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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. How did Cantor finally prove his theory?**
**(a)** By using basic algebra. **(b)** By extension of the infinite series. **(c)** By refining and expanding set theory. **(d)** By extension of the Pythagorean Theorem.

**2. What did Gauss construct?**
**(a)** A proof that demonstrates Newtonian physics. **(b)** A system where the angles of a triangle add up to more than 180 degrees. **(c)** A proof that demonstrated the circumference of Earth. **(d)** A system where the angles of a triangle add up to fewer than 180 degrees.

**3. To how many decimal places did Newton determine the number for pi?**
**(a)** Twelve places. **(b)** Eight places. **(c)** Three places. **(d)** Nine places.

**4. What is one proof that Euler was able to prove?**
**(a)** Newton's method of calculus. **(b)** "little Fermat theorem." **(c)** Bernoulli's principle of lift. **(d)** Descartes' number theory.

**5. What did Euler's sum surprisingly connect?**
**(a)** The area of squares and the area of circles. **(b)** The area under a curve. **(c)** The squares of area and square roots. **(d)** The circumference of a circle and right triangles.

## Short Answer Questions

**1.** Where does the center of mathematical thinking shift to after Italy?

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

**3.** What is the sum of the series 1 + 1/2³ + 1/3³ + 1/4³ ... 1/k³ . . .?

**4.** What sum did Euler find for the series?

**5.** What did George Cantor determine to be true of a set of rational numbers?

## Short Essay Questions

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

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

**3.** Describe the great theorem explained by Dunham in this chapter.

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

**5.** What was Gauss's major unpublished achievement in geometry?

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

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

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

**9.** Explain why Eulers sum of Ï€Â²/6 was in some ways surprising.

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

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