Lecture 6 of David Kung's Mind-Bending Math is called "Infinity Is Not a Number." But before I discuss what Dave Kung has to say about infinity, let me talk about my own misconceptions about infinity as I was growing up. Infinity has always been a favorite topic of mine -- as well as many young students.
When I was a teenager, I noticed that there exist infinitely many integers, and between each pair of consecutive integers there exist infinitely many real numbers. So I concluded that the number of real numbers must be infinity^2, or "infinity squared." But actually, I never though much about infinity beyond there being infinity^2 reals.
Now let's move a few years forward, to my senior year in high school. As I mentioned earlier this week, my school was discarding several old textbooks at this time. One of these books was a old computer programming text. No, the computer language on which this text was based with not Logo, but Pascal. You see, this was around the time that the AP Computer Science test switched from Pascal to C++, and so the school had to discard its old Pascal texts. Even though I didn't have access to a computer that I could program in Pascal, I decided to take the book anyway -- after already knowing BASIC and Logo, I was curious to see what another computer language looked like, and besides. the book was free.
Actually, the Pascal text helped me understand how to program my TI-83 -- which, as I said, I've owned ever since my senior year. Both Pascal and TI-BASIC use While loops -- meaning that the computer or calculator does something over and over while a condition is true. TI-BASIC also has another mysterious loop called a Repeat loop -- it actually means to repeat something until a condition is true (i.e., repeat it while the condition is false). So it's nothing like Logo's Repeat loop, and though C++ has something called a Do-While loop, the C++ loop repeats things while a condition is true, and the TI-BASIC loop repeats it while a condition is false. It's only in Pascal where we find Repeat loops that work the same as the TI-BASIC loop.
Brian Harvey, whose online Logo text I mentioned yesterday, discusses Pascal for those of you who are curious about it. Of course, Harvey only mentions Pascal so that he can describe how to convert the program into Logo. For that matter, he also describes how to convert from BASIC to Logo. Here are the links:
Of course, now you're asking, what does any of this have to do with infinity? Except for the fact that computer loops can be infinite (for example, Repeat until 2 + 2 = 5 in Pascal or TI-BASIC), this seems to completely unrelated to infinity. The answer is that the way the author of the Pascal text describes how algorithms work is indirectly related to the true nature of infinite sets.
An algorithm is simply a recipe -- it's a sequence of steps that tells you how to solve a problem. There has been much discussion about the standard algorithms for arithmetic and Common Core. The standard (or any alternative) algorithm is a sequence of steps that shows how to add, subtract, multiply, or divide two numbers.
But the author describes an algorithm to accomplish something simpler than addition -- suppose I am thinking of a natural number n. Here's a simple algorithm that will allow you to guess my number -- first guess 1, then guess 2, then guess 3, and so on. After guessing m, guess m + 1. Eventually you'll guess my number n, no matter how large n is -- in particular, it will take exactly n steps. This algorithm is a finite algorithm.
Now let's change it -- instead of a natural number, now n can be any integer. Is it possible for you to come up with an algorithm to figure out my integer? The Pascal author suggests, as a first try, to guess 0, then guess 1, 2, 3, and more positive integers, and then move on to -1, -2, -3, and more negative integers. But this algorithm doesn't work -- suppose my integer is -1. You'll have to guess every positive integer before you finally reach -1 -- and since there are infinitely many positive integers, it means that you'll never guess -1. This is not a finite algorithm.
Here's the Pascal author's successful algorithm -- first guess 0, then 1 and -1, then 2 and -2, then 3 and -3, and so on. After guessing the positive integer m, guess -m, and then m + 1. This will allow you to reach n regardless of its sign -- if n is positive, the algorithm will take 2n steps, otherwise it will take 2|n| + 1 steps.
Later on, after reading the Pascal book, I obtained a set theory book that my high school was giving away for free. It's the same text where I found the Barber Paradox (as well as some of the other paradoxes that will appear later in Kung's course). This was a very old book -- it had been written during the 1930's or '40's.
Now the author of this text defines the set S to be a denumerable set if it can be placed in 1-to-1 correspondence with N, the set of all the natural numbers -- that is, if we can identify which element of S is first, second, third, and so on without skipping any elements. He then goes on to show that Z, the set of all integers is denumerable -- and he shows this by noting that we can label 0 as the first integer, 1 as the second, -1 as the third, 2 as the fourth, -2 as the fifth, 3 as the sixth, and so on. In other words, he came up with the exact same sequence as the Pascal author!
The set theorist then concludes that N and Z therefore have the same number of elements. We can use the same notation N(S) for the number of elements of a set that the U of Chicago text uses at the end of Lesson 6-1 -- we have shown that N(N) = N(Z). This came as a surprise to me, as it does to most people learning about this for the first time -- N is a proper subset of Z, yet N and Z have exactly the same number of elements (or as a set theorist would say, the same cardinality)!
Kung begins his lecture by discussing the Hotel Infinity -- a hotel with infinitely many rooms. He states that this is often called Hilbert's Hotel, named after David Hilbert -- the mathematician who first came up with this analogy. (Yes, this is the same Hilbert who came up with a rigorous formulation of Euclidean geometry.) Hilbert's Hotel is so famous that it's easy to find a link to a description of the hotel, such as this one:
Notice that my 1-1 correspondence between N and Z can be thought of in terms of Hilbert's Hotel -- suppose all the natural numbers are already in the rooms, with each natural in its own room. Then a bus with zero and all of the negative integers shows up. So we ask each natural to double its room number, and then the odd numbered rooms are free for zero and its negative friends.
The link above states what to do if there are infinitely many buses. Kung gives an alternative trick that's based on the infinitude of the primes, which was first proved by Euclid. (Yes, this is the same Euclid after whom Euclidean geometry is named.) Each bus is assigned an odd prime, and then each person raises the bus number to the power of his or her seat number. No two passengers are assigned the same room due to the Fundamental Theorem of Arithmetic.
When I was searching Google for links to Hilbert's Hotel, I found, much to my astonishment, that physicists have built a real Hilbert's Hotel. But the "guests" at this hotel are quantum states:
After discussing Hilbert's Hotel, Kung moves on to other paradoxes involving infinity. One of these is Thomson's Lamp, named after the British mathematician James Thomson. Thomson defined a supertask in which a lamp is turned on or off infinitely many times in one minute. If the lamp started off, what is the state of the lamp after a minute? This is sort of like asking, is infinity an odd or an even number? There is no answer -- because, as the title says, infinity is not a number. Notice that one of the Zeno's Paradox links I provided yesterday mentions Thomson's Lamp as well.
The Quick Conundrum for this lesson involves what is known as the Shepard Tone Illusion, named after the Californian mathematician Roger Shepard (who is still alive -- he will turn 97 years old at the end of this month). A Shepard tone is a tone that seems to get higher and higher toward infinity -- but in reality, each tone is actually several tones spaced an octave apart. Higher tones get quieter and lower tones get louder, so that the mind is fooled into thinking that the tone is always getting higher.
Just as with many of Kung's other topics, I could talk about infinity until -- well, infinity. But now we need to get back to Geometry class. Today is an activity day. Last year at the time I posted the inequality lessons, I posted two activities.
The first activity is all about the Euler line -- the line on with the centroid, circumcenter, and orthocenter lie. It's logical to post these lessons now since we'll be covering the triangle centers in next week's lessons. So this activity could serve as a preview -- though unfortunately, it may be a bit too advanced to serve as a preview. It might make more sense to post this next week, but today is the last activity day of the semester, so this is as close to the lessons as I could get it.
The second activity consists of some logic problems. Of course, if you don't like either of these activities, it may be possible to make any Kung lecture this week into an activity (for example, students might have fun with the Hotel Infinity).
This is what I wrote last year about these activities (again including commentary referring to the student I tutored last year):
Now for today's activities. First, I post a worksheet similar to what I showed my student back on Wednesday, with the two triangles with vertices well-chosen as to minimize the use of complicated fractions when calculating the centroid, circumcenter, and orthocenter. Only the centroid of the second triangle has a fraction as its y-coordinate, and the only fractional slope that appears is 1/2 -- but it's its opposite reciprocal, -2, that is needed to find the circumcenter and orthocenter.
At this point, some of you may ask, why shield the students from fractional coordinates, when the PARCC or SBAC exams may have fractions on them? It's because I want the students to understand the concepts first, rather than be intimidated by the fractions. My student, back on Tuesday, saw the questions from Glencoe where fractions appeared inintermediate steps (that is, where y is a fraction and this value of y must be plugged in to find x). I want him to think less about the fractions that might appear and more about what steps he needs to take to find the three centers and why.
Notice that for any triangle, the centroid, circumcenter, and orthocenter are collinear. The proof is somewhat complex -- one might really be tempted just to use a coordinate proof here. This fact was first discovered by the mathematician Euler -- that's right, the same Euler after whom e was named. I also remind you that Euler was also one who solved the Bridges of Konigsberg problem from the first day of school. (Yes, Euler was a very prolific mathematician!) And so we also named the line on which these three centers lie after him -- the Euler line of the triangle.
In my first activity, the students hopefully discover that the three centers lie on the Euler line. This is the ultimate goal.
Now for the other activity, I promised some logic problems, since this is the topic of Section 13-3 of the U of Chicago. Instead of including problems from the text, I chose some problems I found on a message board:
The author of these logic problems goes by the online name "Fireball." The website is devoted to my favorite game show, The Price Is Right. But in the summer, the game show is in reruns, and so those who post there find other topics to discuss. So "Fireball" came up with some logic puzzles. As usual, teachers can decide to include either the logic puzzles or the Euler lines, or a combination of both.
Speaking of The Price Is Right, I wanted to include an activity earlier based on that game show -- "The Triangle Is Right." Students are given right triangles with two legs (or possibly one leg and one hypotenuse) given, and they must guess the length of the remaining side. The winner is the student who comes the closest to the correct length -- without going over, of course. But I didn't include it as an activity, because I liked my puzzle-proof of the Pythagorean Theorem better.
The logic problems from the U of Chicago are about the same difficulty as the Beginner Level problem from "Fireball." Even his Easy Level may take the students some time to solve.
Meanwhile, it's been a while since I posted a topic on the traditionalists. Of course, it goes without saying that the traditionalists won't like anything that I posted this week. Imagine telling my story about the Hilbert Hotel in a Precalculus class (just before they learn about functions and their end behavior), or even worse, devoting an entire day in Geometry to hexaflexagons. I can't see the traditionalists viewing either as anything but a waste of time.
Well, here's a little something that traditionalists might appreciate. For other subjects besides math -- in particular, ELA and history -- some traditionalists recommend Classical Education. Here is a link to a description of Classical Education:
According to the above link, Classical Education divides all of K-12 education into three grade spans:
Grades 1-4: Grammar Stage
Grades 5-8: Logic Stage
Grades 9-12: Rhetoric Stage
Within each of the three stages, students learn about four eras of history:
Year 1: Ancient
Year 2: Middle Ages
Year 3: Renaissance
Year 4: Modern Times
So it's obvious how these would correspond to each of the twelve grades in history. In ELA, students can read literature that aligns with the period of time that they are studying. According to the link, even science fits into the above pattern:
The sciences are studied in a four-year pattern that roughly corresponds to the periods of scientific discovery: biology, classification and the human body (subjects known to the ancients); earth science and basic astronomy (which flowered during the early Renaissance); chemistry (which came into its own during the early modern period); and then basic physics and computer science (very modern subjects).
For subjects such as history, the grades below fifth (or possibly fourth) are a wasteland. Classical Education instead recommends that students in grades 1-4 learn the same history as their older counterparts, just at a more appropriate level of difficulty. The same is true of science -- which means that Classical Education contradicts the current trend of the Next Generation Science Standards. The NGSS (at least here in California) seeks to integrate middle school science, while Classical Education specializes elementary science into those four categories (Biology, Geology, Chemistry, Physics).
Not everyone agrees with Classical Education, though. Here's an article describing an attempt in New York State to teach first graders about ancient Mesopotamia:
Many of the commenters viewed the lessons as age-inappropriate, yet according to Classical Education, first graders (and fifth and ninth graders) should be learning ancient history. Therefore the Mesopotamia lessons are exactly what Classical Education recommends for first graders.
Of course, we can't help but notice a glaring omission at the Classical Education link. One subject is missing -- namely math. So now I wonder, is it possible to fit math into a Classical Education?
We return to the Empire State and notice how the math curriculum is divided into three stages:
Grades K-5: A Story of Units
Grades 6-8: A Story of Ratios
Grades 9-12: A Story of Functions
If we move fifth grade up one level (which makes sense, since in reality fifth graders do learn about fractions and ratios), then the three "stories" correspond to the three stages of Classical Education:
Grades 1-4: A Story of Units
Grades 5-8: A Story of Ratios
Grades 9-12: A Story of Functions
So now we have our three stories/stages. But what should students learn about during each of the four years within each story? We begin with "A Story of Units," and ask, what four things can students do with these "units" (that is, whole numbers)? Well that's obvious -- students should add, subtract, multiply, and divide them. So we should write:
Year 1: Addition
Year 2: Subtraction
Year 3: Multiplication
Year 4: Division
This makes sense for Grades 1-4. I think back to my elementary school's old math texts and recall that the first grade text had an addition problem on the front cover, the second grade text had a subtraction problem, the third grade text had a multiplication problem, and the fourth grade text had -- you guessed it -- a division problem. In practice, though, addition and subtraction are often combined, so first and second graders do a little bit of addition and subtraction each year.
Now if we apply this pattern to Grades 5-8, we see that fifth, sixth, seventh, and eighth graders are expected to add, subtract, multiply, and divide fractions, respectively. This is logical, although eighth grade may be a bit late for division of fractions (seventh, or even sixth, grade is where the Common Core and other curricula place it). Then again, we can argue that considering how much trouble students have with fractions, perhaps we should give them extra time to learn them.
Notice that "ratios" are more than just fractions. Students can learn how to do arithmetic with decimals as the same time as they learn fractions. Also, if by "ratios" we mean "rational numbers" (that is, elements of the set Q), then we need to include negative numbers as well. In this case, addition of integers in fifth grade seems a bit early. Then again, I've seen traditionalists argue that fifth graders really should learn how to add integers. At the website of Dr. Katharine Beals (one of the two main traditionalists I discuss here), the commenter "Auntie Ann" writes:
p.s. Far worse, however, in a packet for rising 8th graders, page 3 has such challenging problems as:
1) 4 + 3
2) -12 + 4
3) 19 + (-3) +6
etc. These are problems I would expect any 5th grader to be able to do. No wonder kids can't hack algebra in 8th grade...or 9th...or college.
Now let's move on to A Story of Functions in high school. Our pattern states that students should learn how to add, subtract, multiply, and divide "functions" during freshman, sophomore, junior, and senior year, respectively. Here we can interpret "functions" as meaning polynomial functions. So this implies that freshmen (that is, Algebra I) should learn how to add x to x to get 2x (in other words, work with linear functions), and juniors (that is, Algebra II) should learn how to multiply x by x to get x^2 (in other words, work with quadratic functions). In senior year, students can divide polynomials to obtain rational functions as part of Pre-Calculus.
The only year that's illogical here is sophomore year -- students don't really separate subtraction from addition in algebra. Of course, if this is the year between Algebra I and II, the only subject that makes sense to put here is Geometry!
But if the point of this scheme is to satisfy traditionalists, then naturally they'll object to this sequence ending at Pre-Calculus rather than Calculus. It may be possible to move all three of these stories down one grade level:
Grades K-3: A Story of Units
Grades 4-7: A Story of Ratios
Grades 8-11: A Story of Functions
(Grade 12: A Story of Calculus)
On one hand, the three stories K-3, 4-7, and 8-11 no longer correspond to the three Classical Education stages 1-4, 5-8, and 9-12. On the other, having each math story begin one year before a history stage begins means that high school math lines up better with the historical eras -- freshmen learn Geometry along with ancient history (as in the ancient Greeks and Euclid). Algebra II now lines up with the Middle Ages -- recall that the word "algebra" was coined by Arabs during this era.
Division of fractions is now moved down to seventh grade, which is reasonable. But now addition of integers is pushed down to fourth grade, and multiplication of units to second grade. I believe I've seen some traditionalists argue that the times tables are a second grade subject.
Kung wraps the lecture by playing another lecture about infinity -- of course, that lecture is itself,
played faster. And in the lecture that Kung plays, Kung plays the lecture even faster. But, as you've figured out by now, all of these infinitely many infinity lectures play in finite time.