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

2. Where did Euler study at the age of 20?
(a) The Academy in St. Petersburg.
(b) Oxford.
(c) University of Moscow.
(d) Cambrigde.

3. Where was George Cantor born?
(a) Germany.
(b) Russia.
(c) Britian.
(d) Switzerland.

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

5. What was similar about both Euler and Gauss as children?
(a) They both were too poor to attend a Universtiy.
(b) They both were late to attend school.
(c) They both showed incredible abilities in mathematics.
(d) They both showed a desire to lead a simple life.

Short Answer Questions

1. What did Dunham describe as the same between artistic movements and mathematical studies in the 19th century?

2. What were the main technique(s) that Euler used to find the sum of the series?

3. How did Gauss feel about his best work?

4. What did Cantor's beliefs lead him to think?

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

Short Essay Questions

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

2. Describe some of Gauss's work.

3. Explain any methods used by Cantor that were unsuccessful.

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

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

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

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 What was Gauss's major unpublished achievement in geometry?

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

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

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

(see the answer keys)

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