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This test consists of 5 short answer questions, 10 short essay questions, and 1 (of 3) essay topics.

## Short Answer Questions

**1.** The author notes in Chapter 6, “Infinity's Twin” that multiplying zero or infinity by itself equals what?

**2.** Jean le Rond d'Alembert came up with what idea that solved the zero problem in calculus?

**3.** Where was Carl Gauss from?

**4.** Bishop Berkeley was a philosopher whose primary achievement was the advancement of a theory he called what?

**5.** What term refers to a region of spacetime from which gravity prevents anything, including light, from escaping?

## Short Essay Questions

**1.** The author states in Chapter 6, “Infinity’s Twin” that before imaginary numbers could be accepted, several developments had to occur. Which was the first?

**2.** How did the discovery of imaginary numbers affect the study of algebra?

**3.** How does the elimination of zero help general relativity theory, according to the author in Chapter 8, “Zero Hour at Ground Zero”?

**4.** What problem do black holes present in physics, according to the author in Chapter 8, “Zero Hour at Ground Zero”?

**5.** How did l’Hopital address the problem of zero, according to the author in Chapter 5, “Infinite Zeros and Infidel Mathematicians”?

**6.** Who was Carl Gauss? What discovery did he make regarding imaginary numbers?

**7.** What discovery did Friedrich Riemann make in the field of projective geometry?

**8.** What problems were encountered in calculus with zero? How did zero apply to the physical world, according to the author in Chapter 5, “Infinite Zeros and Infidel Mathematicians”?

**9.** How did Max Planck address the problem of the ultraviolet catastrophe?

**10.** How many dimensions are required for string theory to work? How are these dimensions described in Chapter 8, “Zero Hour at Ground Zero”?

## Essay Topics

Write an essay for ONE of the following topics:

### Essay Topic 1

Discuss the biography, works, and mathematical developments of Mohammed ibn-Musa al-Khowarizmi. When was his work, Compendious Book on Calculation by Completion and Balancing completed? What did this work entail?

### Essay Topic 2

Describe and discuss the origins and developments of mathematics in Mayan culture. What was the base of the Mayan numeral system? How did this system compare with that of the Egyptians and the Babylonians? How did it deal with zero?

### Essay Topic 3

Discuss Guillaume de l'Hôpital, Bishop Berkeley, and l'Hôpital's rule. How are derivatives used to evaluate limits involving indeterminate forms under l'Hôpital's rule? How did these advancements deal with the problem of zero?

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