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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. On who's work did Euler base his number theory?

2. What is one proof that Euler was able to prove?

3. What did most of 19th century mathematics focus on, as highlighted by Dunham?

4. What was similar about both Euler and Gauss as children?

5. What series was Euler most famous for?

Short Essay Questions

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

2. Describe some of Gauss's work.

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

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

5. Explain how Gottfried Leibniz was able to publish his method of calculus.

6. Describe the connection between Fermat and Euler's work.

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

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

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

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 What was Gauss's major unpublished achievement in geometry?

Essay Topics

Write an essay for ONE of the following topics:

Essay Topic 1

Write a three part essay to explain Archimedes determination of circular area.

Part 1) According to Archimedes what was pi, and why is this value needed to determine circular area?

Part 2) Explain how Archimedes used a right triangle to determine the area of a circle.

Part 3) Why was the determination of circular area useful at the time of Archimedes, and how did it advance the future of mathematics?

Essay Topic 2

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?

Essay Topic 3

Compare Pythagoras's proof of the Pythagorean Theorem and how Heron's formula for triangular area can be used as an alternative proof of the Pythagorean theorem. What are the common assumptions in both proofs? What components of each proof are different?

(see the answer keys)

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