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This test consists of 5 multiple choice questions, 5 short answer questions, and 10 short essay questions.

## Multiple Choice Questions

**1. What did Dunham describe about the following series 1 + 2 + 3 + 4. . .?**
**(a)** The sum converges to infinity. **(b)** The sum converges to a finite term. **(c)** The sum diverges to infinity. **(d)** The sum grows ever smaller.

**2. What did George Cantor determine to be true of a set of rational numbers?**
**(a)** They are non-denumerable. **(b)** They are denumerable. **(c)** They are all prime numbers. **(d)** They are all composite numbers.

**3. What did Dunham describe as the same between artistic movements and mathematical studies in the 19th century?**
**(a)** They are both less concern with reality. **(b)** They are both fascinated on artificial images, such as photography. **(c)** They are both becoming less abstract. **(d)** They are both focused on realism.

**4. What was aleph naught?**
**(a)** A symbol to state the sum of a series. **(b)** A method to numerate terms. **(c)** A symbol to represent the number of items in a set. **(d)** A method to determine the sum of a series.

**5. Where was Euler born?**
**(a)** Germany. **(b)** Switzerland. **(c)** Denmark. **(d)** Finland.

## Short Answer Questions

**1.** What did Gauss construct?

**2.** Where was George Cantor born?

**3.** Which of the following demonstrates the successive squared denominator series?

**4.** What did Cantor's work do to mathematics?

**5.** What did Cantor find after extending the continuum between 0 and 1 into two dimensions?

## Short Essay Questions

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

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

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

**4.** What was Gauss's major unpublished achievement in geometry?

**5.** Describe some of Gauss's work.

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

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

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

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

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

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