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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 did Cantor's cardinal numbers represent?

**2.** What did mathematicians want to perfect in the mid-19th century?

**3.** What was Dunham central theorem for this chapter?

**4.** What did Cantor develop?

**5.** What were the two types of transfinite cardinals defined by Cantor?

## Short Essay Questions

**1.** Give an example of a series who's sum is still unknown.

**2.** Describe who were Jakob and Johann Bernoulli.

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

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

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

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

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

**9.** Explain any methods used by Cantor that were unsuccessful.

**10.** Describe Cantor's difficult personal life.

## 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

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?

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