-- A student asked, "What did Madre do at the circus?"
-- "Remember, only yes or no questions," the teacher cautioned.
This is the second page of the first section in the games chapter, "Mathematical Mystery Tales." As it happens, on page 267, Pappas begins telling us a story -- but we missed that part of the story, since the 267th day was over the weekend.
Actually, this mystery is a type of logic problem. To that end, this fits perfectly with the current Chapter 2 of the U of Chicago text, which is also on logic. We might as well go back to the beginning of the problem on page 267, since there's point in proceeding with the questions on page 268 without reading the setup on the previous page:
It was one of those weeks at the circus, when everything seemed to go wrong. First, one of the horses for the acrobat dancers went lame. Next the head clown threw a fit because the fat lady's child got into his makeup. Then Madre and his wife had an argument over the trapeze artist. The final blow came when Madre was found dead under the big top. Next to his body was his cane, which he occasionally used. An overturned glass of water was on his desk, and a minute pile of sawdust was near his body.
Pappas presents this story from the view of a teacher, Mr. Mason, reading this to his students. The class is to solve the mystery by asking the teacher yes or no questions. This is what the students discover by asking questions on page 268:
-- Madre was not a manager.
-- There was no sign of violence.
-- Something else was near the glass, and it's important to know what it was.
-- It was a pill.
-- There wasn't a pencil sharpener on his desk.
-- He died from a heart attack.
At this point, the students realize that they haven't found out what caused his heart attack yet. The student Gary replies, "Sure we have, he didn't take his medicine. Right Mr. Mason?"
Unfortunately, Mr. Mason's response is on page 269, so you'll have to wait until tomorrow before we can proceed with this mystery. I know, it's just like a TV show that ends, "To Be Continued...."
Chapter 8 of Stanley Ogilvy's Excursions in Mathematical Theory is "The primes as leftover scrap." I begin reading:
"'The prime numbers go forever'; but their frequency decreases. One of the major achievements of analytic number theory has been the discovery and proof of the Prime Number Theorem, which describes the asymptotic density of the primes."
Ogilvy starts this chapter by stating the Prime Number Theorem. According to this theorem, the number of primes less than x is approximately x/log x, and that the probability that a number about the size of x is prime is 1/log x. Actually, these are natural logs, so we should write x/ln x and 1/ln x.
According to Ogilvy, it's easy to find arbitrarily long sequences of composite numbers. For example, here are 99 consecutive composite numbers:
100! + 2, 100! + 3, 100! + 4, ..., 100! + 100.
For the first is certainly divisible by 2, since 100! is; the second by 3; and so on.
The author now asks whether there exists an arithmetic "progression" -- or arithmetic "sequence," as we're likely to call it in a Common Core Algebra I class -- consisting of only composites. There is:
10, 15, 20, 25, 30, ...
all of whose terms are multiples of 5. It's easy to see that if the first term of the sequence is a (or perhaps a_1 with a subscript of 1, as we may see in Algebra I) and the common difference is d, and a and d have a factor in common, then the sequence consists of all composites. Every member of the sequence has that same factor in common.
Just because a conditional is true, it doesn't mean that its converse must be true. According to Ogilvy, it was the 19th century German mathematician P.G.L. Dirichlet who proved the converse -- if every member of an arithmetic sequence is composite, then a and d must have some factor in common. In fact, he proved that not only must a sequence with a and d relatively prime must contain a prime, but it must contain infinitely many primes.
Ogilvy tells us that the number of primes less than x is approximately given by the function Li(x), called the "logarithmic integral" of x (since it's defined by an integral). The author now produces a table, which I don't reproduce here. The table gives, for x the third through seventh powers of 10, the value Li(x) as well as the actual number N of primes less than x and the error. Even though I don't print the table here, I will give its first and last rows:
x = 10^3, N = 168, Li(x) = 178, d = Li(x) - N = 10, d/N = rel. error = .060
x = 10^7, N = 664,579, Li(x) = 664,918, d = 339, d/N = rel. error = .0005
For small values of x, Li(x) is an overestimate and so the error d is positive. The author tells us that the error will become negative at some point before the Skewes' number, which is humongous:
S = e^e^e^79
Note, since Ogilvy's book was written, the upper bound has been reduced significantly from the one found by the South African mathematician Stanley Skewes. The current upper bound is about e^728 (or about 10^316), while the lower bound is 10^19. In my research of the current bounds, many sources use the symbol pi(x) for the what Ogilvy calls N -- the exact count of primes less than N. We notice that "pi" and "prime" both start with the letter p -- which is most likely why it was chosen.
Ogilvy now presents us with the Sieve of Eratosthenes, a classical method of finding primes. We consider a list of numbers from 1 to 100 and cross out all the multiples of 2. Traditionally, we would sieve out multiples of 3 next, but it's easier in decimal to sieve out 5, since this would leave us with numbers ending in 1, 3, 7, and 9. We then cross out multiples of 3 and then 7 -- and since this is the last prime less than sqrt(100), we stop here. All the numbers that remain -- the "leftover scrap" mentioned in the title -- are primes.
The author shows us an alternative way to find the primes. He begins by creating another table, the first row and column of which are the arithmetic sequence with a = 4 and d = 3. The second row and column are the sequence with a = 7 and d = 5, the third row and column are the sequence with a = 10 and d = 7, and so on -- that is, the differences d are successive odd numbers.
Then he claims that if x is any integer greater than 2, x is prime if and only if (x - 1)/2 does not appear in the table. Here is his proof:
n = (x - 1)/2
x = 2n + 1
And so he recreates the table by doubling and adding 1 to each entry. The resulting table resembles a multiplication table -- the upper-left corner is 9 = 3 * 3, and the rows and columns correspond to multiplying by odd numbers (except for 1). Therefore all entries are odd composites, and all the missing odds are prime. (Of course, we don't forget the even prime 2.) QED
Ogilvy wraps up the chapter by defining the lucky numbers. He tells us that they are found using a sieve similar to that of Eratosthenes. I've been omitting the tables for the primes since we know what they are, but let's actually show the sieve for the lucky numbers.
We begin by sieving out all the even numbers. Therefore all lucky numbers are odd:
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57
It doesn't make sense to sieve out 1, so we sieve out 3 by crossing out every third number:
1, 3,
1, 3, 7, 9, 13, 15, 19, 21, 25, 27, 31, 33, 37, 39, 43, 45, 49, 51, 55, 57
The next number is 7, which we sieve out by crossing out every seventh number:
1, 3, 7, 9, 13, 15,
1, 3, 7, 9, 13, 15, 21, 25, 27, 31, 33, 37, 43, 45, 49, 51, 55, 57
The next number is 9, which we sieve out by crossing out every ninth number:
1, 3, 7, 9, 13, 15, 21, 25,
1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 45, 49, 51, 55
Let's cross out the thirteenth, and then the fifteenth number that remains:
1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43,
1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51 (63, 67, 69, 73, 75, 79, 87, 93, 99)
And these are the lucky numbers. The mathematician Stanislaw Ulam, who first named these the "lucky numbers," perhaps named them so since they are "lucky" enough to avoid being sieved out. I notice that 7, the traditional number associated with luck, is a lucky number -- but then again, so is 13, the traditionally unlucky number.
Ogilvy tells us that lucky numbers are similar to prime numbers. The probability that a number near x is lucky is again 1/ln x. The number of twin luckies less than N is similar to the number of twin primes less than N, and there is a Goldbach-like conjecture that every even number is the sum of two lucky numbers.
Here are two links to Lucky Numbers. The first is Wolfram, and the second is the OEIS:
http://mathworld.wolfram.com/LuckyNumber.html
http://oeis.org/A000959
Ulam, who first came up with lucky numbers, suggests that it's the nature of the sieving process that explains why primes and luckies are so similar. As Ogilvy concludes, primes and luckies are both:
"Leftover scrap, perhaps; but it is interesting scrap indeed that organizes itself into two (and possibly more) different and yet strangely similar heaps."
Today marks the start of the review for the Chapter 2 Test. Notice that Chapter 1 of the U of Chicago text had nine sections, and so there was no true review day for the Chapter 1 Test. But Chapters 2 through 6 each have only seven sections, and so the opposite happens -- there are two review days for each section.
In the past, I kept juggling around how I wanted to assess the first three chapters. The worksheet from two years ago prepares the students for a test with 11 questions -- and some of these questions are from Chapter 1, not just Chapter 2. But there's no harm in retesting Chapter 1 material again.
With two days of review, some teachers may use the extra day differently. Some, for example, might choose to cover Lesson 2-7 from the Third Edition of the text today. This section, "Conjectures," doesn't appear in my old Second Edition of the text. There's even an activity -- the Conjectures Game (or "Who Am I?") that I refer to several times on the blog -- that fits here. (How ironic is that? I have a Conjectures Game even though my book doesn't have a conjectures section.) Then the review worksheet can be given tomorrow instead.
This is what I wrote two years ago about today's worksheet:
Here is the rationale for which questions I decided to include on this review worksheet -- just as I did for the Chapter 1 Quiz, these problems come directly from the "Questions on SPUR Objectives" appearing at the end of each chapter.
For Chapter 1, I begin with Question 21, the three undefined terms (point, line, and plane), and then move on to Questions 26 and 32, two of the properties from arithmetic/algebra (Multiplication Property of Inequality and Substitution Property of Equality). Next are Questions 36-37, order on the number line -- except that I made the distances whole numbers, not decimals, and also I omitted point V from the second question, which serves no purpose other than to confuse and frustrate the students. Question 39 directs students to find the two points R on the number line that are the right distance from Q, and Question 41 is another distance question. Finally, I jumped to Question 61, another absolute value question similar to one that appeared on the Chapter 1 Quiz.
For Chapter 2, I begin with Question 16, which asks why the following definition is not a good definition of triangle: "A triangle is a closed path with three sides." The problem is, what exactly is a "closed path"? We're not allowed to give definitions containing words that also themselves need definitions. Question 20 asks the students to rewrite a statement in if-then form, then Question 30 reminds students that just because a conditional p=>q is true, it doesn't mean that its converse q=>p must be true.
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