Journey Through Genius: The Great Theorems of Mathematics Test | Final Test - Medium

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

2. What was true when Euler used n = 5 in the statement 2²ⁿ + 1?
(a) The statement was a perfect number.
(b) The statement was a composite number.
(c) The statement was a prime number.
(d) The statement was not a prime number.

3. Which of the following is not denumerable, proven by Cantor's theorem?
(a) A set of a geometric series.
(b) A set of transcendental numbers.
(c) A set of rational numbers.
(d) A set of imaginary numbers.

4. What were the main technique(s) that Euler used to find the sum of the series?
(a) Trigonometry and basic algebra.
(b) Cubic equations.
(c) Calculus methods.
(d) Quadratic sums,

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

Short Answer Questions

1. What was Dunham central theorem for this chapter?

2. Who was Euler's teacher?

3. Where was Euler born?

4. When was Euler born?

5. What is true about the successive squared denominator series proposed by the Bernoullis?

Short Essay Questions

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

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

3. Describe the connection between Fermat and Euler's work.

4. Explain why Eulers sum of π²/6 was in some ways surprising.

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

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

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 What was Gauss's major unpublished achievement in geometry?

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

9. Describe who were Jakob and Johann Bernoulli.

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

(see the answer keys)

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