- Opening Remarks:
The exercise last week of trying to show the fundamental property of logarithms (log(a)+log(b)=log(ab))
using exhaustion is, I believe worthwhile.
It suggests that the Greeks could have discovered logarithms and their properties.
It also introduces us to the idea that algebraic properties of functions may be determined by transforming integrals, which reappears in
this book, from the sine function to the much more exotic elliptic functions that were a focus (and glory of) of 19th century mathematics.
This week, I will go a easy on you, but you can use this time to research your Term Paper and to try and
attend to the copious reading.
We will read about the mathematics between the Greeks and the Renaissance in Europe.
Stillwell concentrates on a small slice—the number-theoretic work in Asia (China and India).
We see linear Diophantine equations and the fantastic Chinese Remainder Theorem (which was rediscovered later in Europe and continues to
be important today).
The development of Pell's equation by Brahmagupta and Bhâskara II, and then the classification of rational triangles are all
triumphs of Human thought, and interesting topics.
Next week, we will jump into the mathematics developed in Europe that forms the basis for the mathematics of the past 500 years or
so.
Stillwell makes an interesting point at the start of Section 5.1; that while the Greeks influenced many of the main themes of modern
mathematics, there were not neessarily the first to people to consider these ideas, nor did they do it the best.
Stillwell skips two important developments:
- The origins and rise of the Indo-Arabic number system that we all use today, including the introduction of 0, and
- how that number system and the algorithms for multiplication made their way from the Islamic world of the 10th
century (give or take a couple of hundred years) to the medieval world.
Some of this is found in the interesting book
Capitalism and Arithmetic: The New Math of the 15th Century, by Frank Swetz.
The readings from St. Andrews fill in some of these gaps, and give a whole lot more insight on other mathematical developments from Asia.
I learned learned a lot about these topics from two books I own, Berggrens' Episodes in the Mathematics of Medieval Islam and Swetz's
Capitalism and Arithmetic: The New Math of the 15th Century.
- Note: This weekend, I leave Cambridge, and travel to Cardiff and Swansea (towns in Wales).
This includes attneding and speaking at a 1-day conference in Swansea, giving a talk to undergraduates, giving a colloquium talk at
Cardiff, and meeting with several collaborators and old friends in mathematics, and meeting someone who (with me) is on a
committee to select a winner of a mathematics award/prize in the field of combinatorics.
On Wednesday, I fly back to the US.
The point of this, besides lamenting that I am leaving my former college in Cambridge where I lived for 3 years, is that I will not
be able to respond to much from Sunday through Thursday of next week.
- Reading:
- Please watch The History of Mathematics, from minute 54 to
minute 93, on Chinese, Indian, and Islamic mathematics.
- Chapter 5 of Stillwell on Number Theory in Asia (China and India).
- St. Andrews pages on Indian Mathematics.
These are long and comprehensive; look at some of them, perhaps the overview, as well as the history of zero—it is interesting to
see how the greats struggled with this concept.
You can skip the two sub-pages on the number π—while fascinating, it is a bit off topic this week.
As Math history aficinados, do read them sometime.
The link Indian Mathematics—Redressing the balance by Ian G. Pearce, is an extensive treatise on Indian Mathematics, in
which the author argues that Indian Mathematics is wrongly maligned.
You can skip this—I did, even though I read many hours of other sources from this period when I prepared this week.
- With the same advice about skipping π, read the St. Andrews pages on
Islamic Mathematics.
- Assignment: Due Monday, February 12. (HW 4)
Here is a .pdf and a LaTeX source of the assignment.
You will need the image file if you LaTeX this (I have a postscript version, too).
The quiz (on Gradescope) is two exceptionally short questions with even shorter answers that you need to take after you have
completed the reading.
To hand in: We are using Gradescope for homework submission.
- Do exercises 5.4.5 and 5.4.6 in Stillwell.
It helps to recall the definitions.
Write ex for the usual exponential function from Calculus, where Euler's number e is approximately
2.7182818284590452353602874713527.
Then the hyperbolic cosine, cosh(x) of the real number x is
cosh(x):=(ex+e-x)/2.
Similarly, the hyperbolic sine, sinh(x) of the real number x is
sinh(x):=(ex-e-x)/2.
These functions satisfy (cosh(x))2- (sinh(x))2 = 1 (check this!),
and many other identities similar to the classical trigonometric identities.
You may need the sum formulas for hyperbolic functions.
The reason for this is that the hyperbolic functions are almost the same as the trigonometric functions, when viewed properly:
cos(θ):=(eiθ+e-iθ)/2, and similarly
sin(θ):=(eiθ-e-iθ)/2/i.
Here i is the complex imaginary unit, i2 = -1.
- I found the treatment of rational triangles and the proof of Herons's formula in the exercises to Section 5.6 in Stillwell to be
interesting.
Please do all four exercises.
(I have a soft spot for Heron's formula, having learned it in middle school, it is the one bit of my mathematical knowledge
that I have used in a practical consultation. I can tell the story on Piazza if anyone asks.)
- There is a marvelous generalization of the Pythagorean Theorem that I'd like you to try your hand on.
It is on the homework sheet, and was done by a mathematician from one of the traditions we are reading about.
Use similar triangles.
I realize that the figure may pose a problem.