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This test consists of 5 short answer questions, 10 short essay questions, and 1 (of 3) essay topics.
Short Answer Questions
1. What did Cantor develop?
2. What was aleph naught?
3. Where did Euler study at the age of 20?
4. What did George Cantor determine to be true of a set of rational numbers?
5. What did Dunham describe as lacking from calculus previous to the mid-19th century?
Short Essay Questions
1. Explain what was the definition of a series before the Bernoullis, and give examples of what was known.
2. What was Euler able to prove about 2²ⁿ + 1? Why was this a great accomplishment?
3. Summarize in a few sentences, what types of number sets did Cantor prove to be denumerable and non-denumerable.
4. Where did the center of mathematical thinking shift to in the !7th Century, and who are the major scholars of this time period?
5. Why did Euler start working on the sum of series?
6. Describe some of Gauss's work.
7. Who was Georg Cantor, and what was significant about his work in mathematics?
8. Explain any methods used by Cantor that were unsuccessful.
9. Explain why Eulers sum of Ï€Â²/6 was in some ways surprising.
10. Describe the connection between Fermat and Euler's work.
Write an essay for ONE of the following topics:
Essay Topic 1
Write a three part essay to compare Archimedes's and Newton's determination of a number value for pi.
Part 1) Explain what general approach each mathematician used in the determination of pi.
Part 2) What prior knowledge of mathematics is required to use each method?
Part 3) Which number determination for pi was more accurate, Newton's or Archimedes? Explain.
Essay Topic 2
Summarize the puzzles that were presented as a result of Euclid's infinitude of primes. Name and describe some of these puzzles from a historical perspective, then explain what is known about each in modern mathematics. What puzzles remain unsolved today?
Essay Topic 3
Explain the accuracy of Archimedes determination of pi. How did he determine the number value of pi? How accurate is Archimedes's measurement of pi? What other mathematician was able to determine a number value, though more accurate, for pi?
This section contains 884 words
(approx. 3 pages at 300 words per page)