Journey Through Genius: The Great Theorems of Mathematics Test | Mid-Book 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 was NOT true about Archimedes, according to Dunham?

2. Exactly what limit is reached at a quartic equation?

3. Which of Euclid's postulates troubled many of the following generations of mathematicians?

4. What range of values did Archimedes determine for pi?

5. What did Ferdinand Lindeman prove in 1882?

Short Essay Questions

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

2. Describe in two sentences Archimedes's method for determining circular area.

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

4. What was already known about circles before Archimedes?

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

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

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

8. What did Euclid state for his theory on prime numbers?

9. Describe the contents of Cardano's book.

10. Describe what work of Euclid's fascinated Plato and his theory on the shape of the Universe.

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

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

Describe how Gerolamo Cardano became involved in the solution to cubic equations. Why was their a controversy over his publication of the solution to cubic equations? Explain.

(see the answer keys)

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