials. Euclid's algorithm appears in modern algebra in the guise of Euclidean rings (this is a topic in the graduate algebra course I teach). It is also one of the bases for the very modern theory of Gröbner bases, which has led to a computational revolution in algebra in the past 50 years. Euclid's use of the Euclidean algorithm to give foundations for the theory of factorizations of integers, and then to use that to prove that there are infinitely many prime numbers is an example of how mathematical results build on each other. I really appreciated Stillwell's discussion connecting the Euclidean Algorithm to constructive solutions of Pell's equation for N=2. As he notes, we will see more of this when we look at mathematics from the Indian subcontinent.
        It is useful to read and re-read the second page of Section 4.1 in which mentions the Greek method to show that a particular number (e.g. limit of a sequence) could not have anything but a particular value α. This is their method of exhaustion. This indirect, but completely rigorous, method was used by Archimedes in his famous proof that π=π where π on the left is defined to be the ratio of the circumference of a circle to its diameter, and the one on the right, that of the area of a circle to the square of its radius. The Greeks knew that these ratios were constants, but it was Archumedes who showed that they are the same constant. (He was able to relate at least two other ratios involving surface area and volumes of spheres, showing that they too are equal to π.) Keep this in mind when we encounter The Calculus (particularly the meaning of its Latin root).
        I will be asking you to use exhaustion to prove a result about logarithms (via their definition as the value of a particular integral).
• Reading:
  • Chapters 3 and 4 of Stillwell.
• Assignment: Due Monday, February 5. (HW 3) It is OK to discuss this among yourselves. We all have something to learn from each other.
  Here is a .pdf and a LaTeX source of the assignment, and the same .pdf and a LaTeX source of the group assignment.
  The quiz (on Gradescope) is three exceptionally short questions with even shorter answers that you need to take after you have completed the reading.
  Do not hand in (this is for your group to discuss):
  1. Please do Stillwell's exercise 3.3.1, about the factors of a number of the form 2^n-1p, where p is a prime.
  2. 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. How many perfect numbers are there currently known?
  3. Let us call a positive number ample if the sum of its divisors exceeds the number itself. For example, 1+2+3+4+6=16>12, so twelve is an ample number.
     Show that any number of the form 2^n-1(2ⁿ-1) is either perfect or ample.
     Give an ample number not of this form.
  To hand in: We are using Gradescope for homework submission.
  1. Stillwell gives Euclid's elegant proof that there are infinitely many prime numbers, one of the great proofs in mathematics. Find and describe/explain a different proof that there are infinitely many prime numbers.
  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 (3.4.1, 3.4.2, 3.4.3, and 3.4.4). I also give you one computational question about a particular comtinued fraction on the homework.
  3. Please compute the value of the infinite continued fraction with only 1's in it (see the .pdf for the precise statement).
  4. 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?
  5. 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 (4.4.1, 4.4.2, and 4.4.3) 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?
         This is challenging. Do discuss it on Piazza. Let me provide you with a strong hint. Suppose that you want to prove that a particular region R has particular number A as its area. For this, you consider lower- and upper- approximations of the area of R. In this case, you use rectangles (Riemann sums) to get the lower and upper approximations.
     Now here is where the logic of the method of exhaustion comes in. You show that for any number L less than A, there is a lower approximation whose rectangles' areas add up to a number exceeding L, and the same, mutatis mutandis, for any number G greater than A. This shows that the area of R is A, and it is not by an infinitary process.
     I like to think of this method in the following way: To show a number (e.g. the area of R) is some area A, is to show that it cannot be less than A and that it cannot be greater than A.
     Contrast this to the Calculus, where you have Riemann sums that are not necessarily lower or upper approximations to the region R, but we have that the values of the Riemann sums converge to some number. In (this modern vierw of The Calculus), we do not need to know A, we find A in taking the integral, using Newton's Fundamental Theorem of Calculus.
     What makes The Calculus better—and this is what I teach, particularly in Calculus 3—is that by approximating some quantity (say flux across a membrane, or surfaces area, or aggregrate gravitational attraction, or ....) by Riemann sums which converge to some value, we have not only given a way to calculate the quantity we want, but, and this is important, actually defined that quantity.
     Think about this. What is the length of a curve that is not made up of arcs of circles or line segments? Calculus gives the reasonable and rigorous definition that the length is the limit of lengths of approximating polygons (collections of line segments) or arcs of circles. The same is true for other quantities in Science.
     Back to the question you were asked. From the definition of exhaustion, to show two things are equal, say log(a) and log(ab)-log(b), is to show how any lower approximation to one can be transformed into a lower approximation for the other, with the same area, and the same for upper approximations. (Note that, as a mathematician, I use log for the logarithm with respect to the natural base, Euler's number e.)

Last modified: Sat Feb 10 18:18:40 GMT 2024 by sottile