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