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This test consists of 5 multiple choice questions, 5 short answer questions, and 10 short essay questions.
Multiple Choice Questions
1. Which of the following could NOT be included as a step in Euclid's great theorem?
(a) Take a finite group of primes and add them together, plus one.
(b) After summation, the new number can be prime or composite.
(c) If a new number is found to be composite, then it must have some prime as a divisor.
(d) Divide a infinite group of primes by the sum of their composites.
2. What did Euclid state about pi in Elements?
(a) There is no relationship between the area of a circle and its circumference.
(b) The proportion of area to circumference is never equal.
(c) There is a constant relationship between the area of a circle and the square of its diameter.
(d) The proportion of diameter to area is never equal.
3. What did Dunham consider as Archimedes's "masterpiece"?
(a) Archimedes' work on determining a number value for pi.
(b) Archimedes' work on volume to surface area ratios.
(c) Archimedes' work on shperes, cones, and cylinders.
(d) Archimedes' work on determining angular measurements.
4. How did Lindeman prove his conclusion?
(a) Lindeman proved that square roots are irrational numbers.
(b) Lindeman proved that some numbers are constructable without the use of a compass.
(c) Lindeman proved that all numbers are constructable with a compass and ruler.
(d) Lindeman proved that some numbers are not constructable with only a compass and straight-edge.
5. Which of the following is true in modern math about twin primes?
(a) They are not considered whole numbers.
(b) We don't know if they are finite or infinite.
(c) Their sum is always another prime number.
(d) They are infinite.
Short Answer Questions
1. How many sides did the pentadecagon have, as presented by Euclid?
2. Who was Eratosthanes?
3. What did Apollonius work with in mathematics?
4. In what time period did mathematicians find a solution to cubic equations?
5. What shape was NOT demonstrated in the Elements as having a relationship to other shapes?
Short Essay Questions
1. Describe in two sentences Archimedes's method for determining circular area.
2. Explain when the knowledge of ancient scholars was rediscovered.
3. What was Euclid's definition of composite and perfect numbers?
4. How did Heron find the area of a triangle, and what did Dunham state about Heron's work?
5. Explain what Neils Abel proved.
6. Explain who was Gerolamo Cardano, and how did he become involved with the solution to the cubic.
7. What did Dunham explain about the shift in learning from the West to the East?
8. Describe the contents of Cardano's book.
9. Describe why Dunham infers that Euclid's Elements was an evolutionary book not so much for what it said, but in how it was presented.
10. What did Euclid state for his theory on prime numbers?
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