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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 under a curve.
(b) The circumference of a circle and right triangles.
(c) The squares of area and square roots.
(d) The area of squares and the area of circles.

2. What is the sum of the series 1 + 1/2³ + 1/3³ + 1/4³ ... 1/k³ . . .?
(a) 0.
(b) Nobody has determined the sum.
(c) 2.
(d) Infinity.

3. What did Cantor find after extending the continuum between 0 and 1 into two dimensions?
(a) That the set is still equal to c.
(b) That the set is equal to 1.
(c) That the series is infinite.
(d) That the set is infinite.

4. What did Gauss do with his best work?
(a) He did not publish it.
(b) He gave it to his students.
(c) He gave it to his son to publish.
(d) He published it.

5. What did Cantor define as the continuum?
(a) Real numbers between 0 and 1.
(b) All imaginary and real numbers.
(c) The square root of any real number.
(d) All imaginary numbers.

Short Answer Questions

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

2. Which of the following is not denumerable, proven by Cantor's theorem?

3. When was Euler born?

4. How did Cantor finally prove his theory?

5. Who encourages Newton during his studies at Cambridge?

Short Essay Questions

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

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

3. Describe who were Jakob and Johann Bernoulli.

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

5. Describe some of Gauss's work.

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

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

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 What was Gauss's major unpublished achievement in geometry?

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

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

(see the answer keys)

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