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. Which of the following was a major part of Gauss' work in mathematics?

2. What did Cantor suspect about transfinite cardinals?

3. What was Dunham central theorem for this chapter?

4. How did Euler prove if the number 4,294,967,297 was prime or composite?

5. What is true about the successive squared denominator series proposed by the Bernoullis?

Short Essay Questions

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

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

3. Describe some of Gauss's work.

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

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

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

7. Describe Cantor's difficult personal life.

8. Explain what was the definition of a series before the Bernoullis, and give examples of what was known.

9. Describe what the Bernoullis discovered about series, and give an example.

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

Essay Topics

Write an essay for ONE of the following topics:

Essay Topic 1

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 2

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.

Essay Topic 3

Summarize the discoveries on series made by the Bernoulli brothers and later, Euler. How did the Bernoullis advance mathematical understanding of an infinite series, and how did Euler even further advance this knowledge? Give examples. What principles about series and the sum of series are still being worked out in modern mathematics?

(see the answer keys)

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