Journey Through Genius: The Great Theorems of Mathematics Test | Mid-Book Test - Easy

William Dunham (mathematician)
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This test consists of 15 multiple choice questions and 5 short answer questions.

Multiple Choice Questions

1. Dunham showed that Heron's proof could also be used as which of the following?
(a) A proof of Archimedes' number theory.
(b) A proof of the Pythagorean Theorem.
(c) A proof of Euclid's number theory
(d) A proof of Hippocrates' squared areas.

2. Which of the following was NOT true about Archimedes, according to Dunham?
(a) His death was described as being by the hands of a Roman invader.
(b) Archimedes died because he refused to follow orders until he completed a math problem.
(c) He died during the fall of Syracuse.
(d) He died as a soldier.

3. In Elements, how many postulates must be accepted as given?
(a) Twenty-two.
(b) Five.
(c) Eighteen.
(d) Twelve,

4. What shape was NOT demonstrated in the Elements as having a relationship to other shapes?
(a) Pentagon.
(b) Hexagon.
(c) Hyperbola.
(d) Triangle.

5. How many definitions were stated in Elements?
(a) Five.
(b) Thirty.
(c) Twenty-three.
(d) Eighteen.

6. Which shapes as described by Euclid, inspired the Greek philosopher Plato?
(a) Regular polyhedrons.
(b) Manifolds.
(c) Triangles.
(d) Spheres and cones.

7. 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) If a new number is found to be composite, then it must have some prime as a divisor.
(c) Divide a infinite group of primes by the sum of their composites.
(d) After summation, the new number can be prime or composite.

8. Which of the following is true in modern math about twin primes?
(a) They are infinite.
(b) Their sum is always another prime number.
(c) We don't know if they are finite or infinite.
(d) They are not considered whole numbers.

9. Which of Euclid's postulates troubled many of the following generations of mathematicians?
(a) Euclid's postulate on creating an arc.
(b) Euclid's postulate on right triangles.
(c) Euclid's proof on right triangles.
(d) Euclid's postulate on parallel lines.

10. Which of the following becomes an important definition in mathematics that was first presented in Elements?
(a) 180 degree angle.
(b) Circle.
(c) Parallel line.
(d) Intersection.

11. What was true about Heron's theorem as described by Dunham?
(a) It was to determine the area of a circle by measuring a right triangle inside the circle.
(b) It was to find the area of a triangle when only the length of the sides are known.
(c) It was to determine the volume of a sphere without measuring the circumference.
(d) It was to solve equations were only two varibles are known.

12. What provided most of the content in the book Elements?
(a) Hypotheses.
(b) Postulates.
(c) Propositions.
(d) Notions.

13. How did Archimedes demonstrate his theory of pi?
(a) He demonstrated that the area of the circle is never less than the area of the triangle.
(b) He demonstrated that the area of the circle is never equal to the area of the triangle.
(c) He demonstrated that the area of the circle is neither greater than nor less than the area of the triangle and therefore must be equal to it.
(d) He demonstrated that the area of the circle is always greater than the area of the triangle.

14. Which of the following was NOT one of Gauss' discoveries?
(a) That there is no apparent contraction to the assumption that the sum of angles in a triangle can have fewer than 180 degrees.
(b) That angles in a triangles can not add up to more than 180 degrees.
(c) "Non-euclidean" geometry.
(d) That under Euclid's definition parallel lines can intersect.

15. Which of the following is an example of a postulate that must be accepted in Elements?
(a) It is possible to draw a straight line between an infinite number of points.
(b) It is possible to connect any two points with a line and make a circle.
(c) It is possible to draw an arc with any three points.
(d) It is possible to draw a circle that contains no lines.

Short Answer Questions

1. What name did Euclid give for numbers that could be divided by numbers other than themselves and one?

2. What did Dunham consider as Archimedes's "masterpiece"?

3. Who was Heron?

4. In what century did Archimedes live?

5. What was true about Hippocrates's proof?

(see the answer keys)

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