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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 mathematicians want to perfect in the mid-19th century?

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

4. Who were Johann and Jakob Bernoulli?

5. Who else, besides Newton, independently discovered a calculus method?

Short Essay Questions

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

2. Describe some of Gauss's work.

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

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

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

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

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

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

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

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

Essay Topics

Write an essay for ONE of the following topics:

Essay Topic 1

Write an essay to explain what is meant by "infinitude of primes." Use the following questions to guide your writing. What was Euclid's definition of a prime number? How were prime and composite numbers related according to Euclid?

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

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?

(see the answer keys)

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