This post is dated April 23rd, one week after Easter and the last day of spring break. This is the 113th day of the year, and so we read page 113 of Pappas.
This is what Theoni Pappas writes on page 113 of her Magic of Mathematics:
"Georg Cantor's set theory and transfinite numbers were brilliant accomplishments. His proof of the countability of the rational numbers was elegant."
As it turns out, a new section begins on page 113 -- "Cantor & the Uncountable Real Numbers." Here "Cantor" refers to the great 19th century German mathematician, Georg Cantor.
We don't need to read page 113 to figure out what Pappas is talking about here. Eugenia Cheng writes about this in Chapter 6 of her latest book, Beyond Infinity. The name of this chapter is "Some Things Are More Infinite than Others." She begins:
Children sometimes have this kind of argument:
"I'm right."
"I'm more right."
"I'm right times a hundred."
"I'm right times a million."
"I'm right times a billion."
"I'm right times infinity!"
"I'm right times two infinity!"
"I'm right times infinity squared!"
However, as Cheng points out, "two infinity" and "infinity squared" are equal to just infinity. This is implied in Cantor's proof that there exist only countably infinitely many rationals (that is, just as many rationals as whole numbers). In a sense, there are "infinity squared" rationals, since there are infinitely many numerators and infinitely many denominators. But "infinitely squared" is infinity, and so there are only (countably) infinitely many rationals.
Cheng continues:
"There are more irrational people than rational people in the world, probably. In fact, most people are a bit irrational -- that, to me, is an important part of being human and not a computer."
She uses this as an analogy about numbers -- just as more people are irrational than rational, more numbers are irrational than rational. In some ways, this is counterintuitive, because if one were to think of a number, that number most likely be a rational number -- indeed a natural number.
And so Cheng asks the reader to find the area of a circle whose radius is the chosen number. She points out that if the original chosen number was rational, the resulting area must be irrational:
"If you multiply a rational number by an irrational number, the result is irrational. Likewise, if you add a rational and an irrational number, the result is rational."
She points out that in order for the area to be rational, we must choose some contrived irrational number for the radius, something like 1/sqrt(pi):
pi * (1/sqrt(pi))^2 = 1.
Bringing this back to Pappas, notice that on her Mathematical Calendar, she often comes up with such contrived problems in order to make the answer be the date. So on the first of the month, she might indeed ask for the area of a circle with radius 1/sqrt(pi).
Cheng also shows us how to find the radius of any circle whose area is the rational number a/b:
r = sqrt(a/(pi b))
pi r^2 = pi(sqrt(a/(pi b)))
= (pi a)/(pi b)
= a/b.
She suggests that the reason that we must come up with such contrived radii just to make the area rational is that there are so many more irrational numbers than rational numbers.
Cheng proceeds:
"We're going to 'count' the irrational numbers, except that we don't have a way of saying what the irrational numbers are. Irrational numbers are defined by what they're not: they're the real numbers that are not rational."
She adds that she'll actually define irrational numbers in a subsequent chapter. (I still remember the college class where I first learned about what irrational numbers are.)
Cheng lists the following things that we already know:
1. The rational numbers are countable.
2. If you put two countable sets together, you get another countable set (as in the two-floor Hilbert Hotel).
The Hilbert Hotel is the dominant analogy used to explain the sizes of infinite sets. Indeed, the "room" I mentioned last year when describing a David Kung DVD lesson is in a Hilbert Hotel:
We begin as Cantor did -- let's imagine what a solution would look like. We would have every decimal in a room -- and we'd be able to prove it by giving a list of all of the decimals. So all we have to do is check the list -- if all the decimals are listed then we have a 1-1 correspondence, and if there's a decimal missing then we don't have a 1-1 correspondence.
And Cheng is getting ready to do the same thing in her book. She concludes that as soon as we know how many reals there are, then we also know how many irrationals there are:
rationals if reals then irrationals
countable (known) countable countable
uncountable uncountable
She continues:
"The real numbers are difficult to pin down, but for now let's say they are all the possible decimal numbers, where the decimals are allowed to go on forever, repeating or not repeating. (In fact, all decimals go forever if you put 0's on the end; we just don't usually bother writing all those 0's.)
Our goal is to determine whether it's possible to move every guest from a hotel whose rooms are numbered with all the reals, to one whose rooms are numbered all the rationals. She eliminates simple possibilities, such as placing the smallest real in room 1 (because there is no smallest real), and just reversing the real number's digits to form a natural number (because some real numbers have infinite representations such as 1/6 = 0.16666..., but there is no infinite natural number ...66661).
And so Cheng comes up with a list and applies Cantor's famous diagonal argument:
new room no. old room no.
1 0.238795317...
2 0.984718573...
3 0.389716438...
4 0.777362889...
5 0.444317895...
6 0.879000001...
7 0.892225673...
8 0.191919234...
She explains, "Then the digits we ask for when we knock on each person's door are the ones in bold here, along the diagonal."
new room no. old room no.
1 0.238795317...
2 0.984718573...
3 0.389716438...
4 0.777362889...
5 0.444317895...
6 0.879000001...
7 0.892225673...
8 0.191919234...
Cheng then adds one to each digit marked in bold, to produce the following number:
0.39042174...
She adds, "and we can show that this person is not in any room n by looking at their nth digit, which is the different from the person who is actually in room n. So this person has not been evacuated to any room, and the smarty-pants [who claims to have matched every real with a natural] has failed."
And this is a contradiction, completing the indirect proof that the set of real numbers is uncountable.
Most of the time, the uncountability of the real numbers is the end of the story. But Cheng moves on with another possible example of an uncountable set:
"Another way in which something can be 'more infinite' than the natural numbers is more subtle, and perhaps relates more to the word 'uncountable'"...another way in which we can fail to evacuate a hotel is if we get decision fatigue."
She assumes that the original hotel has double rooms, and the destination hotel has single rooms. We would like to be able to give the following instruction:
"If you're currently in room n, then one of you go to room 2n and the other to room 2n - 1."
but there isn't necessary any way to determine which guest goes to 2n and which goes to 2n - 1.
The following analogy is usually attributed to the British mathematician Bertrand Russell. We first assume that the guests in the rooms are pairs of shoes -- left shoes and right shoes. Then it's easy to empty them into a single room hotel. We could go:
-- left shoe 1, right shoe 1, left shoe 2, right shoe 2, left shoe 3, right shoe 3, ...
and so on. Written mathematically:
-- The left shoe n will go to position 2n - 1.
-- The right shoe n will go to position 2n.
But if these were socks, then there's no such thing as a "left sock" or a "right sock." And indeed, as obvious as it might seem, it's impossible to prove that we can make infinitely arbitrary choices unless we assume an extra postulate, or axiom, in set theory. As Cheng explains:
"The question of whether or not this is possible is not exactly resolved in mathematics, but is a tricky and subtle point that still worries mathematicians. It is called the Axiom of Choice."
She tells us that if we assume the Axiom of Choice, then the socks fit into the hotel -- and so there are only countably many socks. But without AC, then the socks might not fit -- they'd be uncountable!
As it happens, AC is set theory's Parallel Postulate. Trying to prove AC from the other axioms of set theory is as futile as trying to derive Euclid's Fifth Postulate from the first four. There are some versions of set theory in which AC holds, and others in which it fails -- just as there are some versions of geometry in which parallels uniquely exist, and others in which they aren't unique.
But let's return to the main result of this chapter. There exist uncountably many reals, yet all but countably many of them are irrational. So the irrationals must be uncountable as well. Actually, because of AC issues, it might be that the rationals are countable and the irrationals are countable, yet their union, the reals, are uncountable. As it turns out, this isn't the case, because we can prove that the irrationals are uncountable without using AC. Here's how:
One of my favorite questions is, is there a 1-1 correspondence from R to R \ Q? As it turns out, there does exist such a 1-1 correspondence, and we can prove it. Let's think of in terms of a Hilbert's Hotel -- except it's an uncountable hotel with a room for every irrational. Now a bus arrives containing all of the rational numbers. How can we fit them into the hotel?
Well, let's take our favorite irrational, pi, and so we add pi to every rational. We see that the sum of a rational and an irrational is irrational, and so we can ask each rational to go to the room where that irrational is presently sitting. And where do those irrationals already in those rooms go? By now we should know the Hilbert's Hotel pattern -- they add pi to themselves to get new irrationals, and then go to the room where those new irrationals are. In other words:
The number x + n * pi (x, rational, n a whole number) goes to room x + (n + 1)pi.
Of course, we must be careful that x + (n + 1)pi isn't accidentally rational. But there is an indirect proof that it's always irrational: assume that x + (n + 1)pi = y is rational. Then rearranging the equation, we obtain pi = (y - x) / (n + 1), the quotient of two rationals, so is itself rational. So we get that pi is rational -- a contradiction since pi is irrational. Therefore x + (n + 1)pi is irrational. QED
We notice that in this case, there are uncountably many rooms, yet only countably many occupants need to change rooms. Most irrationals don't have to move -- in particular, sqrt(2) doesn't move, neither does e, and neither does -pi.
Here's another thing I've noticed -- sets of numbers that are special, or fit some sort of pattern, tend to be countable, while sets of numbers that don't fit patterns tend to be uncountable. The natural numbers fit the ultimate pattern -- 1, 2, 3, so they are countable. The rational numbers are countable (they fit the pattern a / b), while the irrationals are countable. The algebraic numbers are countable (they fit into polynomials), while the transcendental numbers are uncountable.
What about the set of all numbers consisting of the digits 0 and 1 only -- is it countable? Even though this may sound like a pattern, it really isn't. After all, in binary, all reals consist of only two digits, so consisting of two digits isn't special enough to make the set countable. The set of reals with only two digits is uncountable, regardless of which digits or the base. (In the special case of 0 and 1, we can use Cantor's diagonal argument to see why.) If the digits are 0 and 2, and the base is changed to ternary (base 3), we obtain a famous uncountable set -- the Cantor fractal set!
The idea, though, that most real numbers are irrationals is profound. After all, at one time, ancient mathematicians were surprised that irrational numbers exist at all.
Here is a link to a common indirect proof that sqrt(2) is irrational:
http://www.math.utah.edu/~pa/math/q1.html
Neither the U of Chicago nor Glencoe gives the proof outright. But both hint at it -- I just mentioned the U of Chicago's square root proofs. The Glencoe text asks the students to prove that if the square of a number is even, then it is divisible by four. As we can see at the above link, this fact is directly mentioned in the irrationality proof.
I remember once reading the proof of the irrationality of sqrt(2) in my textbook back when I was an Algebra I student. Until then, I had always heard that sqrt(2) was irrational, but I never realized that it was something that could be proved. So I was fascinated by the proof. Naturally, the text only included this as an extra page between the main sections, so it was something that the teacher skipped and most students probably ignored.
The irrationality of sqrt(2) has an interesting history. It goes back to Pythagoras -- he was one of the first mathematicians to use sqrt(2), since his famous Theorem could be used to show that the diagonal of a square has length sqrt(2). The website Cut the Knot, which has many proofs of the Pythagorean Theorem, also contains many proofs of the irrationality of sqrt(2):
http://www.cut-the-knot.org/proofs/sq_root.shtml
Now there is a famous story regarding sqrt(2) and Pythagoras. At the following link, we see that Pythagoras was the leader of a secret society, or Brotherhood:
http://nrich.maths.org/2671
Now Pythagoras and his followers believed that only natural numbers were truly numbers. Not even fractions were considered to be numbers, but simply the ratios of numbers -- numberhood itself was reserved only for the natural numbers. In some ways, this attitude resembles that of algebra students today -- when the solution of an equation is a fraction, they often don't consider it to be a real answer, even though modern mathematics considers fractions to be numbers. (The phrases real number and imaginary number reflect a similar attitude about 2000 years after Pythagoras -- that some numbers aren't really numbers.) So of course, the idea that there were "numbers" that weren't the ratio of natural numbers at all was just unthinkable.
Pythagoras and his followers must have spent years searching for the correct fraction whose square is 2, but to no avail. Finally, one of his followers, Hippasus, discovered the reason that they were having such bad luck finding the correct fraction -- because there is no such fraction! And, as the story goes, Pythagoras was so distraught, afraid that the secret that sqrt(2) was irrational would be revealed, that he ordered to have poor Hippasus drowned at sea!
But as I said, nowadays students simply complain when they have a fractional, or worse irrational, answer to a problem. No one has to drown any more just because of irrational numbers.
Question 10 on my test review, therefore, is actually the final step of that proof, since that's the step where the contradiction occurs. They are given a triangle with sides of length 3 and 8, and two angles each 40 degrees (one of which is opposite the side of length 3). The students are to use the Converse of the Isosceles Triangle Theorem to show that the missing side must also be of length 3, and then the Triangle Inequality to show that 3 + 3 must be greater than 8, a contradiction.
When I wrote this problem, I had trouble deciding how difficult I wanted my indirect proof to be. For example, I considered giving 100 as the measure of the angle opposite the side of length 8, and give only one 40-degree angle instead. Then the students would have to use the Triangle Angle-Sum Theorem to find the missing angle as 40 degrees before applying the Isosceles Converse.
Or, to go even further, we can derive a contradiction without making the angle isosceles at all. For example, we could make the angle opposite the 8 side to be, say, 90 degrees instead of 100. Then the missing angle would be 50 instead of 40. If the triangle is drawn so that 50 degrees is opposite the 3 side, then by the Unequal Angles Theorem, the missing side would be less than 3, so the sum of the two legs would still be less than the longest side.
But this might confuse the students even more -- especially if the 90-degree angle is marked with a box (to indicate right angle) rather than "90." A right triangle might lead a student to use the Pythagorean Theorem to find the missing leg. Although this still eventually leads to contradiction -- the missing side would be sqrt(55), which isn't less than 3 -- that irrational side length might still cause some students to drown.
And so I wrote my Question 10 on the review so that it will actually help the students prepare for the corresponding question on the test. I balance out this tough question with some easier questions about logic (converse, inverse, etc.). Hopefully the test won't be too hard for the students.
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