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 was described as true about the series 1 + 1/2 + 1/6 + 1/10 + 1/15 + 1/21?

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

3. How did Cantor finally prove his theory?

4. Where did Euler study at the age of 20?

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

Short Essay Questions

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

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

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

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

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

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

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

8. Describe Cantor's difficult personal life.

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

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

Essay Topics

Write an essay for ONE of the following topics:

Essay Topic 1

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.

Essay Topic 2

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?

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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