Pick a number
Odd coincidences happen all the time. The other day I was on my way to Tulsa and had stopped at the so-called “World’s Largest McDonalds” that is on I-44 down around Vinita. I was in the mood to put another coat of cholesterol on my arteries and my family wasn’t along to perform an intervention. As a consequence I was in line to get a Quarter-Pounder combo with Fries, when I looked up an saw Bitty Bubba in the next line over.
Bitty Bubba, as you may recall, is one of my old friend Bubba’s nephews. He is the great hope of the family, so, as a consequence, the sum total of the family’s desires and fears are resting on his frail shoulders.
He was alone there too, so we sat down to eat together and began to talk. As it turns out, he’s taking a class in elementary probability and had learned a game he wanted to show me,
“Guess a number between one and three,” he directed.
“Inclusive or exclusive?” I asked. Whatever other interests I might have, I will be a mathematician until I die, so I make sure I know the rules before I get involved in a game. Bitty Bubba was a little confused by my question, however.
“What?” he asked.
“Inclusive means you include the endpoints—one and three in this case—and exclusive means you exclude them.”
“Uh...inclusive I guess,” he said.
“Two?” I replied.
“No,” he laughed. “I was thinking of 2.5.”
He had me. I’d been careful to ask about the endpoints, but I’d neglected to ask what he’d meant when he said “number,” assuming that he meant integer. You know what they say, if you assume you make an ass of you and me.
“That’s pretty funny,” I said. I appreciated his humor, but I didn’t want to waste a teaching opportunity. “Suppose, though, we’d stuck to the integers. What would’ve been my probability of getting a correct answer then?”
“Well,” he said, drawling it out to give himself time to think, “it would be one-third because there were three numbers to choose from—one, two, and three—but only one right number.”
“You are correct,” I said. I wasn’t surprised. In my experience as a math teacher, most students just know this. “Now, I’ve got a harder question to ask. You increase the numbers to those with a decimal representation. What is the probability of my choosing correctly then?”
His eyebrows nit for a moment like he was thinking very intently.
"I want to say zero," he finally answered, "but I don't know why."
"You are correct," I answered him. "The probability is zero. You can think about it like this. If you had ten numbers, the probabilty would be one tenth and if you had 100 numbers, it would be one one-thousandth, but there are infinitely many decimal numbers between one and three. When you say there are infinitely many numbers, that means whatever number you name there are more decimal numbers than that. That means the probability is smaller than one tenth or one one-thousandth or one one-millionth and so on. The probability is smaller than any number you can think of, so it's zero."
Bitty Bubba got that look in his eyes that so many of my students get when they are really fascinated with what I am saying. Oddly enough, it reminds a lot of people of a deer caught in headlights.
"Well, that's really interesting, but..."
"Do you know what is more interesting?" I asked. I hated to waste this teaching opportunity.
"But I really..."
"The probability of choosing a decimal number at random and getting a repeating decimal is zero as well," I said.
"Repeating decimals?" Bitty Bubba asked.
"Yes, repeating decimals are those like 1.222222... where the 2s go on forever or 2.31234343434...where the 34s repeat forever. There are infinitely many repeating decimals, there are even more that don't repeat. In fact, the number that are repeating is somehow insignificant to the the number that doesn't."
I was waxing eloquent now and look on Bitty Bubba's face was more mesmerized than ever.
"Indeed, we need new concepts to make sense of this. Instead of cardinality that we use when we count the members of a set, we need to use the concept of measure..."
"Oh!" Bitty Bubba cried out. "My bus is leaving."
With that he rushed from the dining area.
I thought this was strange because I hadn't even seen any buses. I thought it was even stranger when I saw someone that looked a lot like Bitty Bubba riding down the interstate below me on a motor cycle.
Monday, November 3, 2008
Merle Bunyan, who is a good ole boy from the same neck of the woods where I grew up, still thinks that going out to the section line roads and spotlighting deer is a legitimate means of entertaining himself, or he did, I should say. A few years back he was out late at night doing just that and mysteriously disappeared for five days. This experience, whatever it was, changed him.
When he turned up, everyone who knew him, myself included, simply assumed he had been out on a drunk. This is what we call Ockham’s Razor where I come from, which is to say it’s simplest to assume they were drunk.
This assumption didn’t hold up over the long haul because after a while Merle started exhibiting what his wife interpreted as psychiatric symptoms. Last year during the Super Bowl he turned off the TV and started tidying up the house. His wife took him into the free clinic the very next morning.
His problems turned out to be deep seated. He started showing up for work on time and attending PTA meetings. It was then his psychiatrist started having sessions with him under hypnosis. During these sessions, it became apparent that his change in behavior stemmed back to the time he’d disappeared for five days.
I became involved when his wife, who I knew from back home, brought a tape of some of his sessions.
“Why are you bringing this to me,” I asked. “I’m not a psychiatrist. I am just a mathematics professor.”
“I know that,” she said, “and that is exactly why Merle and I need you. He would have been here himself, but he’s cleaning out the oven for me, poor man.” A little tear trickled down her cheek.
“Why do you need a mathematician?” I asked.
“Because of what’s on this tape,” she said. It was at that point she pulled out a cassette recorder and pressed the play button. What follows is my interpretation of what I heard on that tape and, as such, is necessarily incomplete.
During the early part of the tape, it became clear that Merle was describing being taken aboard a spaceship. I am a skeptic about such things. The alien abduction has entered our modern mythology, so I believed Merle was just describing what he’d seen in some movie. And, in fact, the details that he recounted of his abduction matched the prevailing mythology to an astonishing—or perhaps not so astonishing—degree.
In spite of this, I came to believe his case might be an authentic alien abduction because of what I heard on the tape. The reason I changed my mind was the same reason Merle’s wife had brought me the tape, the mathematics on it.
You need to understand Merle was never any good at the mathematics he learned in school or what he had to use in life. He once bought cigarettes on special: One pack for 3 dollars, two for seven. On the tape, however, he said some things that required a precise knowledge of higher-level mathematics—at least a higher level than we would ever expect from him.
The first of these things was his description of a breed of alien creature that I will refer to as “termites” for reasons that will become apparent later in the story. From his description, I believe the termites were what followers of science fiction might refer to as nanites. These are little self-replicating robots.
The termites were creatures that reproduced by something very much like binary fission, which is to say they divided in half like bacteria. Each of the daughter termites was no more than half as large as the mother termite. Merle said that the daughter termites didn’t grow before they reproduced and they were twice as fast as the mother. The original birth he witnessed took about a minute, the second generation only took about thirty seconds to reproduce, the third generation only took about 15 seconds to reproduce, and so forth.
The psychiatrist who was conducting the interview asked how he could follow them after the first few divisions because they would be getting quite small by that time, and Merle said the aliens were displaying it on the screen of a fancy microscope for him.
The first point at which I began to become suspicious that what I was hearing might be true was a place in the interview when Merle said something that surprised me.
“They divided infinitely many times, and it was over in two minutes.”
The first thing that caught my attention was Merle using the word “infinitely,” and, in fact, pronouncing it correctly. His English is not much better than his math, and I’ve already explain how bad that was. The second was saying that it was over in two minutes, which would be true for the situation he described. The difficulty of understanding this was illustrated by the confusion in the psychiatrist’s next question.
“If there were infinitely many divisions, wouldn’t that take an infinite amount of time?”
The answer to that is no. The reason for this is that each generation of termites only took half as long to produce as the preceding generation. This produces what mathematicians call a geometric series. Geometric series of the type Merle described are known to sum to finite quantities. In this particular case it’s not difficult to see. Take a piece of paper and draw a line segment on it. Mark the endpoints and the center point. Label one endpoint with a zero, the other with “2,” and the center point with a “1.” We will use this line segment to keep a running total of the time it takes for all of the generations of termites to reproduce.
The first generation reproduced in one minute, and we can denote that one minute by the “1” in the center. The next generation reproduces in a half a minute. Adding this half minute to the half minute already noted splits the difference between 1 and 2 taking us to three quarters. The next generation takes one quarter of a minute which splits the difference again, and so on. At the end of each generation the total time expired splits the difference between the previous total and 2. Consequently, after infinitely many generations only two minutes would have expired.
There were details about this that bothered me one of which was the following. If the termites divided exactly in two at each stage, then, after an infinite number of divisions, they would have been of zero volume. However, for the sake of argument, I decided perhaps the termites might only be intrusions into our space of creatures that existed in higher dimensions.
Before I could explore this thought, the psychiatrist asked a very intelligent question.
“Reproduction requires energy,” he began. “Typically creatures which divide by binary fission spend time between each generation taking in energy so they can reproduce. How did the creatures you describe do this?”
It was then that Merle told about the “toothpicks.” The aliens who had abducted Merle laid out toothpicks—or some edible substance that looked like toothpicks—for the termites to eat. That’s why I’ve take to calling them termites.
The termites consumed the toothpicks in the following strange way. The first termite was at the center of the toothpick and consumed the middle one-third in a very precise fashion. This left two pieces. Each of the daughter termites then consumed the middle third of those two pieces making for a total of four pieces of the original toothpick. After the next generation, there were eight pieces and so forth.
“So,” the psychiatrist asked, “was the toothpick entirely consumed after the two minutes?”
It was Merle’s answer to this that convinced me something strange had happened.
“Did I say something funny?” I psychiatrist asked.
“All of it was gone, but there was an infinite amount left,” was all Merle said. The psychiatrist didn’t understand, but I did.
The process that Merle described was very similar to the construction of what mathematicians call the Cantor set. The construction of the Cantor set begins with a line segment. At each stage the middle third of each remaining segment is removed leaving twice as many segments that are one-third as long as the segments in the previous stage of the construction. The net affect is the total length of the remaining segments is two-thirds of the previous total length.
When one takes two-thirds of the previous length at each stage of an infinite process, the amount remaining will tend to zero. However, at no stage are the endpoints of any interval removed, so even after all of the length of the interval is removed, there are an infinite number of points remaining. Indeed, the Cantor set is said by mathematicians to be uncountable which is more intensely infinite than the set of natural numbers.
Return to earth
At that point in the interview, Merle told how the aliens communicated with him telepathically. Having read his mind, and sensing certain deficiencies in him as a sapient being, they implanted the termites into his brain in order that they might improve his character. Then they let him go.
Since his return, he has exhibited a remarkable change in character, so I am convinced his strange tale must be true. Certainly his is not the Merle I remember.