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Math 331 Homework: Day 2

Quote of the Day The Greeks reportedly suffered some intellectual indigestion when they discovered the irrationality of square roots, and with good reason: putting the theory of irrational numbers on a firm foundation requires a lot of work. Gerald B. Folland

From Wikipedia Pythagoras was very interested in music, and so were his followers. The Pythagoreans were musicians as well as mathematicians. Pythagoras wanted to improve the music of his day, which he believed was not harmonious enough and was too hectic. According to legend, the way Pythagoras discovered that musical notes could be translated into mathematical equations was when one day he passed blacksmiths at work, and thought that the sounds emanating from their anvils being hit were beautiful and harmonious and decided that whatever scientific law caused this to happen must be mathematical and could be applied to music. He went to the blacksmiths to learn how this had happened by looking at their tools, he discovered that it was because the anvils were “simple ratios of each other, one was half the size of the first, another was 2/3 the size, and so on.” (See Pythagorean tuning.) The Pythagoreans elaborated on a theory of numbers, the exact meaning of which is still debated among scholars. Pythagoras believed in something called the “harmony of the spheres.” He believed that the planets and stars moved according to mathematical equations, which corresponded to musical notes and thus produced a symphony.

Reading Read: Chapter 1.2. The material on Dedekind cuts is among the most difficult in the text. Read and reread it several times. We will work through this together in detail. Also: Review the material on partial orders from Math 135 (see Section 4.2 in the textbook Chapter Zero). See the Practice on the back.

Hand in Wednesday Notation: Often I will refer to problems in using the form Problem x.y.z meaning problem z at the end of Section x.y. Be sure to review these problems (most are at the end of Section 1.1 on page 9 and following) over the weekend and then come in for help if you need it. (Work in pencil, be neat, staple pages.) 1. Problem 1.1.1 (easy, but neat!). 2. (a) Problem 1.1.7. (b) Problem 1.1.8, where k ∈ and k ≥ 2. (Use Problem 1.1.7 even if you get stuck on the proof of N √ 1.1.7). In particular, when k = 2, this shows that p is irrational. 3. Problem 1.1.10. For part (b) first show sa is irrational, then use part (a). 4. Assume that b, c ∈ Q and that the polynomial x2 − bx + c has two distinct roots, r and s. Prove: If one of the roots, say r, is rational, then so is the other root s. Hint: If r and s are the roots, how else can you write the polynomial? 5. Problem 1.1.16(b). 6. Every rational number is supposed to be a real number. But a rational is a ratio of integers while a real number is a Dedekind cut. These are two very different types of objects. So for each rational s, we need to ‘identify’ the cut Ess that corresponds to the rational s. We do the ‘obvious’ thing. The rational s corresponds to the cut: Ess = {q ∈ Q | q < s}. Show that Ess is, in fact, a cut by verifying that Ess satisfies properties i–iii of Definition 1.2.1. (You may use any basic property of the rationals, e.g., Q is closed under addition and the usual properties of inequalities.)

7. Let α1 and a2 be Dedekind cuts. Define a new set γ = {p1 + p2 ∈ Q : p1 ∈ α1, p2 ∈ α2}. Show that γ satisfies Property 3 of a Dedekind cut. (What do you need to show?) 8. Challenge: Problem 1.1.23. Over 2

In Class, Practice, and Journal Work Make sure you have good proofs of these problems. Keep them in a journal or your notebook. 1. EZ: Show that if a is irrational, then −a and 1/a are both irrational. 2. One nice property of integers is that when added or multiplied together, these sums or products are again integers. We say that the integers are closed under multiplication and addition. (a) Prove that the rationals are also closed under multiplication and addition. That is, when two rationals are added or multiplied the result is again a rational. (b) Show that the irrationals are not closed under either addition or multiplication. √ √ 2 3. Show that an irrational raised to an irrational power can be rational.√ (Consider ( 3) . This number is either is rational or it is not. In the later case, try raising it to 2.) √ √ 4. Prove (by contradiction) that 2 + 3 is irrational. Hints: √ √ √ √ (a) Assume 2 + 3 is rational. Prove that 2 − 3 must also be rational. (Consider the product of both numbers.) (b) Now show that both numbers in (a) cannot be rational. (Consider their sum and problem 2.) 5. Let α be a cut. Then by definition we know that there is a rational r not in α. Let s be any rational so that r < s. Show that s∈ / α. 6. Let α and β be Dedekind cuts. (a) Prove that α ∪ β is a cut. (The previous problem is useful.) (b) Prove or disprove: α ∩ β is a cut. (c) Prove or disprove: αC is a cut.