M425 Assignments/Study Guide 1 - Spring 2002

Here L1, L2, etc. refer to lectures (i.e. chapters) in the text. The dates shown are when assignments are made. Class discussions are based on the assigned readings, problems, and questions. You should know when, where, and what for the highlighted mathematicians.

Day 1 - Monday January 14
- Read L1, L2
- Write up exercises 1.6, 1.7, 2.1, 2.3
- Know the concepts:
Day 2 - Wednesday January 16
- Write up exercises 2.7, 2.8, 2.9
- Know formulas for:
- Study questions:
  - Show that the set of all positive integers is equivalent to the set of all positive integral multiples of 5.
  - What are the first two stages of geometric development? Describe the characteristics of each and give examples.
  - Describe the arithmetic, geometric, and heronian mean of two numbers. How do they compare? Why is the heronian mean relevant?
Day 3 - Friday January 18 -- Thales
- Read L3
- Write up exercises 3.3, 3.4, 3.9 (make a table for the given examples)
- Study questions:
  - What is the third stage of geometric development? Describe its characteristics and give examples.
  - What is the Greek mystery?
  - Ontogeny recapitulates phylogony. What does this mean?
Day 4 - Wednesday January 23 -- Pythagoras
- Read L4
- Write up exercises 4.1, 4.2 - notebooks are due Friday for a preliminary check
- Study questions:
  - Is the Pythagorean theorem aptly named? Give arguments yes and no.
  - Give at least two of your favorite proofs of the pythagorean theorem.
  - What geometric facts do you need to assume to make your proofs work?
  - Define the concepts congruent by addition and congruent by subtraction.
Day 5 - Friday January 25 -- Pythagoras
- No new reading assignment -- but note that Monday's assignment is long and complex.
- Study questions:
  - Explain how the law of cosines is related to the pythagorean theorem.
  - Explain how Pappus's theorem generalizes the pythagorean theorem and prove it.
Day 6 - Monday January 28 -- Pythagoras
- Read L5
- Write up exercises 5.5, 5.7a, 5.8
- Study questions:
  - Show how sums, differences, products, quotients, and square roots can be found geometrically.
  - What was the great crisis for the Pythagoreans, and how was it brought on?
  - Prove that the square root of 2 is not rational.
  - What does it mean to say that numbers are incommensurable?
  - How are rational and irrational numbers recognized by their decimal expansions?
  - Given a periodic decimal, find the corresponding rational number.
  - Which numbers can be constructed geometrically?
  - Which regular polygons can be constructed? How?
Day 7 - Wednesday January 30 -- Eudoxus
- Read L6
- Write up exercises 6.1, 6.3, 6.5
- Study questions:
  - Give a proof that areas of triangles are to each other as their bases are to each other, assuming that the bases are commensurable.
  - How did Eudoxus prove that fact, when it was pointed out that some line segments are incommensurable?
  - How would we prove it, given our knowledge of limits?
Day 8 - Friday February 1
- Read L7
- Write up exercises 7.1, 7.7, think about term project
- Study questions:
  - What are the steps of material axiomatics?
  - Why are technical terms useful in such a theory?
  - What is the difference between axioms, postulates, and theorems?
Day 9 - Monday February 4 -- Euclid
- Read L8
- Write up exercises 8.1 (give reasons), 8.4(a,b,c), 8.9, think about term project
- Study questions:
  - Why the name Elements? Who was Euclid and when did he live?
  - What are some similarities between the Elements and the Bible?
  - Was Euclid's Elements the first? What happened to it after Euclid?
  - What is in the Elements?
  - What is the Euclidean algorithm? How does it work?
  - How can the roots sometimes be found, geometrically, for a quadratic equation?
Day 10 - Wednesday February 6 -- DO FOR MONDAY'S CLASS -- Archimedes
- Read L9
- Write up exercises 9.1, 9.6, 9.10, think about term project
- Study questions:
  - About when did Archimedes live? Where? Where did he study?
  - Tell some stories about Archimedes' life and work.
  - What remarkable facts did Archimedes discover about the cylinder vs the sphere?
  - What method did Archimedes use to discover such geometric facts? Explain that method.
  - What method did he use to prove those facts? Explain that method.
  - What is the theorem of the broken chord? What trig identity does that imply?
Day 11 - Friday February 8 -- -- EXAM 1

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