Thursday 10 November class

• Activity: "Dodge Ball" from The Heart of Mathematics

Topics: How to count/measure infinite sets!

First we have to examine closely what we do when we count a finite set. Essentially we are making what is called a one-to-one correspondence between the first however-many counting numbers (the natural numbers) and the items that we are counting.

The idea of one-to-one correspondence is what will carry over into infinite sets.

• David Hilbert's Hotel, and some theorems about the sizes of some infinite sets: See the Science4All article. (Warning: it contains math notation that may not display correctly on a phone. Here is another article which uses less math notation but has fewer details)

If an infinite set of numbers (or anything else) can be put into a one-to-one correspondence with the st of natural numbers (that is, it can be put into the form of an infinite list), we say it is a countable set. In the Science4All article, countable sets are called listable, which is a good name for them but is not standard.

First we discussed Hilbert's infinite hotel. Even if every room is occupied, we can accommodate an additional guest, or even a countably infinite number of guests! So you can say "infinity plus 1 is infinity" or "infinity plus infinity is infinity" - but wait! That second statement needs some care, because it turns out there are different sizes of infinity.

We showed the following in class:
 • The whole numbers are countable (listable)
 •  The even numbers are countable (listable)
 •  The integers are countable (listable) - this is also proved in some of the videos linked below
 •  The rational numbers are countable (listable) - this is also proved in some of the videos linked below
 • The real numbers are not countable. In other words, the set of real numbers is larger than the set of natural numbers (or even the set of rational numbers)!
The implication of this is that the set of irrational numbers is uncountable, which means that it is reasonable to say that there are more irrational numbers than rational numbers. (More on this next time.)

Look for the proofs of the following theorems in the article or in this video and this video (for the Cantor proof) , and here is another video as well. (all three of them are worth watching.)

    • Theorem: The integers are countable (listable)
    • Theorem: all countable (listable) sets have the same size (same number of elements)
    • Theorem: The rational numbers are countable (listable)
   And then the big one:
    • Theorem: The real numbers are uncountable (they are not listable).
The last one is Cantor's diagonal proof, and it relies on the strategy that Player 2 can use for "Dodge Ball" to make sure they always win.


Homework:
• Journal assignment: write the proofs of the Theorems listed above in your own words.
• Prove that the odd numbers are countable (listable) by pairing them up with the natural numbers. (Make a one-to-one correspondence with the natural numbers. Watch the videos linked above if you have doubt about what this means.)

• Next time: Power sets and the Continuum Hypothesis. (The continuum is another name for the real number line.)
  See Math is Fun on sets for some set notation that we will be using.

Rational and Irrational numbers UPDATED - Tuesday 1 November class

Update: here are some links to good sources about how people in Euclid's time and place thought about numbers, and about the Pythagoreans in particular:
Euclid's Proof of the Infinitude of Primes (from the Prime Pages, a great source for all things related to prime numbers. This discussed how the numbers were viewed at Euclid's time and place.)
Pythagorus - Greek Mathematics (from The Story of Mathematics website)
Pythagoras - The Music and Math Connection (video, from the Science Channel: Warning: video may autoplay)

• Rational and Irrational numbers:
We begin by defining some important sets of numbers (some we have seen before):
the Natural numbers - which are also called the positive integers
the Whole numbers (which include the Natural numbers) - also called the non-negative integers
the Integers (which include the Whole numbers)
the Rational numbers (which include the Integers)
If we think of "number" as a directed distance  ( a vector) starting from 0 on the number line, then there are points on the number line that correspond to these numbers we have defined so far. But it turns out that there are ponts (distances) which cannot be represented as a ratio of integers. We call these numbers Irrational numbers.

Examples of numbers which are irrational include π and the square root of 2. We will prove later that the square root of 2 is irrational.

We showed how the square root of 2 appears as a length (distance), namely, the hypotenuse of a right triangle whose sides both have length 1. Using the Pythagorean Theorem, the hypotenuse has length the square root of 2.

For any rational number, we can prove (using the Pigeonhole Principle) that its decimal expansion will either terminate or (eventually) repeat (here's another description of the proof from A Math Less Travelled): also, conversely, a terminating or repeating decimal will always represent a rational number. So from this we know that an irrational number must be represented by a nonterminating, nonrepeating decimal. That decimal is called the decimal expansion of the number.

More on the decimal expansions of rational numbers: how long can they go on before they repeat? (Here is a discussion similar to what I said in class which also uses the Pigeonhole principle: scroll down at that link until the header "Decimal representations of rational". )


How to write a repeating decimal as a ratio of integers: videos from Khan Academy part 1, part 2
  So 0.9999999... = 1
[A better way to prove that 0.9999999... = 1 uses the idea of limit and the sum of an infinite geometric series - this is usually done in pre-calculus classes. The fact is that an infinite decimal expansion is really an infinite sum: 0.9999999... means 9/10 + 9/100 + 9/1000 + 9/10000 + ... (the sum has infinitely many terms).]


Homework:
(Journal assignment pending)

Problems to do:
1) By using long division by hand, find the decimal expansions of these rational numbers:
    $\frac{1}{250}$
    $\frac{3}{11}$
    $\frac{7}{12}$
2) Rewrite the following decimals as ratios of integers: reduce to lowest terms
    2.3
    0.056
    0.1717171717...
    0.2555555...

 
Next up: proving that the square root of 2 is irrational; other irrational numbers; more on decimal expansions; the Real Numbers