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February's homeworks

• Homework 6: 2.2.2.5, and two extra problems (due 21 February)
• Homework 5: 2.2.1.4; 2.2.2.6, 12 (due 14 February)
• Homework 4: 2.2.1.1(a, e, f), 2; 2.2.2.1(a, b, d), 2, 7; and an extra problem (due 7 February)

22 February. Have a good spring break!

Homework for Thursday was three extra problems. (In particular, notice that the extra problem handout now has five problems, so that you should re-download it if your copy has only two problems.) Please notice that the problem discussing squares greater than 2 is stated on the handout slightly differently to the way it was stated in class. You should solve the problem as it is worded on the handout. I have also changed the wording slightly to clarify what you may use in your proofs (both the axioms of order and arithmetic will be necessary).

The solutions for Homework 5 are now available (but require authentication).

In Thursday's class, we formalised the idea about the order on real numbers which we had started to discuss on Tuesday as the axiom of completeness, which discusses the existence of greatest lower bounds, or ‘infima’. (There is a corresponding notion of a least upper bound, or ‘supremum’.) It is no exaggeration to say that this axiom is why everything you know about the real numbers is true. As a first application, we used it to prove that there exists a real number whose square is 2 (a fact which we showed by geometric means earlier). The proof pointed out the interesting (if familiar) fact that changing a number does not change its square much—in particular, that it is possible to “bump” slightly a number whose square is (say) greater than 2 to get a new number whose square is also greater than 2. This was our first introduction to the concept of continuity, which we will explore, together with other fundamental ideas of calculus, over the next few weeks.

20 February. Homework for Tuesday was two extra problems.

In Tuesday's class, we discussed some familiar consequences of the axioms of order that hold in all ordered number systems (for example, the integers, the rational numbers, and the real numbers), then looked at what happens in more specific settings. We used well ordering (in its guise as the principle of infinite descent) to show that 1 is the smallest positive integer. We then formulated the Archimedean principle, which relates the orders on the integers and the real numbers (and has its origins in Archimedes's famous assertion “Give me only a long enough lever and a place to stand, and I can move the Earth”), and used it finally to provide the necessary background for our long-ago proof that the rational numbers are dense in the real numbers. We finished by discussing what axiom might replace well ordering (which doesn't hold there) on the real numbers, and briefly discussed, but did not formalise, the idea of considering the whole set of ‘smaller’ (instead of just a smallest) numbers. This somewhat unintuitive idea will be formalised next time, when we will also explore its consequences.

17 February (3 PM). The solution set for Exam 1 is now available (but requires authentication).
17 February. The solution set for Homework 4 is now available (but requires authentication). Also available is a statistics sheet for exam 1.

I apologise for the delay in posting solution sets. I will try to have up a solution set for exam 1 by later this evening.

14 February (7:30 PM). Homework for today is an extra problem. (In particular, notice that the extra problem handout now has two problems, so that you should re-download it if your copy has only one problem.)

In today's class, we used the axioms of integer arithmetic to prove such familiar statements as 0a = 0 and (-1)a = -a for all integers a, and to prove ‘additive cancellation’. We showed that multiplicative cancellation could be proven for integers by enlarging to rational numbers, but not internally to the integers using only the axioms of arithmetic. See page 38 of the natural numbers notes for more details on how one can prove it entirely within the integers (using the order axioms which we will discuss below). We also stated an analogue of the statement that “the product of two negative numbers is positive”—namely, that the product of the additive inverses of two integers is the same as the product of the integers themselves. (This springs from the (false) idea of a negative integer as “one with a minus sign”.)

We also broadened the idea of positivity and negativity of integers to the idea of order (i.e., inequalities). We stated a few basic axioms that hold in all the familiar ordered number systems, but also the important axiom of well ordering, equivalent to the principle of infinite descent, which is special to the integers (and is the basis for proof by induction).

14 February. Homework for Tuesday was problem 2.2.2.5 and an extra problem.

In (last) Thursday's class, we discussed how the formula for complex multiplication in polar coördinates makes it very easy to take powers of complex numbers. At first, we only discussed positive integer powers, but then we showed that the same recipe (raise the length to the desired power, then multiply the argument by it) worked as well for negative integer powers (and 0). In particular, we saw how to divide. The natural next question is whether it works as well for rational powers, e.g., for extracting roots, and we saw that this is almost true—except that we have to remember that a complex number has more than one argument, hence (usually) more than one root (indeed, every non-zero complex number has n distinct nth roots). We also turned to another kind of ‘root’ (really a generalisation of the same concept), namely, the roots of a real quadratic polynomial, which, familiarly, come in pairs (corresponding to the + and signs in the quadratic formula). Any two real numbers can be so paired, but, we observed, only certain complex numbers can.

In Tuesday's class, we found the specific complex number, namely 7 − 3i, which can appear as the other root of a real quadratic polynomial which vanishes at 7 + 3i. This motivated the general idea of the conjugate, of a complex number, which can be defined as the number with the same real part but opposite imaginary part; the same length but opposite argument; or the reflection of the (point in the complex plane representing the) original number through the x-axis.

In the rest of Tuesday's class, we returned to an earlier topic, namely, the ‘size’ of the rational numbers. We have previously showed that the rational numbers are small in various senses, and a homework problem (extra problem 1 on Homework 2) showed that they are big in another sense—namely, that there is a rational number between any two real numbers. (We say that the rational numbers are dense in the real numbers.) However, the problem in question assumed two facts, and we will now address the problem of how the two facts may be proven. Accordingly, we tabulated the principles of arithmetic on the integers which we will take as axioms (that is, tools for proof that are themselves not proven): the integers are closed under addition, addition is commutative and associative, there is an additive identity, and every element has an additive inverse; similarly for multiplication (except that few integers have integer multiplicative inverses); and multiplication distributes over addition. We discussed the extent to which these axioms hold or fail in other number systems.

The material from (last) Thursday's class, and the first part of Tuesday's class, finished Section 2.2.2. The second part of Tuesday's class covers some of the notes on natural numbers linked in the handouts section above. Specifically, the axioms themselves appear on p. 19. In (this) Thursday's class, we will prove some (familiar) consequences of these axioms, covering parts of pp. 23–25, 29–35, and 38–39.

11 February. Homework for Thursday was problems 2.2.1.4 and 2.2.2.6, 12.
5 February. Office hours today (Tuesday) will be held 4–5:20 PM, not just 4–5 PM.
3 February. Thanks to Nichole Erickson for pointing out that some of the practice problems for the first exam involve terminology that has yet to be defined in class. Specifically, for problem 6, a subset of the real line is dense if and only if every interval on the real line contains an element of it (for example, by extra problem 1 on homework 2, the set of rational numbers is dense). For problems 7–9, the conjugate z of a complex number z = a + ib is a − ib (that is, the complex number with the same real part as z, but opposite imaginary part). I'm sorry for any confusion this caused.

Any unfamiliar terminology that occurs on the exam itself will be defined there.

1 February. The grader has agreed to try to return the homeworks by Monday. If all goes well, they will be available for pick-up in my office hours on Monday, or by appointment.

In Thursday's class, we discussed the geometric significance of complex multiplication, motivating this with the cases of multiplication by a positive real number (which stretches a vector without changing its direction) and by i (which rotates a vector counterclockwise by π/2 radians without changing its length). These considerations motivated the idea of identifying a complex number not by rectangular coördinates (its real and imaginary parts), but by polar coördinates (its length, or absolute value, and argument). As part of our work today, we

• named the quantities
• Re(z),
• Im(z),
• |z|, and
• Arg(z)
(the real part, imaginary part, absolute value or length, and argument, respectively, of z) associated to a complex number z;
• described the important formula |z|2 = a2 + b2 (when z = a + ib); and
• noted that
• |zw| = |z||w|,
• |z + w| ≤ |z| + |w| (the triangle inequality), and
• |z − w| is the distance between z and w
for any complex numbers z and w.
The material covered finished Section 2.2.1, and covered much of Section 2.2.2. No new homework was assigned on Thursday. After the exam on Tuesday, next Thursday's class will cover the rest of Section 2.2.2, after which we will move on to the postponed axiomatic discussion of the integers.