Day 14: Heart of Mathematics

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

A job well done proving that there are an infinite number of irrationals. We showed that there is an irrational between any two rationals. Since there is an infinite number of "rational gaps", there is an infinite number of irrationals.

• Reminder: Project Assignment 1.

  By Friday, 10/3, give me your tentative plans. I'll expect a full, one-page description of what you plan to do by then.

• Did you get out your 3-D glasses, and have a look at the figure on p. 126? What does the figure show?

• Did you understand the binary representation of the square root of two? What's so special about 10?

• 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.

• Why is 1 = .9999999999....?

• Note the use of the pigeonhole principle (p. 129) in the argument that a rational has a repeating decimal expansion.

• Let's look at some problems:
  • I.2
  • I.5
  • I.15
  • I.18
  • I.20