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 did British scholars accuse Leibniz of?

2. What was true when Euler used n = 5 in the statement 2²ⁿ + 1?

3. What did Cantor suspect about transfinite cardinals?

4. Where was Euler born?

5. What did Dunham describe as lacking from calculus previous to the mid-19th century?

Short Essay Questions

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

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

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

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

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

6. Describe Cantor's difficult personal life.

7. What was Gauss's major unpublished achievement in geometry?

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

9. Explain why Eulers sum of π²/6 was in some ways surprising.

10. Describe what mathematical and artistic movements are focused on in the second half of the 19th century.

Essay Topics

Write an essay for ONE of the following topics:

Essay Topic 1

Summarize the parts of Euclid's Elements. Use the following questions to guide your writing. What were the basic components to Elements? What were the most important theorems and proofs in Elements? How did Elements change the future of mathematics. Name a few mathematicians after Euclid who used Elements in their own work.

Essay Topic 2

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 3

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?

(see the answer keys)

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