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January's homeworks

• Homework 3: 2.1.4.5 and extra problems (due 31 January)
• Homework 2: 2.1.3.7 (using a calculator!), 11; 2.1.4.1(b–d); and extra problems (due 24 January)
• Homework 1: 2.1.1.3, 9, 12(a, b); 2.1.2.4; 2.1.3.3(a), 6, 8; and extra problems (due 17 January)

31 January. The comments below originally (incorrectly) said that Problem 2.2.2.4 had been assigned for this week. It is not part of this week's homework.
30 January (10 PM). A review sheet for exam 1 is now available. It will be a good idea to read the review sheet and practice problems on Thursday, and to come to class prepared to ask questions about those handouts or the course material.

Homework for Tuesday's class was 2.2.1.1(a, e, f) and 2; 2.2.2.1(a, b, d), 2, and 7; and an extra problem.

In Tuesday's class, we began our discussion of complex numbers, motivating them by the idea of solving cubic (not quadratic!) equations. Unlike the quadratic formula, the cubic formula can involve taking cube roots of complex numbers even when used to solve an equation which has real solutions. We defined a complex number as the sum of a real part and a “nonsense part” (a real multiple of a fixed square root of -1), and showed how to perform arithmetic on such objects. By recording the real part and the imaginary part, we obtain from a complex number an ordered pair of real numbers, which can also be thought of as a point or a vector on the real plane. This allows us to connect complex arithmetic to vector arithmetic, which we began to do.

The axiomatic treatment of the integers has been postponed until after the exam. Tuesday's class covered some of the material from Section 2.2.1. In Thursday's class, we will finish Section 2.2.1 and move on to some of the material from Section 2.2.2.

30 January. A list of practice problems for exam 1 is now available (but requires authentication). I will post a review sheet later tonight.
27 January. The solution set for Homework 2 is now available. This week, a mail was sent out to the entire class announcing availability. In future, you can opt in or out of such notifications on the My Workspace/Preferences/Notifications tab in CTools.
24 January. Homework for Tuesday and Thursday was problem 2.1.4.5, and two extra problems.

In Tuesday's class, we finished our descriptions (and proofs) of the delays and periods of the decimal expansions of rational numbers in terms of their denominators. We also mentioned that the reason why 2 and 5 are important to our discussion is that they are precisely the prime divisors of 10, the base (as etymologically suggested!) to which we are computing the decimal expansions. As an illustration of this point, we briefly discussed the analogues of decimal expansions in other bases, and the special primes that crop up there. We introduced the idea of “listability” (usually called countability) of a set, and showed (by arranging its elements in a two-dimensional array and reading them off in a spiral pattern) that the set of rational numbers is listable.

In Thursday's class, we showed further that the set of real numbers is not listable by giving a recipe to produce, from any given list of real numbers, a new real number not on it. (Our recipe involved taking some care to account for the fact that there are some real numbers with two different decimal expansions.) As an interesting application of this, we defined the algebraic numbers and mentioned (without proof) that they also form a listable set, so that we know not all real numbers are algebraic—even though we don't yet have the tools to write down any specific non-algebraic numbers! (The subject of transcendental numbers will explored further in next week's Math Club talk.) Further, we discussed a basic notion of length for subsets of the set of real numbers, and showed that, for this notion, the set of natural numbers, or indeed any listable set, has length 0, while the set of real numbers has length .

The material covered in Tuesday's class finished Section 2.1.3 of the text and moved on to Section 2.1.4, which we continued to discuss in Thursday's class. We will use Section 2.1.4 as a jumping-off point for an exploration of an axiomatic treatment of the natural number notes which are also part of our course text. Students are encouraged to read the pages numbered 19 and 20 in the text (physical pages 20 and 21) before next class.

21 January. Solutions for homework 1 are now available (but require authentication).
18 January. Homework for Thursday was problems 2.1.3.7 and 11, problems 2.1.4.1(b–d), and some extra problems. These extra problems were written on the board in class, but students should consult the handout for the exact statement. (In particular, the problem about finding a rational number between any two real numbers is best approached using an auxiliary result which appears on the handout but was not written on the board.)

In Thursday's class, we showed that the decimal expansion of the reciprocal of any number of the form 2m5n (for integers m and n) terminates after max{m, n} digits, and we guessed that the integers bigger than 1 whose reciprocals have immediately repeating decimal expansions are precisely those integers indivisible by 2 and 5. Our reasoning used “magic denominators”—special denominators which make it particularly easy to compute a decimal expansion of the fraction in which they appear. For the first sort of number, the magic denominators were powers of 10, whereas, for the second, the magic denominators were numbers 1 less than a power of 10.

Thursday's class continued with material from Section 2.1.3.

16 January. The homework that is collected each Thursday will include all problems which were assigned up to, and including, the preceding Thursday. In particular, unless otherwise announced, problems assigned on a given Tuesday will be due nine, not two, days later.

Although it is not officially part of the homework (in particular, will not be collected), students were also encouraged on Tuesday to try to work out Problem 2.1.3.7.

In Tuesday's class, we finished the proof of the principle from last class by showing that every rational number is represented by an eventually repeating decimal. Our proof, which was essentially just long division, also showed that the repetition in the decimal expansion of a/b will begin after at most b − 1 digits (possibly fewer). This allowed us to give a new answer to the question of whether R is larger than Q; we alluded to two other ways of answering the question, which will be explored a bit in Section 2.1.4. We also saw that the corresponding question of whether R is larger than C is answered not by purely algebraic methods, but rather also by using the concept of order. This suggests that the proper way to view R is as an ordered field, a concept explored further in Section 6.2. At the end of class, students were asked to compile a chart listing the decimal expansions of 1/n as n ranged from 2 to 13, noting, in each case, the delay before the repeating block starts and the length of that repeating block.

Tuesday's material came mostly from Section 2.1.3 (although with reference to Sections 2.1.2 and 2.1.4). We will try to finish Section 2.1.3 on Thursday, and then move on to Section 2.1.4 (probably next week).

15 January. The syllabus has been updated to include the office hours and midterm exam dates. Since I do not have complete control over the date and time for the final exam date, I cannot yet announce it definitely.
11 January. Office hours are now set for M, 5–6 PM, and Tu and W, 4–5 PM. Homeworks will be collected weekly on Thursday.

Homework for Tuesday was problems 3, 9, and 12(a, b) from 2.1.1; problem 4 from 2.1.2; and some extra problems. Homework for Thursday was problems 3(a), 6, and 8 from 2.1.3. (In future, we will write “problems 2.1.3.3(a), 6, 8” instead of “problems 3(a), 6, and 8 from 2.1.3”.) Since this week's homework will be particularly long, students are encouraged to start working on it as soon as possible.

In Tuesday's class, we discussed that the need for closure under (non-0) division forces us to enlarge our number system from the set of integers to the set of rational numbers. Although some more equations can be solved in the real numbers than in the rational numbers, there are still some equations without solution, so we didn't resolve yet the question of why we enlarge to the real numbers. However, it is reasonable to view the process of enlarging to the complex numbers as allowing solutions to all reasonable (meaning non-constant polynomial) equations. We also discussed various closure-type results: The sum of a rational number and an irrational number is never rational, and similarly for differences and (non-0) sums and quotients. (These facts are proven on the homework.)

In Thursday's class, we explored the gap between the set of rational numbers and the set of real numbers—that is, we considered the irrational numbers. Much of the class was spent in a proof of the familiar fact that √2 is irrational. One proof of this uses the existence and uniqueness of prime factorisations of integers, which we will prove in Chapter 5. We closed by beginning to characterise rational numbers in terms of the decimals which represent them. We articulated the principle that a number is rational if and only if it is represented by an eventually repeating decimal (where “eventually repeating” includes immediately repeating and terminating—the repetend being viewed as the digit 0), and proved half of this principle, namely, that an eventually repeating decimal represents a rational number.

Tuesday's and Thursday's classes finished up Section 2.1.1, covered what is of interest to us from Section 2.1.2, and began Section 2.1.3. Next Tuesday's class will continue with, and possibly finish, Section 2.1.3.

6 January. Please be sure to read the handouts Errors to avoid and A note on proofs. These discuss some important guidelines for writing up arguments on your homework. The tips on the first will affect your grade, so require particular attention.

The concepts discussed in Thursday's class came from Section 2.1.1 of the text. We will continue Section 2.1.1 (with a glancing look at 2.1.2) and move on to Section 2.1.3 next week.

3 January. In today's class, we discussed some definitions of the concept of ‘number’, eventually obtaining a hierarchy of numbers starting with the whole, or counting, numbers, and building up all the way to the complex numbers. We discussed practical reasons why it might be appropriate to enlarge our number system beyond the most basic, and easiest-to-understand, domain of the whole numbers, and then sought out mathematical reasons. Doing so led us to the concept of closure of a set under an operation (which has nothing to do with closed and open intervals). In particular, the natural numbers are closed under addition and multiplication, but not subtraction or division. Enlarging to the integers yields a structure which is still closed under addition and multiplication, but now closed under subtraction.

Students are encouraged to think about similar mathematical reasons why we might want to enlarge from the integers to the rational numbers, from the rational to the real numbers, and from the real to the complex numbers.

1 January. The course web site for Math 486 in Winter 2008 is now live, though most of the links are not active yet. Further content will be available soon.