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This test consists of 15 multiple choice questions and 5 short answer questions.

Multiple Choice Questions

1. Which of the following was one of Gauss' early discoveries?
(a) A method to simplify Newton's calulus.
(b) A proof that the Pythagorean Theorem was correct.
(c) A way to construct a regular 17-sided polygon.
(d) A demonstration of the sum of the series 1 + 1/2³ + 1/3³ + 1/4³ . . . 1/k³ . .

2. Which of the following was a major part of Gauss' work in mathematics?
(a) Simple proofs to demonstrate Bernoulli's series.
(b) Proofs to show that Archimedes' number theory was wrong.
(c) Proofs on the area of a square.
(d) Elemental proofs related to the foundations of algebra.

3. Which of the following is a quote from Bertrand Russell included by Dunham?
(a) "Mathematics, rightly viewed, posses not only truth, but supreme beauty."
(b) "Mathematics is truth."
(c) "Matematicians are the gatekeepers of knowledge."
(d) "The study of mathematics is the study of the universe."

4. What is true about the successive squared denominator series proposed by the Bernoullis?
(a) The sum converges.
(b) The sum converges to 2.
(c) The sum diverges into infinity.
(d) The sum diverges.

5. What did mathematicians want to perfect in the mid-19th century?
(a) The method of finding the volume of spheres.
(b) The method of finding the area under a curve.
(c) The definition of infinite.
(d) The definition of pi.

6. What didn't Euler attempt?
(a) A series starting with the number 1.
(b) A series where exponents are odd.
(c) A series of sequencially smaller terms.
(d) A series where exponents are even.

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

8. What did Cantor suspect about transfinite cardinals?
(a) That the sum of the series is infinite.
(b) That the sum of the series is finite.
(c) That there are transfinite cardinals even greater than c.
(d) That there are transfinite cardinals much less than c.

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

10. How did Gauss feel about his best work?
(a) He was unceratin if it would be accepted by his collegues.
(b) He was confident that it would change mathematics.
(c) He was uncertain if it was useful.
(d) He was confident that his students would find it of great importance.

11. What was the same about the series proposed by Leibniz and the series proposed by Bernoulli?
(a) Both series were composed of successively smaller terms.
(b) Both series were convergent.
(c) Both series were divergent.
(d) Both series were composed of successively larger terms.

12. What did Cantor's cardinal numbers represent?
(a) Infinite sets.
(b) Finite series.
(c) Sets of all imaginary numbers.
(d) Series of prime numbers.

13. What is one proof that Euler was able to prove?
(a) Newton's method of calculus.
(b) "little Fermat theorem."
(c) Descartes' number theory.
(d) Bernoulli's principle of lift.

14. What did Newton's calculus involve?
(a) Proving the cubic equation.
(b) Determining the volume of a sphere.
(c) Proving the existance of pi.
(d) Determining the area under a curve.

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

Short Answer Questions

1. Which word best describes Newton's childhood?

2. Who else, besides Newton, independently discovered a calculus method?

3. Which name does NOT belong?

4. What hindered Euler's work as he grew older?

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

(see the answer keys)

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