Instructor: Loren Spice
E-mail: firstname.lastname@example.org. Please include the string ‘486’ in any e-mail to this address.
Office phone: 7347632423
Office hours: M 5–6 PM, Tu and W 4–5 PM, and, on 17 April only, 3–3:50 PM
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Announcements and handouts will be available on this page. Since announcements may be posted at any time, you should check this page frequently. However, I will try to make sure that important announcements and handouts are mentioned in the RSS feed. Make sure you hard refresh this page to ensure that you are not reading a cached version. Handouts will be PDF files, but are available to students in other formats upon request.
I include below a summary of the last two days of class. I will post one more announcement when the final grades are available.
We stated (last) Thursday's class by discussing another interpretation of what's so great about the greatest common factor of two numbers—namely, that it's divisible by any other common factor—and mentioning its connection to the least common multiple. We then moved on to modular arithmetic, giving many definitions and examples of congruences, which are systematisations both of
In Tuesday's class, we discussed the fact that, as congruences behave well with respect to multiplication, they also (by induction) behave well with respect to exponentiation. This allows one both to compute remainders of uncomputably large powers upon division by small numbers, and (more practically) to create divisibility tests. (We discussed the tests for divisibility by 3 and 9, but the test for divisibility by 11 is only a little more complicated.) Although we did not have a chance to discuss it, this behaviour is also the basis of a primality test using Fermat's little theorem (Theorem 6.4).
Please notice that, while I will not solve the practice problems in office hours, I will be happy to discuss them, and any approaches you have considered or difficulties you have had. Office hours may also be a handy opportunity to talk to other students about their solutions.
The final will be on Thursday, 17 April, 4–6 PM, in Dennison 110. Note that it is not at the usual class time and not in the usual class room (or even in the usual building).
Since we did not cover everything on the review sheet, a few items have been removed. Specifically, students are no longer required to prove the uniqueness of prime factorisations; to prove the existence of infinitely many primes; or to use complete sets of residues. The reiew sheet has been updated to reflect these changes. Accordingly, I removed from the list of practice problems problem 18.104.22.168, since it involves only complete sets of residues, which are no longer covered on the exam.
Note that I also removed from the review sheet the requirement to find multiplicative inverses in modular arithmetic. However, students should be aware that this is really just a special case of the Euclidean algorithm, and the omission of this topic does not mean that they need not understand the Euclidean algorithm.
After a small amount of new material in Tuesday's class, we will have a review. In particular, we will discuss in some detail what new topics will and will not be covered on the exam. Students should also come prepared to ask questions about specific problems.
No further graded homework will be assigned, but students are strongly encouraged to solve the practice problems (not least because one of them will appear on the final exam!).
Evaluations will be handed out at the end of class on Tuesday.
In Tuesday's class, we completed our discussion of the iterated procedure for computing gcfs. (This procedure is called the Euclidean algorithm.) Our large chart featured, together with a computation of the gcf, considerable additional information, which we used to write gcf(a, b) in the form ax + by for some integers x and y, and to compute the continued fraction expansion of a/b.
The material from Tuesday's class corresponds to Sections 5.2.2 and 5.2.3 of the text, and to pages 11–15 and 25 of the number theory notes. I have also put links to some external resources above (but note that I have not checked all the content of these resources, and they should not be regarded as part of the official course material).
In Thursday's class, we will move on to a discussion of modular arithmetic, corresponding to Section 6.1.1 of the text. Modular arithmetic has numerous applications, but we will content ourselves with trying to use it to understand and manufacture divisibility tests. In this connection, see pages 30–31 of the number theory notes. The external readings linked above also offer content on modular arithmetic (which extends far beyond what we will discuss).
In yesterday's class, we discussed integer division (i.e., division with remainder), showing that any integer can be divided by any other. Our proof actually outlines a semi-practical approach to performing the division, so that the result is usually referred to as the division algorithm. We observed the interesting fact that, if the remainder of a upon division by b is R, then the common factors of a and b are the same as those of b and R. Thus, to compute the greatest common factor (or gcf) of two given numbers, it suffices to compute the gcf of two smaller numbers. Iterating this procedure gives a very rapid procedure for calculating gcfs, which we began to implement for a = 14022 and b = 90210. We did not complete this calculation in class, so students are asked to complete it themselves over the weekend and be ready to discuss it in Tuesday's class.
The material from yesterday's class corresponds to Sections 5.2.1 and 5.2.2 of the text, and to pages 8 and 10–13 of the number theory notes.
In Thursday's class, we showed how to compute the derivative of the square root function, and proved the product rule for derivatives (using the limit laws that we had already proven). The sum and quotient rules will be proven on the homework. We gave examples of functions which are not continuous, and of continuous functions which are not differentiable (making formal the idea that a differentiable function is one whose graph doesn't have ‘corners’), but showed that every differentiable function is continuous. (Informally, this is because the graph of a differentiable function looks like a line near each point, and lines are graphs of continuous functions. Formally, our calculation involved computing the limit that appears in the definition of continuity.) Although there is much more to calculus—even to differential calculus—we leave a deeper discussion of it to other classes. At the end of Thursday's class, we moved back into the integers, defining the notion of divisibility (and its many associated synonyms and symbols).
In today's class, we introduced the notion of a factorisation. At its most basic, a factorisation is just an expression of a number—usually an integer—as a product of—usually two—other numbers—usually integers. We singled out the positive integers which (in a sense) can't be factored, calling them prime and the remaining ones (other than 1!) composite. The primes are ‘atomic’ (for multiplication), both in the sense that they can't be broken down, and in the sense that every other integer is ‘made up’ of them. The precise meaning of this is that every positive integer has a prime factorisation. We formalised the idea of trial divisions to prove this by appeal to the well-ordering principle (in two different disguises). The proof brought up the idea of divisibility and primality tests, of which we will see examples later in the course. We also mentioned the somewhat surprising fact that it is easier to find the greatest common factor of two numbers than to factor either of the numbers individually—a somewhat paradoxical-seeming fact that we will justify next class.
The material from today's class corresponds to parts of Sections 5.2.2 and 5.2.4, both of which we will explore in more detail in future classes. This material is also discussed well in the number theory notes. The material we have covered so far is discussed in pages 2–5, 17–19, and 26 of those notes.