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Name: _________________________ | Period: ___________________ |

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

## Multiple Choice Questions

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

**2. Where did George Cantor live in the 1860s and 1870s?**
**(a)** Germany. **(b)** Russia. **(c)** Scotland. **(d)** Britian.

**3. What did most of 19th century mathematics focus on, as highlighted by Dunham?**
**(a)** Algebra. **(b)** The theoretical. **(c)** The immediately practical. **(d)** Geometry.

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

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

**6. How did Euler prove if the number 4,294,967,297 was prime or composite?**
**(a)** He used Newton's calulus methods. **(b)** He factored it. **(c)** He used his own rule of squares. **(d)** He divided it by 2.

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

**8. Which of the following did Dunham concentrate on as one of Newton's great advances?**
**(a)** Binomial theorem. **(b)** Area of a sphere. **(c)** Quintic theorem. **(d)** Quadratic equation.

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

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

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

**12. What did George Cantor discover?**
**(a)** A method to measure a curved area. **(b)** A way to compare the relative sizes of infinite sets. **(c)** A way to determine the accuracy of a calculation. **(d)** A method to measure infinity.

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

**14. What is true about real numbers between 0 and 1?**
**(a)** They are not denumerable. **(b)** There is no set for these numbers. **(c)** No sum can be determined. **(d)** They are denumerable,

**15. To how many decimal places did Newton determine the number for pi?**
**(a)** Three places. **(b)** Twelve places. **(c)** Eight places. **(d)** Nine places.

## Short Answer Questions

**1.** What did Cantor develop?

**2.** What did Cantor struggle with later in his life?

**3.** What did Cantor's cardinal numbers represent?

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

**5.** What did Cantor find after extending the continuum between 0 and 1 into two dimensions?

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