Wednesday, April 19, 2017

Lesson 13-6: Uniqueness (Day 136)

This is what Theoni Pappas writes on page 109 of her Magic of Mathematics:

"Many, many years ago, there was a big party being given for all the numbers of the time. One was there in all its glory."

That's right -- this is another story with a party to which numbers are invited. Last week, I read the story that Pappas wrote about a new concept -- the quaternion. Compared to the quaternions, the subject of today's story is the ordinary number pi.

Yes, I know that Pi Day was a month ago. We're even almost a week past "reversed Pi Day," which is 4/13 or April 13th. Still, the next three pages of the book are all about Pappas and her "Parable of Pi," and so I'll be writing about it the rest of this week.

At the number party, there are natural numbers like 1 and 2, rational numbers like 1/2, 1/4, and 2/3, and even algebraic numbers like sqrt(2) and sqrt(7). But then pi crashes the party.

The other numbers ask pi to prove that he belonged there. The others point out that they all lie on the number line, even irrationals like sqrt(2), which can be placed using a straightedge and compass. As it turns out, such numbers are call constructible. But, as it turns out, pi is not constructible.

The number 1 then asks pi to tell his story, and so pi obliges. The story that pi tells is given on the next few pages in Pappas.

Note: This spring break post has been edited so that I can write about Eugenia Cheng's new book, Beyond Infinity. The "Eugenia Cheng" label has been added to this post. Today I selected a random chapter to discuss.

Chapter 3 of Eugenia Cheng's Beyond Infinity is called "What Infinity is Not." She begins:

"When babies first learn to climb up a step, they are enthralled. They do it again, and again, and again, and they get higher and higher and higher, possibly all the way to the next floor unless some mean adult removes them, which they usually do."

Cheng tells us this story to compare walking up steps with walking up the natural numbers. With steps, there's a limit as to how high we can go -- a top step. But what about the naturals? Is there a top step to the natural numbers, called infinity? In other words, is infinity a natural number?

In her first book How to Bake Pi, Cheng gives several food analogies. She does the same here:

"It's a bit like the infamous lawsuit about whether Jaffa Cakes are cakes or cookies (because for some obscure reason cakes and cookies are taxed differently in the UK)."

Believe it or not, the U of Chicago text tells an eerily similar story. It appears at the beginning of Chapter 2, and I wrote about it last year on the blog (in September 2015):

Lesson 2-1 of the U of Chicago text deals with definitions. But the introduction to the chapter mentions a 1986 USA Today article concerning a non-mathematical definition: cookie. Normally, as teachers we'd ignore this page and skip directly to the first lesson, except that this article is mentioned all throughout 2-1, even including the questions!

The U of Chicago text is referring to an American lawsuit while Cheng is discussing a British lawsuit, so they are clearly two different cases. Yet the idea is the same -- whether giant food companies potentially owe millions hinges on an exact definition of "cookie."

"Now I know what the word cookie means and so do you. My 2-year-old Jamie knows what a cookie is and can ask for it by name. But the definition turns out to be so important in this case that here are these high-priced lawyers, these learned counselors asking the please tell them what a 'cookie' is...."

This passage comes from the U of Chicago text, but it could have been inserted into Cheng's Chapter 3 and no one would have batted an eyelid. In both cases, whether a certain confection is a "cookie" depends on an exact definition of cookie. Likewise, in the U of Chicago text, whether a certain figure is a rectangle depends on an exact definition of "rectangle," and in Cheng's book, whether infinity is a natural number depends on an exact definition of "natural number."

According to Cheng, for some terms, we can determine whether an object satisfies the definition of that term by checking whether it appears on a list of all such objects. But we can't do that for terms such as "prime number," because there are too many. She writes:

"The largest known prime number at time of writing has more than 22 million digits."

I wrote about this huge prime in my "Day in the Life" post from the day after winter break.

Likewise, we can't check the list of all natural numbers to see whether infinity is on the list. So Cheng begins by attempting to define "natural number":

"The most basic kind of numbers are also the first kind that small children learn to use by counting things: 1, 2, 3, 4, and so on. In mathematics these are called the natural numbers, because they're the most natural."

But this definition is inadequate since, for example, we can't tell whether the next natural number after 4 is 4 1/2 or 5.

As an aside, Cheng also writes about whether 0 is a natural number. She writes:

"Personally I can just see that arguments about whether or not 0 is a natural number can easily put people off the entire subject of mathematics altogether. But some people think it's so important that I will probably receive hate mail for suggesting it isn't; I have already had people shout at me about it at the end of talks."

Don't worry, Eugenia -- traditionalist Katharine Beals agrees it's unimportant whether 0 is a natural number or not. I wrote about this on the blog two years ago (in April 2015).

Anyway, Cheng decides that the natural numbers are "what we get if we start at 1 and keep adding 1", so by definition, here are the naturals:

1 + 1
1 + 1 + 1
1 + 1 + 1 + 1
1 + 1 + 1 + 1 + 1

Of course, we give 1 + 1 the name "two," and so on. Cheng now goes off on a tangent regarding how to pronounce the various natural numbers in different languages. She explains that in Cantonese, the words for naturals with two digits are simple -- 24 is "two-ten-four." So there are no special words for two-digit numbers as "eleven" and "twelve" and "twenty" are in English. It's sometimes suggested that this is the reason why Chinese students fare so well in math -- less time trying to figure out the words for numbers means more time to learn arithmetic. (Author Malcolm Gladwell is the most famous advocate of this theory.)

So now we want to determine once and for all whether infinity is a natural number or not. And here Cheng writes:

"So far we've seen that we can make all the natural numbers by starting with 1 and adding 1 repeatedly. But how do we know we won't get to infinity like this?"

To answer this, she provides us the following recursive definition of natural number:

The natural numbers are all the numbers that are either

-- 1, or
-- n + 1, where n is itself a natural number

She writes that infinity is obviously not 1, and so the only hope for infinity to be a natural number is it to be n + 1 for some natural number n. But since infinity + 1 = infinity, we instead reach the unhelpful circular statement "If infinity is a natural number, then infinity is a natural number."

Indeed, Cheng lists some properties that we expect infinity to have:

-- Adding 1 should not make it any bigger.
-- Adding it to itself should not make it any bigger.
-- Multiplying it by a natural number should not make it any bigger.

On the other hand, she lists some properties that we expect natural numbers to have:

-- It doesn't matter what order you add them up, because you always get the same answer. For example, 3 + 2 = 2 + 3.
-- You can subtract natural numbers from each other (as long as you're careful about not going negative, as we haven't introduced negative numbers yet).
-- If you do the same thing to both sides of an equation, the equation still holds.

These, of course, are some of the postulates from Lesson 1-7 of the U of Chicago text -- the Commutative Property of Addition and the Addition (Subtraction, etc.) Property of Equality.

Anyway, Cheng starts with:

1 + infinity = infinity.

Now we subtract infinity from both sides, which gives:

1 = 0.

This is simply not true. To Cheng, this produces a proof by contradiction -- in other words, an indirect proof -- that infinity is not a natural number!

We are already familiar with indirect proofs -- for example, today's Lesson 13-6 of the U of Chicago text derives Playfair's Postulate from the Corresponding Angles Consequence indirectly. Cheng, afraid that indirect proofs may appear circular to novices, gives a few more examples:

Suppose I were a bunny rabbit,
Then I'd have a fluffy tail.
But I don't have a fluffy tail.
Therefore I am not a bunny rabbit.

Suppose infinity were a natural number.
Then we'd be able to subtract it from both sides of the equation.
But we can't subtract it from both sides of the equation.
Therefore infinity is not a natural number.

Here's another example of an indirect proof involving a theorem from Lesson 13-6:

The Glide Reflection Theorem only works when the preimage and the image have opposite orientation, not the same orientation. If a figure and its image have the same orientation, then we know that the isometry mapping one to the other is either a translation or a rotation. This case may be a bit tricky -- it could be that the easiest way is simply to translate A to A' and see whether this translation maps B to B' -- if not, then a rotation is necessary. But how do we find the center?

We know that the center of rotation is equidistant from A and A'. Thus it lies on the perpendicular bisector of AA'. For the same reason, the center lies on the perpendicular bisector of BB'. So where these two points intersect is the center of rotation. Notice that if these two perpendicular bisectors are parallel, then the above reasoning constitutes an indirect proof that there is no rotation mapping one to the other -- that is, there is a translation map instead.

Today's worksheet covers all of Lesson 13-6. This means that not only are there questions about the Glide Reflection Theorem, but also about uniqueness in general. A modern form of Euclid's original five postulates are given.

2017 Update: Returning to Pappas, we can prove that pi is not constructible. This is related to a very famous theorem -- the impossibility of Squaring the Circle. Its proof is quite complicated, but it follows a familiar format -- the indirect proof. Basically, we assume that pi is constructible and show that this assumption leads to a contradiction. Therefore pi is not constructible.

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