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

William Dunham (mathematician)
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This test consists of 15 multiple choice questions and 5 short answer questions.

Multiple Choice Questions

1. What did George Cantor determine to be true of a set of rational numbers?
(a) They are all composite numbers.
(b) They are all prime numbers.
(c) They are denumerable.
(d) They are non-denumerable.

2. What were the two types of transfinite cardinals defined by Cantor?
(a) 1 and pi.
(b) c and pi.
(c) אₒ and c.
(d) pi and אₒ.

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

4. Where did Newton go to school before he went to Cambridge?
(a) Cambridge Prep.
(b) Oxford Grammar School.
(c) Charles II Grammar School.
(d) The King's School.

5. In what area was Gauss especially interested?
(a) The elements of geometry.
(b) The proof of the infinite series.
(c) The circumference of Earth.
(d) The elements of number theory.

6. Who else, besides Newton, independently discovered a calculus method?
(a) John Napier.
(b) Pierre de Fermat.
(c) Isaac Barrow.
(d) Gottfried Leibniz.

7. Who was Euler's teacher?
(a) Isaac Newton.
(b) Jakob Bernoulli.
(c) Gottfried Leibniz.
(d) Johann Bernoulli.

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

9. 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 infinite.
(c) That the set is equal to 1.
(d) That the series is infinite.

10. What did Euler prove about 2²ⁿ + 1?
(a) That the statment is sometimes prime and sometimes composite.
(b) That the statement is neither prime nor composite.
(c) That the statement is always a composite number.
(d) That the statement is always a prime number.

11. Where does the center of mathematical thinking shift to after Italy?
(a) To Turkey and Russia.
(b) To Germany and Russia.
(c) To Britian and Scotland.
(d) To France and Britian.

12. What was Dunham central theorem for this chapter?
(a) That the sum of a set of real numbers is finite.
(b) That there are other transfinite cardinals greater than c.
(c) That the sum of an infintire series is always infinite.
(d) That the area of a circle is fundamentally related to the square of its area.

13. What was most noticeable about Euler at a young age?
(a) He was not very quick with arithmatic.
(b) He had an aptitude for literature.
(c) He was very athletic.
(d) He had a remarkable memory.

14. What was described as true about the series 1 + 1/2 + 1/6 + 1/10 + 1/15 + 1/21?
(a) It's a divergent series squared numbers.
(b) It's a convergent series of cubic numbers.
(c) It's a convergent series of triangular numbers.
(d) It's a divergent series with a sum of 2.

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

Short Answer Questions

1. What did Cantor suspect about transfinite cardinals?

2. What did Newton's calculus involve?

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

4. What did Cantor develop?

5. Who eventually solved the sum of the successive squared denominator series?

(see the answer keys)

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