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

## Multiple Choice Questions

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

**2. What did Dunham describe as the same between artistic movements and mathematical studies in the 19th century?**
**(a)** They are both less concern with reality. **(b)** They are both fascinated on artificial images, such as photography. **(c)** They are both focused on realism. **(d)** They are both becoming less abstract.

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

**4. Which of the following demonstrates the successive squared denominator series?**
**(a)** 1 + 1/2 + 1/6 + 1/10 + 1/15 . . . **(b)** 1 + 1/4 + 1/9 + 1/16 . . . **(c)** 1 + 1/2 + 3/4 + 4/5 . . . **(d)** 1 + 1/2 + 1/3 + 1/4 + 1/5 . . . 1/1000 . . .

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

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

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

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

**9. What were the two types of transfinite cardinals defined by Cantor?**
**(a)** c and pi. **(b)** 1 and pi. **(c)** pi and Ã—ÂÃ¢â€šâ€™. **(d)** Ã—ÂÃ¢â€šâ€™ and c.

**10. Which of the following is a series that the Bernoullis proposed did not converge on a finite sum?**
**(a)** 1 + 1/2 + 1/3 + 1/4 + 1/5 . . . 1/1000 . . . **(b)** 1 + 1/2 + 1/6 + 1/10 + 1/15 . . . **(c)** 1 + 2 + 3 + 4 + 5. . . **(d)** 1 + 1/2 + 3/4 + 4/5 . . .

**11. What was similar about both Euler and Gauss as children?**
**(a)** They both showed incredible abilities in mathematics. **(b)** They both were too poor to attend a Universtiy. **(c)** They both were late to attend school. **(d)** They both showed a desire to lead a simple life.

**12. Who eventually solved the sum of the successive squared denominator series?**
**(a)** Leonhard Euler. **(b)** Jakob Bernoulli. **(c)** Johann Bernoulli. **(d)** John Napier.

**13. What hindered Euler's work as he grew older?**
**(a)** His increasing blindness. **(b)** He had a stroke. **(c)** He had very bad arthritis. **(d)** His hearing was getting worse.

**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 with a sum of 2. **(b)** It's a convergent series of cubic numbers. **(c)** It's a convergent series of triangular numbers. **(d)** It's a divergent series squared numbers.

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

## Short Answer Questions

**1.** In his later life, what position did Isaac Newton hold?

**2.** When was Euler born?

**3.** Which of the following was one of Gauss' early discoveries?

**4.** What is the sum of the series 1 + 1/2³ + 1/3³ + 1/4³ ... 1/k³ . . .?

**5.** Who was Euler's teacher?

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