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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 concept did Dunham end his book with?
(a) Cantor and his voyage into the infinite.
(b) Archimedes and the infinite series.
(c) Heron's triangulated area.
(d) Newton's method of calculus.

2. Who were Johann and Jakob Bernoulli?
(a) Brothers and students of Leibniz.
(b) Cousins and students with Newton at Cambridge.
(c) Cousins and students with Leibniz in Paris.
(d) Twin brothers and students of Newton.

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

4. What is true about real numbers between 0 and 1?
(a) They are not denumerable.
(b) No sum can be determined.
(c) They are denumerable,
(d) There is no set for these numbers.

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

Short Answer Questions

1. What series was Euler most famous for?

2. What did Cantor's work do to mathematics?

3. Where did Newton go to school before he went to Cambridge?

4. What did Cantor develop?

5. Where did Euler study at the age of 20?

Short Essay Questions

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

2. Who was Georg Cantor, and what was significant about his work in mathematics?

3. Describe who were Jakob and Johann Bernoulli.

4. Explain what was the definition of a series before the Bernoullis, and give examples of what was known.

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

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

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

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

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

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

(see the answer keys)

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