Day 13 in Mat115

Today:

• Collect homework for sections 2.2 and 2.3

• Reminder: Project Assignment 1, due Friday:

  Give me your tentative plans (especially if you'll be working with someone else). I'd like a full, one-page description of what you plan to do by then, or several options of things that you'd like to consider.

• Rationals, Irrationals and Reals....
  • First of all, what is a rational number? And what are these so called lowest terms?

  • ...an irrational number...lies hidden in a kind of cloud of infinity. -- Michael Stifel

  • What makes numbers "irrational"? (The story of the Pythagoreans - a local copy) Hippasus - the Prometheus of numbers?

  • The proof that the square root of 2 is irrational (here's an alternative proof).

    The proof that the square root of 2 is irrational is based on contradiction: we assume that it is rational, and then discover that it leads to an impossibility. We could use the same idea to show that the square root of 5 is irrational.

  • Rational numbers have terminating or repeating decimal representations (i.e., rational representations!); Irrational numbers do not - there is no pattern!

  • Why is 1 = .9999999999....?

  • Pi is perhaps the most famous irrational number, but historically people have had trouble with that. For example, in the Bible, 1 Kings 7:23, we find this:

    And [Solomon] made a molten sea, ten cubits from the one brim to the other [diameter = 10]: [it was] round all about [a circle], and his height [was] five cubits: and a line of thirty cubits did compass it round about [circumference = 30].

    Pi = circumference/diameter = 3!?! Even so, that's better than the Indiana legislature did, a few thousand years later, as our authors note in the text (p. 119).

    (See also http://www.yfiles.com/pi.html for the Biblical calculation of pi.)

    Archimedes did a lot better, born in about 287 BC in Syracuse, Sicily -- check out this applet that shows how he did it!

  • Irrational Problems:
    • Find a rational number between 1001/1003 and 1002/1003

    • Find an irrational number between 1001/1003 and 1002/1003!

    • Can you find a rational number between two irrationals?

  • The real numbers are all the rationals and the irrationals together, usually represented on the "real number line". Did you get out your 3-D glasses, and have a look at the figure on p. 126? What does the figure show?

  • Why does every interval on the real line contain an infinite number of rational numbers?

  • Why does every interval on the real line contain an infinite number of irrational numbers?

  • Describe the argument given for the claim that there are more irrationals than rationals.

Next time: Problems pause