On One Hand But Then on the Other

By Bobby Neal Winters

I was approached yesterday by a friend of mine at church in Opolis in the following way.

“You’re a math professor, so I got a question for you,” he said.  “It’s from my grandson.”

I remained calm outside, but on the inside I had the reaction a gunfighter in the Old West had whenever someone said, “They say you’re pretty fast.”  You never know when the person asking the question might be faster.

I decided to take my chances and listen.  He held out his right hand and began to count extending his fingers one by one:

“One, two, three, four, five,” he said, extending his pinky last. “And five is ten,” he said extending all of the fingers of his left hand at once.

I nodded because I knew we weren’t at the hard part.

He then held out the fingers of his left hand and started extending them thumb first:

“Ten, nine, eight, seven, six,” he said, giving emphasis to the six. “And five is eleven, so you got eleven fingers.  How can I explain to my grandson why this is wrong.”

And my friend knows it’s wrong.  But knowing it’s wrong and being able to explain why are two different things.  And I could go on to a digression about politics here, but my point is math, or at least the explaining of math.

My friend knows this is wrong because we have ten fingers and ten does not equal eleven, no matter how fast you talk.

Let’s first analyze how this is presented because that is very important to how the confusion comes in.  We start off with something that is true: one, two, three, four, five, and five is ten.   What has happened there?  We’ve listed off the names of the first five numbers that we learned when we learned how to count.  We’ve set up a correspondence between those names and our fingers.  

The next part is where something subtle is done which sets us up for the confusion. When he says, “And five makes ten,” he’s made a very subtle shift.  He’s using the name five to refer to the quantity of fingers on his left hand.”  The five he said when he held out his pinkie was referring to a place in order; the five he said when he held out all his fingers at once was referring to the quantity of fingers.  Mathematics call the first one ordinality and the second cardinality; think of these as order and quantity.

In listing the numbers in standard order, “one, two, three, etc,” the name of the last number listed is also the name of the quantity of the items in the list.

This does not work when you count backwards, and that is one of the things that the example my friend brought me illustrates. In counting backwards from ten, there is no point at which the number counted with also be the quantity of things counted. (Ironically, if he’d started counting at eleven it could be made to work, but you can’t do that because you have ten fingers. It is the center of this trick that you start counting backwards at ten because you know there are ten fingers.)   

When you hold out the pinkie on your left hand as you count backwards from ten until you get to six, you are giving its order as if all the fingers were counted beginning with the other hand.  It can refer to a quantity, but that quantity would be the pinkie itself and the fingers on the left  hand.

Now, you must understand that I didn’t tell my friend all this.  I said, “It’s a confusion between cardinality and ordinality.  Don’t let your grandson play poker with Bill.”

Bill is a man who’s given me poker lessons. After $15 worth, I learned: don’t play cards will Bill. But that is a different story.

Notions of Sameness

Notions of Sameness

By Bobby Neal Winters

Mathematicians use sets and structures on sets to model realities that they perceive only in their minds.  I dealt with this in the matter of topology in one of my recent essays.  These structures induce special properties that we want to preserve.  I am still interested in this mainly in the case of topological spaces, but I will work my way up to it via a series of examples from other areas of mathematics.

The first example is that of sets with no structure in which we can only discuss the concept of cardinality.  This is to say, how many elements does a set contain?

With finite sets we can say that a set is composed of five elements or seventeen elements or two trillion five hundred fifty four billion two hundred twenty seven million three hundred twelve thousand five hundred and two elements.  There is a number we can associate with them.  

For infinite sets, the above does not work; we have to do a flanking maneuver. We avoid the word number and use the term cardinality.  We say that two sets have the save cardinality if they can be put into one-to-one correspondence with each other.  

We use the language of functions to make such notions precise.  Functions are thought of as having a sending set (the domain) and a receiving set (the range).  The range is like a target.  When two sets have the same cardinality, there is a function from one set to the other that is one-to-one and onto.  A function is onto if every point in the receiving set is matched with a point in the sending set.  A function is one-to-one if no set in the receiving set is matched twice.

A function that has both of these properties at once is christened with the high-falutin’ label of bijection.  It’s technical and pretentious but remember it anyway because it’s important.

If two sets have a bijection between them, they have the same cardinality.  If one (and therefore the other) of those sets is finite, they will also have the same number of points.  By this mathematical slight of hand of using a particular type of function to create a new notion of “sameness” we’ve expanded the notion of number/size into the notion cardinality/size.

This is done in the realm of algebra as well.  And here I must be careful because I am using the word “algebra” differently than many use it.  If you had pre-algebra and algebra in middle and/or high school, or if you have had intermediate algebra or college algebra in college, then rest assured that I am talking about something that is related to that in the way that a Tomahawk Missile is related to, well, a tomahawk.

Algebraists, and I am over-simplifying, study sets with binary operations on them.  For example, they might study the real numbers with the operation of addition or the positive real numbers with the operation of multiplication.  (Notice I put the word “might” in there because they aren’t really interested in those two particular sets, but I will use them because I don’t want to talk about a lot of algebra now.)

The real numbers numbers with addition and the positive real numbers with multiplication are examples of groups.  Groups are sets with binary operations that obey certain rules.  Don’t worry about what those rules are; if you can’t help it, look it up on Wikipedia.

Those of you who have had a course in college algebra might remember the logarithm.  I put emphasis on the word might there because I’ve no doubt that you have seen the logarithm.  You might remember it; you might wake up in cold sweat screaming it at the top of your lungs; your lover might have attempted to comfort you afterwards asking, “What is a logarithm, Sweet heart?” only to have you deny ever having heard the word.

In any case, the logarithm is a function from the positive real numbers to the real numbers that is bijective, i.e. it is one of those important one-to-one correspondences.  The important thing about it is that it changes the operation of times to that of plus: log(a times b) is equal to log(a) plus log (b). In mathematical ages B.C.--before calculators--the logarithm was used as a means of doing multiplication by turning it into addition.  You may not have notice, but addition is a lot easier than multiplication.  

Such a function that is bijective and preserves the operations on the groups--in this case turning multiplication into addition--is called an isomorphism.  The groups are said to be isomorphic to each other and a mathematician in the capacity of algebraist can’t tell the difference between them.

I could go on listing the various areas of mathematics and the functions they use to preserve there important concepts, but instead of doing that let me focus on topology.

A topological space is a set along with certain special subsets that we call “open” sets.  Topological concepts are those concepts that can be defined in terms of open sets.  As a consequence of this, we are interested in functions that preserve open sets.  A bijective function between two topological spaces that takes an open set in the first to an open set in the second is called a homeomorphism.  A pair of spaces between which there is a homeomorphism are said to be homeomorphic.  Spaces that are homeomorphic are the same as far as a topologist is concerned.  

Most frequently, I see homeomorphisms defined differently.  We usually define a homeomorphism to be a continuous function which is invertible and whose inverse is continuous.  This is mathematically equivalent to the other definition I’ve just given, but it’s shorter.  

Continuous functions are interesting in their own right.  Indeed, one might argue that topology was created in order that we might better study continuous functions.  There is also quite a well-developed theory of continuous functions that allows us to check whether a particular function is continuous.  For example, it is quite easy to use calculus to check that the logarithm and its inverse the exponential function are both continuous; it might be less easy to show they preserve open sets.

But in our desire to be efficient with our definitions, we should take care not to obscure for the beginner the very thing that homeomorphisms do: they preserve open sets.  As they preserve, nay set up a one-to-one correspondence between the open sets of each space, any properties defined in terms of open sets will be preserved.  That is the point.

Pick a number

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.