Announcements
Final answers updated
I elaborated my answers to the "true-false" problems, so you might want to reread the answers: final1_answers.pdf.
Final exam and answers
There were four different final exams.
I wrote solutions for the first one final1_answers.pdf. The others are very similar.
Have a great break and see you next semester!
Answers to the practice final
Here are my solutions to the practice final practice_final_answers.pdf.
Answers to Homework 9
Here are my solutions to homework 9: hw9a.pdf.
Also, to prepare for class tomorrow, be sure to finish reading chapter 3.
Practice Final Exam
The final exam will be on Tuesday, December 17 from 4:00-6:00pm in Kiely 065 (the regular classroom).
As a reminder, we covered Chapters 1, 2, and 3 (except for sections 2.9-2.17) in the book, as well as the entire introduction and other supplementary topics on sets, logic, and functions.
I thought it might be helpful for you to have a practice final exam that includes samples of the kinds of problems that I expect you to be able to solve on your final exam. Don't restrict your studying to just the practice exam since it doesn't have all the topics we covered.
Answers to Problem Set 3
Here are my solutions to Problem Set 3: ps3a.pdf.
Lecture Tuesday Night
Professor John Conway will present a talk on Tuesday, December 10 at 8:30 p.m. in the Science Building, C205. Professor Conway is an excellent mathematician and an entertaining speaker. I encourage everyone to attend. Here's the abstract:
Abstract: The Hebrew calendar is very complicated. It uses the fact that the mean distance between new-moon instants is 29 days, 12 hours, 44 minutes and 10/3 seconds, and that 235 of these ”lunations” is almost exactly equal to 19 years. Despite this, I shall describe an algorithm by which one can convert mentally between the Hebrew and Roman calendars.
Also, let me reiterate the invitation to the Math department's end of semester party on Monday (Dec 9) at 12:30pm in Kissena I Room 120. You are all welcome!
Homework 9
Here are a few problems to work on over the weekend. hw9.pdf.
Work hard on them. Similar problems will appear on the final exam.
Math Department end of semester party
The Department of Mathematics cordially invites you to its end of semester party.
- Monday, December 9 12:30pm-1:30pm
- Room 120 - Kissena Hall I
Final Exam
The final exam will be on Tuesday, December 17 from 4:00-6:00pm in Kiely 065 (the regular classroom).
Three things
First, here are my solutions to Problem Set 2: ps2a.pdf.
Second, since I didn't have time to prove it in class today, please work through the proof of Theorem 2.9 in the book.
Third, the due date for Problem Set 3 will be extended from Tuesday December 3 to Thursday December 5. That's in nine days, but get started early since you'll have to read sections 3.1-3.5 carefully to do problems 1, 2, 3a, and the bonus (and you'll have to at least skim a few other sections to find the theorems you're asked to state in 3b,c,d).
Problem Sets 2 and 3
I wrote two problem sets ps2.pdf and ps3.pdf.
Problem set 2 is due on Tuesday, November 26. I will collect it and grade it. The first question is about trigonometry, the relevant material is in sections 2.5, 2.6, and 2.7 and the rest involves integration and material that has been covered already, although you may find the exposition in section 2.18 about the integral as a function of the upper limit of integration helpful. There is a bonus question.
Problem set 3 is due on Tuesday, December 3. I will collect it and grade it. It is about continuous functions, the subject of Chapter 3. There is a bonus question.
Love and Mathematics
In the most recent New York Review of Books, the philospher Jim holt begins his review of Ed Frenkel's book Love and Math: The Heart of Hidden Reality
For those who have learned something of higher mathematics, nothing could be more natural than to use the word “beautiful” in connection with it. Mathematical beauty, like the beauty of, say, a late Beethoven quartet, arises from a combination of strangeness and inevitability. Simply defined abstractions disclose hidden quirks and complexities. Seemingly unrelated structures turn out to have mysterious correspondences. Uncanny patterns emerge, and they remain uncanny even after being underwritten by the rigor of logic.
Later, he adds
Mathematics is abstract and difficult; its beauties would seem to be inaccessible to most of us. As the German poet Hans Magnus Enzensberger has observed, mathematics is “a blind spot in our culture—alien territory, in which only the elite, the initiated few have managed to entrench themselves.” People who are otherwise cultivated will proudly confess their philistinism when it comes to mathematics. The problem, says Frenkel, is that they have never been introduced to its masterpieces. The mathematics taught in school, and even at university (through, say, introductory calculus), is mostly hundreds or thousands of years old, and much of it involves routine problem-solving by tedious calculation.
That bears scant resemblance to what most mathematicians do today. Around the middle of the nineteenth century, a sort of revolution occurred in mathematics: the emphasis shifted from science-bound calculation to the free creation of new structures, new languages. Mathematical proofs, for all their rigorous logic, came to look more like narratives, with plots and subplots, twists and resolutions. It is this kind of mathematics that most people never see.
Read the whole review at http://math.berkeley.edu/~frenkel/Love-and-Math-NYRB.pdf and check out the book.
Homework for the weekend
For the weekend, read sections 2.1-2.11 in Chapter 2. That's pages 88--111. Here are some exercises:
- Exercises 7, 9, 15, 17 in section 2.4 on page 94
- Exercises 4, 5, 15, 23, 26 in section 2.8 on pages 105-107
- Exercises 1, 5 in section 2.11 on page 111
We'll start class on Tuesday with a short quiz on this material.
Exam 3 and solutions
Here is the third exam and solutions.
3:00-4:00 office hour cancelled
I have a meeting today until 4:00 today so unfortunately, that office hour is cancelled. I will be taking questions in class today and if you don't get your question answered then, I'll be available for the hour after class.
Solutions to Homework 8
Here are solutions to Homework 8: hw8a.pdf.
Solutions to old midterm
Last week, I posted an exam from 2004: 157midterm_2004.pdf. This morning, I posted solutions: 157midterm_2004_answers.pdf.
Homework for the weekend
We've pretty much wrapped up all of the material in Chapter 1 "The concepts of Integral Calculus". Be sure you've carefully read through sections 1.27. Here are some exercises:
- Exercises 10, 11, 15, 16, 18, 20, 21, 23, 25, 26 in section 1.26 on pages 83-84.
Reminder: Exam next week
We will have an exam in class on Thursday, November 14 on the material in Chapter 1 "The concepts of Integral Calculus".
I posted an exam from 2004: 157midterm_2004.pdf. But that exam doesn't quite cover everything, so be sure you've worked through the whole chapter and done the exercises posted above. We'll reveiw on Tuesday, and I'll answer questions.
Selected answers to Homework 7
Here are some selected solutions to Homework 7: hw7a.pdf.
Homework 8
Here are a few problems to work on over the weekend. hw8.pdf.
The first problem is review. Then, spend significant amount of energy reading and studying the sections in the text. The exercises that follow the reading assignments should be fairly short tests of the reading. Approach them thoughtfully.
Problem Set 1 answers
Here are my solutions to Problem Set 1: ps1a.pdf.
Problem Set 1
I wrote a problem set ps1.pdf. It's due on Thursday, October 31. I will collect it and grade it. Half of the problems (the first three) have been assigned already and involve the material in sections 1.8-1.14 (pages 60-70) of Apostol's book, and half of the problems involve products and unions of sets and are new. I called this assignment a "problem set" instead of a take-home exam to signal that it's okay to work together on the problems.
Homework 7 and answers to homework 6
I collected the problems that I already assigned this week, including the exercises I gave during the class on Tuesday about functions and called it homework 7. Complete it by Tuesday, October 29: hw7.pdf.
Also, I revised the solutions to homework 6, adding a bit more. In particular, I wrote a complete solution to the problem from the textbook about the zeroes of polynomials. See the updated version hw6a.pdf.
Homework for the week
To accompany the lectures this week, read sections 1.8-1.14 (pages 60-70) in Apostol and do the following exercises:
- Exercises 1, 3 in 1.11 on page 63.
- Exercises 1, 2, 5, 11, 13-17 in section 1.15 on pages 70-72.
Exam 2 and Exam 2 Answers
Here is the first exam and solutions.
Selected solutions to homework 6
Here are some selected solutions to homework 6:
Reminder
I made the announcement in class on Tuesday, but I want to remind you that we'll have an exam in class on Thursday October 17. It will be fairly short-it probably won't take the entire class period. The material on the exam will be from what we've covered in the class so far. That's everything that was on the first exam, plus the material on Homework 6 about functions.
Homework 6
Here is some homework for Thursday:
Exam 1 Answers
Here are my solutions to the first exam exam1-answers.pdf.
Exam 1
Here is the first exam exam1.pdf.
It is due at the beginning of class on Tuesday, October 8. If you have a questions (or find a typo) post it to the forum so that everyone in the class can read the question and answer.
Solutions to practice Exam 1
The first exam, originally scheduled to be in class on October 3, will be a take home exam, due on Tuesday, October 8. Many of the problems from the practice exam were discussed in class on Tuesday, October 1. Here are some written solutions practice_exam_1_answers.pdf.
Practice Exam 1
As indicated on the syllabus, the first exam will be in class on October 3. To help you prepare, I wrote a practice exam.
The practice exam, and the selected solutions to the homework problems that I wrote, are collected on the exams page.
Selected answers to homework 5
Here are some selected answers for homework 5, including the problem about binomial coefficients.
Homework 5 and answers to homework 4
Here are some selected answers for homework 4 as well as a new assignment (homework 5). Be sure to get started on the new assignment to prepare for class on Thursday, September 26.
- Answers to homework 4. In pdf: hw4a.pdf and LaTeX: hw4a.tex.
- Homework 5. In pdf: hw5.pdf and LaTeX: hw5.tex.
Also, as indicated on the syllabus, there will be an exam in class on Thursday, October 3.
Homework 4
Here is some homework for the weekend:
- Read all of part 3 of the introduction (up to page 32).
- Do exercise 5 in I3.5 on page 21. This has already been assigned, but this is a good problem to practice writing down a careful solution.
- Do exercises 1, 3, 7, 8 in I3.12 on page 28. For these, you'll want to have worked through sections I3.9, I3.10, I3.11 carefully beforehand.
Remember, you can post questions and answers (and edit them) on our Forum.
Solutions to homework 3
Homework 3
For Tuesday, September 17, keep working on the reading and problems from homework 2. I also wrote some problems about propositional logic and sets for you to work with.
I'll post answers in over the weekend also. I will try to finish the preliminary introductory material next week, so we can get started on calculus!
Homework 2
For Tuesday, September 10, read through sections I3.1-I3.5 on the field and order axioms of the real numbers and do exercises 1-10 in Section I3.3 (on page 19) and exercises 1-10 in Section I3.5 (on page 21). Be sure you've worked through the reading and exercises from Homework 1 as well.
Remember, there is no class on Thursday, September 5. I will go over Homework 1 and 2 in detail in class on September 10.
(Partial) solutions to Homework 1
Here is my solution to problem 20 on page 15. In pdf: hw1a.pdf and in LaTeX: hw1a.tex
I added a solution to problem 1d on page 8 and also updated my other solutions. In pdf: hw1b.pdf and in LaTeX: hw1b.tex
Homework 1
So that you can get started studying right away, I've reproduced, with permission of John Wiley & Sons, Inc., the first thirty pages of the introduction of the textbook here: intro.pdf. Watch for typos!
Read through Parts 1 and 2 of this introduction (pages 1-16) and do some exercises to check your understanding
- Exercises 1, 3 in I1.4 on page 8.
- Exercises 2, 6, 20 in I2.5 on page 15
Information about Piazza
The math 157 discussion forum is hosted on Piazza. Questions about the course should be posted on the forum. The link to the forum is piazza.com/qc.cuny.edu/fall2013/math157.
After you've signed up, you can ask and edit questions and answers using the Q & A page.
Welcome to Math 157
This page will contain announcements for Math 157. For now, take a look at the syllabus, which has practical information such as the textbook, office hours, and important dates.