## 18.100A -

### 18.100A Recent Syllabus (Fall 2015)

(Adobe Acrobat -- pdf file)

### 18.100A Recent Information and Rules (Fall 2015)

( Adobe Acrobat -- pdf file )

This course is an introduction to devising mathematical proofs and learning to write them up. It is primarily for students with no prior experience with this.
The class usually contains students from years 2,3,4,and G (grad students) -- about equal numbers of each. Sometimes freshmen also take it.

Graduate students (not in Course 18 (math)) should register for it using the number 18.1001 to get graduate credit for the course. The two numbers both represent a single course with one room, one set of rules and assignments.

Students in the class are from different majors -- about 1/4 math majors, the rest have been in various types of science and engineering majors, also economics and management.

The subject matter for the first 2/3 of the syllabus (up to Exam 2) is the proofs of one-variable calculus theorems and arguments which use these theorems.
The emphasis is on the use of inequalities to do estimation and approximation, two basic tools of analysis.
It is assumed that students know ordinary calculus fairly well, or once knew it and will review it when they need to. Calculus is used from the beginning as a source of examples.

The last third goes beyond calculus, getting into uniform convergence of series of functions, to justify differentiation and integration of infinite series term-by-term.
There is similar work involving integrals depending on a parameter, to justify differentiating under the integral sign with respect to the parameter.
(Differentiating the Laplace transform F(s) = L(f(t)) with respect to the s-variable is an example.)

Toward the end, there is a brief introduction to point-set topology (18.100B begins with this), which is used in upper-level courses having an analysis prerequisite.
For students who are interested, there is also an even briefer introduction to sets of measure zero and the Lebesgue integral.

#### 18.100A or 18.100B?

The best way of getting a feeling for the difference between 18.100A and 18.100B is to look at the corresponding textbooks.

The book for 18.100B is Rudin: "Principles of Mathematical Analysis", available for inspection at the Kendall Square Tech Coop; or on Amazon for sample pages.

The book for 18.100A is described below; it is available in an inexpensive printing at the Tech Coop or online at Quantum Books or Amazon.
In content it follows the 18.100A description above closely.
Three links are given below, so you can get some idea of its style, what's in it, and its rigorous but hopefully simplified approach to limits, as described toward the end of the Preface, and in Chapter 3.

#### 18.100A Textbook: Mattuck --- Introduction to Analysis

(CreateSpace, 2013; available on Amazon), (previously pub. by Pearson/Prentice-Hall div., 1999) Website: Introduction to Analysis

Its five-page Preface can give some idea of the book's origins and its (hopefully) user-friendly style: Preface .

The Table of Contents tells you what's in it, and what's not, and the order of topics: Table of Contents .

Leafing through some of the early chapters of this book and of Rudin will give you a further idea of the differences in style and content between the A and B versions of Introductory Analysis.
Here are a few sections from Chapters 1-3 -- about 15 pages in all -- to give you a sample of the writing style: Sample sections.

Chapter 1: Real Numbers and Monotone Sequences
Chapter 2: Estimations and Approximations
Chapter 3: The Limit of a Sequence

Corrections to the first seven printings by Pearson are on the book's website: List of Corrections.
The eighth printing by CreateSpace is the same book as the earlier printings, with the following changes:
--- All significant corrections have been incorporated;
--- A three-page Appendix F has been added;
--- The book's current price on Amazon is \$15.

Helpful features of the book: These are described briefly on the Amazon website for the book (Google: Mattuck Introduction to Analysis and select the topmost link) or in more detail in the "Preface" link given earlier.
On the Amazon website mentioned above, there are several reviews that you can read.

It also gives a description for students, instructors, and general readers of what's in the book and who it's for.
You can also find this description at Introduction to Analysis .

Posted on the book's website (see the link just given) are the first three chapters and Appendix A from the 8th printing, for the convenience of students who will not have a copy of the book before the first few class meetings.

What's below is more MIT-specific about how the subject was run in the Fall 2015 semester.
Further details are given in the "Information and Rules" handout (see link to it at the beginning of this page). This will be revised for the Fall 2016 semester.

### Fall 2015 Information

Exams: There are two 80-minute exams, and a 3-hour final.

Assignments: The current homework assignment is posted here after class. This allows for some flexibility in content and difficulty, and for feedback from the class members.
Earlier posting will be done when possible.

Homework is due in class twice weekly, on Monday and Friday, and returned graded at the following class meeting.
There are usually from 3-6 problems, depending on their difficulty or length, or whether it's an assignment due Monday or Friday.
Sometimes "Questions" are included (exercises having solutions at the end of the chapter), as an aid in learning how to write up solutions, and as a source of hints.
Since the homework is really where the learning takes place, and timely feedback is essential to improving, handing in 3/4 of the assignments when they are due is a requirement for passing; any exceptions have to be for cause, and arranged in advance. Students who are accepted into the class late have to make up the missed assigments.

The textbook is by and large an adequate substitute for class attendance; students in the past have found it sufficiently clear. A few just read the book, get the assignments online here, hand them in and retrieve their graded assignments in class, just before it starts. Others slip the homework under my door before class, retrieving the returned homework from a box outside my door. See the link near the top "Information and Rules" for more details.

Subject Prerequisites: The Subject Catalog says Calculus I and II; or 18.014 (Calculus I with Theory) and Calculus II (Corequisite: taken simultaneously with 18.100A). This latter alternative is incorrect for 18.100A: since 18.014 and 18.100A have similar material, students who have had 18.014 should not also take 18.100A; they should take 18.100B or C instead. Further explanation is given below.

Upperclassmen (i.e., students in years 2,3,4,G) will almost always have credit for Calculus I and II, and will have had at least one or more additional math subjects, like Differential Equations, Linear Algebra, Complex Variable, Probability, Statistics, or an introductory Discrete Math subject like 18.062J. Any of these will help provide "mathematical maturity", which will help with 18.100A, but they are not prerequisites.

Even the requirement of Calculus II is primarily for mathematical maturity; almost nothing is used in 18.100A except what's in 18.01 (Calculus I) and high-school mathematics; only in the last two weeks is reference made to 18.02 (vector and double integral topics in Calculus II)) and 18.03 (the Laplace transform topic in Differential Equations).

Freshmen taking Calculus II with a standard AP BC Calculus I background and a desire to see and make up proofs can therefore also take 18.100A instead of 18.014 (Calculus I with Theory).
18.100A and 18.014 are similar; the difference is that 18.100A assumes Calculus I, but 18.014 does not; the corresponding textbooks are written to reflect this diffference. From the above, it follows that students (at all levels) with a good background in proofs (from summer "math camps" for instance, or a proof-oriented "honors" Calculus course in high school, or 18.014, or more advanced math subjects) should study Analysis in the 18.100B or C versions.

Book Prerequisites: The textbook assumes knowledge of standard K-12 mathematics, plus one-variable calculus -- differentations and easy integrations (including the natural log, exp, and trig functions), with standard applications to finding rates, maximum points, areas, and simple volumes, plus an intuitive idea of what a limit is. These things are used from the beginning to provide examples, but the aim of the book is to describe the theory beind all of this -- to teach you how to read mathematical proofs and how to find and write them down yourself, using real analysis as the mathematical material.