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 Gauss do with his best work?

2. Who encourages Newton during his studies at Cambridge?

3. How did Gauss feel about his best work?

4. Which of the following was one of Gauss' early discoveries?

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

Short Essay Questions

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

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

3. What were the two transfinite cardinals discovered by Cantor, and what method did he use to determine them?

4. Describe who were Jakob and Johann Bernoulli.

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

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

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

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

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 What was Gauss's major unpublished achievement in geometry?

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

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

Dunham presented a number of controversies over publication and intellectual property rights in the history of mathematics. Explain what seems to be the common issues historically in the publication of one's discoveries. Describe what conditions often lead to controversy over intellectual property rights.

Essay Topic 3

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.

(see the answer keys)

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