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Name: _________________________ | Period: ___________________ |

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

## Multiple Choice Questions

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

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

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

**4. Which phrase best describes Newton as a student at Cambridge?**
**(a)** Quiet recluse of no intelligence. **(b)** Tolerant, mildly interested in science. **(c)** A highly praised genius. **(d)** Unnoticed, but remarkable.

**5. What was aleph naught?**
**(a)** A symbol to state the sum of a series. **(b)** A method to determine the sum of a series. **(c)** A symbol to represent the number of items in a set. **(d)** A method to numerate terms.

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

**7. 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.

**8. What did Cantor's work do to mathematics?**
**(a)** It caused a reevaluation of basic algebra. **(b)** It raised arguments on the origins of geometry. **(c)** It caused much agreement among mathematicians on the use of calculus. **(d)** It forced the reexamination of set theory.

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

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

**11. Who encourages Newton during his studies at Cambridge?**
**(a)** John Napier. **(b)** Henry Briggs. **(c)** Isaac Barrow. **(d)** Henry Stokes.

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

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

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

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

## Short Answer Questions

**1.** What did Cantor define as the continuum?

**2.** In what area was Gauss especially interested?

**3.** Which of the following is a series that the Bernoullis proposed did not converge on a finite sum?

**4.** Which word best describes Newton's childhood?

**5.** When was Euler born?

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