Journey Through Genius: The Great Theorems of Mathematics Test | Final Test - Hard

William Dunham (mathematician)
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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 was true when Euler used n = 5 in the statement 2²ⁿ + 1?

2. What is true about real numbers between 0 and 1?

3. What concept did Dunham end his book with?

4. Where did Newton go to school before he went to Cambridge?

5. What did Gauss construct?

Short Essay Questions

1. Give an example of a series who's sum is still unknown.

2. Describe the controversy that Newton was caught in with his publication of his calculus methods.

3. What great theorems and work of Newton did Dunham highlight?

4. Describe the great theorem explained by Dunham in this chapter.

5. Describe Newton's days in Cambridge and what he eventually came to discover.

6. Where did the center of mathematical thinking shift to in the !7th Century, and who are the major scholars of this time period?

7. Summarize in a few sentences, what types of number sets did Cantor prove to be denumerable and non-denumerable.

8. Explain any methods used by Cantor that were unsuccessful.

9. What was the great theorem of this chapter? Describe it briefly.

10. What was Euler able to prove about 2²ⁿ + 1? Why was this a great accomplishment?

Essay Topics

Write an essay for ONE of the following topics:

Essay Topic 1

Write a a three part essay to explain how Cantor's work in mathematics also fit into the cultural movements of the time.

Part 1) Describe the artistic movements of the 1860s and 1870s. In general, how could they be related to mathematical philosophy at the time?

Part 2) What was Cantor's work in mathematics in the 1860s and 1870s?

Part 3) Explain how Cantor's philosophy and mathematical theorems fit with the new artistic and philosophic ideas of the time.

Essay Topic 2

Write an essay to describe what was known about mathematics and geometry before and in the time of Thales.

Part 1) Describe what we know about geometry before Thales.

Part 2) In what ways were mathematics and geometry used before Thales?

Part 3) Explain what we know about Thales' work. How did it advance mathematics?

Essay Topic 3

Describe Gauss's work on what was to be known as non-euclidean geometry. What was Gauss's system for triangles where angles added up to fewer than 180 degrees? What were some of his conclusions? Did he publish his work? Was their any controversy surrounding his work on this system? Explain.

(see the answer keys)

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