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

2. Which of the following could NOT be included as a step in Euclid's great theorem?
(a) After summation, the new number can be prime or composite.
(b) Take a finite group of primes and add them together, plus one.
(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.

3. What did Gauss set out to prove?
(a) That a right angle is always equal to 90 degrees.
(b) That a circle can have less than 360 degrees.
(c) That the sum of the angles in a triangle is 180 degrees.
(d) That Euclid's postulate on straight lines was incorrect.

4. What did Plato use his inspiration from Euclid for?
(a) To construct his theory on the shape of the Universe.
(b) To create a new theorem of algebra.
(c) To prove Euclid's number theory was incorrect.
(d) To classify geometric shapes by their complexity.

5. After working on pi, what did Archimedes continue with in his study of mathematics?
(a) He studied the relationship of sine to cosine.
(b) He studied the relationship between ratios in triangles.
(c) He studied the volume to surface area ratios of cubes.
(d) He studied the volume and surface area of spheres, cones, and cylinders.

6. Who's method did Tartaglia's challenger use in the contest to solve cubic equations?
(a) del Ferro's method.
(b) Cardano's method.
(c) Fontana's method.
(d) Pacioli's method.

7. What did Dunham consider extraordinary about the Elements?
(a) The content was totally unique.
(b) The content was not based on previous authors' work.
(c) How geometric proofs were presented.
(d) How Hippocrates ordered the book.

8. What did Euclid do in his 48th proposition?
(a) Euclid proved the Pythagorean Theorem.
(b) Euclid proved the converse of the Pythagorean Theorem.
(c) Euclid demonstrated how to use the Pythagorean Theorem.
(d) Euclid demonstrated the faults of the Pythagorean Theorem.

9. Exactly what limit is reached at a quartic equation?
(a) The limit of algebra.
(b) The limit of the Pythagorean Theorem.
(c) The limit of the decompressed cubic method.
(d) The limit of logical geometric proofs.

10. What was the same about Apollonius and Erosthanes?
(a) They both studied the Universe.
(b) They both calculated the circumference of the earth.
(c) They were both mathematicians.
(d) They were both born in the same year.

11. Where was the modern number system developed?
(a) In the West.
(b) In ancient Rome.
(c) In the East.
(d) In ancient Alexanderia.

12. What was known about pi, during Archimedes' time?
(a) Nothing, the concept of pi was unknown.
(b) That it was never the same number value for a given circle.
(c) That it could not be assigned a relationship between measurements in a circle.
(d) That is was the relationship between the diameter and circumference of a circle.

13. Who wrote a treatise that supposed that cubic equations may be impossible to solve?
(a) Niccolo Fontana.
(b) Luca Pacioli.
(c) Scipione del Ferro.
(d) Gerolamo Cardano.

14. That properties of specific shapes were early Egyptians aware of?
(a) Parallelograms.
(b) Pi and the diameter of a circle.
(c) Right triangles.
(d) Irregular solids.

15. According to Euclid, when is a triangle a right triangle?
(a) When a triangle can be constructed with three unequal sides.
(b) When a triangle does not have a side which can be considered a hypotenuse.
(c) When a triangle has a side whose square is the sum of the squares of the two legs.
(d) When a triangle has three sides whose squares are equal to the area of the triangle.

Short Answer Questions

1. Which words best describe how solid proofs were developed in Elements?

2. What provided most of the content in the book Elements?

3. "Straight lines in the same plane that will never meet if extended forever" is a definition of what?

4. What did Archimedes manage to prove using Euclid's ideas?

5. Who was Eratosthanes?

(see the answer keys)

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