Suggested Exercises

• Section and page numbers are from the textbook The History of Mathematics: An Introduction by David Burton.

Suggested Exercises 1 (covering Lectures 1-2): 

1. Sec 1.2: 1, 2

2. Sec 1.3: 1, 2, 3, 4, 6, 7, 10, 11, 13, 14, 17

3. Sec 2.3: 21, 22, 23, 24

4. Sec 2.4: 1, 2, 4, 11

5. Give two examples for the mathematics perception of animals.

6. What are the two systems that control animals' number sense in cognitive psychology?

7. What is the importance of the Rosetta stone?

8. Give three examples for the mathematical questions from Mesopotamian tablets.

9. Summarize important Chinese mathematical findings.

10. Explain some of the mathematical findings of the Indian mathematician Aryabhata.

Suggested Exercises 2 (covering Lectures 3-4): 

1. Sec 5.5: 1, 2, 5, 8, 10, 14

2. Explain the contributions of Millet school.

3. Who was the leader of a cult about numbers in ancient Greece? Explain his contributions.

4. Explain overall Hellenistic period and its affect on mathematics.

5. Summarize contributions of Euclid.

6. Define Diophantine equations.

7. Explain overall golden age of Islam and its affect on mathematics.

8. When and where was the first madrasa founded?

9. Summarize contributions of Omar Khayyam.

10. How did Islam effect European mathematics?

Suggested Exercises 3 (covering Lectures 5-6): 

1. Give two examples of Turkish center of culture (cities).

2. Explain important Turkish zijes.

3. Who is considered the first Ottoman mathematician?

4. Compare Taqi ad-Din's and Brahe's observatories.

5. Why Salih Zeki is crucial for Turkish science and philosophy?

6. Explain in details the university reform 1933.

7. Give two examples of scientific terms which are named after recent Turkish scientists.

8. Sec 6.2: 1, 4, 11

9. Sec 6.3: 1, 4

10. Sec 8.1: 7, 9, 11

11. Explain overall dark ages of Europe.

12. What are the contributions of Fibonacci?

13. Summarize Italian algebra during renaissance.

14. Give three examples of the history of modern notation.

Suggested Exercises 4 (covering Lectures 7-8): 

1. Explain the general idea of Platonism.

2. What are the ideas of Kant about philosophy of mathematics?

3. Explain the history of set theory from Riemann to ZFC set theory.

4. What are the three main foundational schools in mathematics? Explain them.

5. Explain the contributions of Frege and Russell.

6. Explain the contributions of Gödel and Cohen.

7. What is modularity theorem and how is it related to Fermat?

8. Give three examples of number systems "bigger" than real numbers.

9. Who proved that higher order (> 4) equations are non-solvable in radicals?

10. Give some details about the life of Galois.

11. Explain Euler's contributions on number theory.

12. Who is the Turkish mathematician who contributed to search of the proof of twin prime conjecture?

Suggested Exercises 5 (covering Lectures 9-10): 

1. Explain the ideas of Aristotle on motion.

2. What is the heliocentric model?

3. Are there epicycles in the model of Copernicus?

4. Explain Kepler's laws.

5. From which law one can derive Kepler's laws?

6. Why method of exhaustion is important for calculus?

7. Who proved the Poincare conjecture on dimension 3?

8. Write 5 axioms of Euclidean geometry.

9. What is the importance of the 5th axiom of Euclidean geometry?

10. Compare Euclidean, hyperbolic and elliptic geometries.

11. Explain the history of aether.

12. What do special and general relativity achieve?