Math 4433

(Intro. to Analysis I)

Summer 2010

Nine-banded Armadillo (Dasypus Novemcinctus). A family of these is currently living under the floor of the shed behind our house. Nine-banded armadillos always give birth to identical quadruplets: the original zygote splits, and then splits again, forming four copies which each grow into a baby armadillo. This image taken from

Instructor: John Albert
Office: PHSC 1004
Office hours: Mondays, Tuesdays and Thursdays from 2:30 pm to 3:30 pm (or by appointment)
Phone: 325-3782

Course information

Answers to problems on the third exam.

Answers to the questions in problem 4 of Quiz 3.

Writeup of a solution to one of the example problems I put on the board at the end of the last lecture.

Review sheet for the third exam.

Here are copies of Exam 3 from the last three summers: Exam 3 from 2007, Exam 3 from 2008, and Exam 3 from 2009. Students tell me they find these old exams useful for preparation, but you should keep in mind that I teach the course differently each year.

Review sheet for the second exam.

Review sheet for the first exam.

Syllabus for the course.



Sometimes I post an assignment in advance but change it in class the day before it's due. If you miss a class you should check this web page after class for the final version of the next day's assignment.


Due Date


1 Tuesday, June 8 2.1.15(b), 2.1.20(a,b)
2 Wednesday, June 9 2.2.2, 2.2.5
3 Thursday, June 10 2.2.16(a)
4 Tuesday, June 15 2.3.1, 2.3.10
5 Wednesday, June 16 2.3.5, 2.3.6
6 Thursday, June 17 2.4.1, 2.4.6
7 Friday, June 18 3.1.11, 3.1.12
8 Monday, June 21 3.1.15, 3.1.16
9 Tuesday, June 22 3.1.8, 3.1.9
10 Wednesday, June 23 3.2.3
11 Thursday, June 24 3.2.5
12 Monday, June 28 3.2.7, 3.2.9
13 Thursday, July 1 3.3.1, 3.3.2
14 Friday, July 2 3.3.4
15 Tuesday, July 6 3.3.15
16 Wednesday, July 7 3.3.11, 3.3.12
17 Thursday, July 8 3.3.13, 3.4.12
18 Friday, July 9 3.4.14
19 Tuesday, July 13 3.6.3
20 Wednesday, July 14 3.7.8
21 Wednesday, July 15 3.7.12, 3.7.14(a,b)
22 Wednesday, July 22 4.1.7
23 Friday, July 23 4.1.4
24 Monday, July 26 4.2.4
25 Tuesday, July 27 4.1.14
26 Wednesday, July 28 5.3.1, 5.3.2
27 Thursday, July 29 6.1.1(a,b), 6.1.2

References and Links

You can consult these for more information on analysis and the foundations of mathematics.

The founder of the subject of analysis, as we learn it in this class, was the French mathematician Augustin-Louis Cauchy (1789-1857). Here is a nice article about what Cauchy did, and why.

The distinction between rational and irrational numbers has a long history, going back to the Greeks in the time of Plato and before, but the Greek mathematicians of those times also thought about numbers very differently than we do today. In particular, they did not have our modern concepts of multiplication and division: it would not have made sense to them to think of 2/3 as a number that could be added to 7/8. In "The Mathematics of Plato's Academy", by D. H. Fowler, it's suggested that the Greeks of Plato's day would have based their idea of the ratio of two quantities on the process of "anthyphairesis", or successively removing squares from rectangles in the way we demonstrated in class. (The meaning of the Greek word "anthyphairesis" seems to be something like "two things taking turns removing parts from each other".) A Google search on "anthyphairesis" turns up several interesting commentaries on this suggestion, including a couple of favorable ones ( Plato's Theory of Number, by I. Bulmer-Thomas, and F. Gouvea's review of Fowler's book) and a not so favorable one (a review of Fowler's book by S. Unguru).

The article which I referred to in class, containing fourteen proofs of the fact that the series generated by 1/n^2 has sum pi^2/6, is this one by Robin Chapman. You can also find a link to this and other articles on the interesting Wikipedia page on this topic, titled "Basel Problem".