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 is a series that the Bernoullis proposed did not converge on a finite sum?

2. What did Cantor suspect about transfinite cardinals?

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

4. What did Cantor's cardinal numbers represent?

5. What did Gauss do with his best work?

Short Essay Questions

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

2. Why did Euler start working on the sum of series?

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

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 What was Gauss's major unpublished achievement in geometry?

6. Describe some of the characteristics of Leonhard Euler, and what made him successful.

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

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

9. Who was Georg Cantor, and what was significant about his work in mathematics?

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

Essay Topics

Write an essay for ONE of the following topics:

Essay Topic 1

Write a three part essay on the lives and work of both Euler and Gauss.

Part 1) What was similar about Euler and Gauss from an early age? What characteristics did they have in common?

Part 2) Give some examples of great theorems proposed by Euler and by Gauss. Do you think their work had anything in common mathematically?

Part 3) What could be considered Euler's greatest work? What could be considered Gauss's greatest work? How did both mathematicians advance the study of mathematics?

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

In Dunham's book, there were many descriptions of mathematicians and their personal lives. Write an essay to describe the common elements or characteristics that seem to be shared among these historically famous mathematicians. Also, explain what personal characteristics made them all excellent mathematicians.

(see the answer keys)

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