November's comments.
30 November. In Monday's class, we discussed Section 6.1, on the definition of fractions. In particular, we discussed the area, (linear) measurement, and set models of fractions, showed how to compare the sizes of fractions, and considered renaming fractions to produce different-looking but equal fractions. Depending on what kind of renaming one does, this process corresponds to simplifying, or to putting over a common denominator.

In Wednesday's class, we discussed most of section 6.2, namely, the part dealing with addition of fractions. We also considered the naive model of addition of fractions in which one adds numerators and denominators, and showed that this (wrong) kind of addition produces interesting results when we repeatedly ``add'' 0/1 and 1/1. This is called ``Farey addition''.

Students should read the remainder of section 6.2 for Monday, especially the parts dealing with mixed numbers, and fractions as division. They should also read section 6.3, on multiplication of fractions.

Homework for Monday was set 24, and for Wednesday was set 25.

26 November. In Wednesday's class, we finished our discussion of Chapter 5 by giving another picture of the Euclidean algorithm, and using it to prove that a number has only one prime factorisation. The key idea is that breaking up a number into two parts doesn't ``lose'' any prime factors. (By way of contrast, notice that breaking up 12 into 3⋅4 ``loses'' the factor of 6.)

No new homework was assigned on Wednesday. The reading for Monday is Sections 6.1 and 6.2 of the text.

I have posted a sheet with some statistics for Exam 2. Since I just realised I never posted it, I have also included a similar sheet for Exam 1.

16 November (6 PM). I have posted a practice sheet for Exam 2. It contains several exponentiation problems, and a few other problems.
16 November (1 PM). I have posted the review sheet for Exam 2. It contains essentially the information from yesterday's class, but with a little more detail. I will post the handout on exponentiation later today.
15 November. In Monday's class, we discussed primality testing and the infinitude of primes. Specifically, one tests for primality by performing trial divisions (or using divisibility tests, if possible), but only with divisors up to and including the square root of the number being tested. Since performing a few more trial divisions than necessary is OK, one can overestimate the square root instead of computing it exactly. That there are infinitely many primes is shown by contradiction. One assumes there are only finitely many, then manufactures a number which is not divisible by any of those finitely many primes, contrary to the half of the Fundamental Theorem of Arithmetic which we have already proven. We also started our discussion of G(reatest)C(ommon)F(actor)s and L(east)C(ommon)M(ultiple)s, and showed how to compute GCFs by writing down one or two factor lists.

In today's class, we showed how to compute GCFs using the exponential forms of the prime factorisations of the numbers in question. This is faster than writing factor lists, but can still be slow for very large numbers. Thus we discussed Euclid's algorithm, which allows one to compute GCFs without factorisation. We discussed a way to organise the steps of Euclid's algorithm into a four-column table which not only finds the GCF, but also shows how to write the GCF as a difference of multiples of the original numbers in question. Finally, we discussed the LCM, which can be found by writing lists of multiples or using prime factorisations, or by using the formula lcm(a, b) = a⋅b/gcf(a, b).

Homework for this week is sets 22 and 23 from the text. Although it is not part of the homework, students are also strongly encouraged to use the procedure from class to find the GCF, d, of a = 11523 and b = 4163, and to find integers m and n such that m⋅a + n⋅b = d. (One of m or n will be negative.)

A copy of the review sheet discussed in class, as well as a handout discussing simplifying exponential expressions, will be available tomorrow. I will send out an e-mail when it can be downloaded.

8 November. In today's class, we proved the lemma that every whole number greater than 1 is divisible by a prime, then used this lemma to prove the theorem (half of the Fundamental Theorem of Arithmetic) that every whole number greater than 1 is a product of (not necessarily distinct) primes. Underlying both these proofs was the Well-Ordering Principle, an axiom which says that any non-empty collection of whole numbers has a smallest element (in other words, that there is no infinite descending sequence of whole numbers).

Office hours for today have been extended to 3--5 PM, not (as was originally announced) 2:30--4:30 PM. Office hours for tomorrow are cancelled.

Since we did not discuss factor trees in class, students should be sure that they have read the material about them in Section 5.3.

Homework for today is the following extra question:

Find all the divisors of 22⋅32⋅72.

7 November. In Monday's class, we discussed the tests for divisibility by 9 and 11 (the test for divisibility by 3 being easy once one understands the one for 9). We then moved on to the notion of prime and composite numbers, and discussed the sieve of Eratosthenes, a method for writing down large lists of prime numbers.

For Wednesday's class, students should read Sections 5.3 and 5.4 of the text.

Homework for Monday was problem 1 on quiz 7, problem 7 on homework set 20, and homework set 21. Problem 1 on quiz 7 reads as follows:

Suppose that a and b are two whole numbers, both of which have remainder 2 upon division by 3. Prove that ab has remainder 1 upon division by 3.
2 November. Problem 7 on homework set 20 involves the test for divisibility by 9, which we haven't covered yet. (Thanks to Kate Miller for pointing this out.) Therefore, I am dropping this problem from the homework.
1 November. In Monday's class, we discussed Section 5.1, which defines even and odd numbers and introduces the basic idea of a proof (a logical demonstration showing that the truth of some statement follows from the truth of accepted, or previously proved, statements). The important steps to remember in a proof are: Define your terms, name your quantities, write down your knowns, then write down what you want to prove. With this structure in place, it will often be obvious what to do next.

We did some example proofs, showing that the sum of two even numbers is even (algebraically and pictorially), and that the square of an odd number is one more than a multiple of four (algebraically).

In today's class, we gave a picture proof of the statement above about squares of odd numbers. We then moved on to the section on divisibility tests. After reviewing these tests (for divisibility by 2, 3, 4, 5, 8, 9, 10, and 11), we discussed also the test for divisibility by 6 (namely, divisibility by 2 and 3) and then noted that the key idea of all these tests is to subtract something from a large number, leaving a smaller number which we can test for divisibility instead. We proved half of the divisibility lemma, which is the key result showing that this subtraction is legal (i.e., doesn't change the answer).

In class, we were trying to show:

Suppose that A, B, and k are whole numbers such that k divides A. Then k divides B if and only if it divides B − A.
Remember that an `if and only if' statement is really two statements. We proved one of them:
With notation as above, if k divides A and B − A, then it divides B.
Students should state and prove the other as part of this week's homework.

Students should read and be familiar with the proofs of the correctness of the tests for divisibility by 2, 4, 5, 8, and 10. We will discuss the proofs of the correctness of the tests for divisibility by 9 and 11 on Monday.

Homework for this week is sets 19 and 20, as well as the extra problem described above.

Permanent version of this page created on 8 December 2006.