MAT 101: Survey of Mathematical Reasoning
Final Research Topics
You may choose any of the available topics on a first come, first served basis. If it is
crossed out, that means that someone else has selected that topic.
1. In the first chapter of Michio Kaku’s book, Visions, the author makes several predictions
about things that will occur before the year 2020. Find at least three of his predictions that have
already come true. Include when the prediction came true and who made the prediction come
true.
2. In the first chapter of Michio Kaku’s book, Visions, the author discusses how the quantum
theory was the basis for many of the advances of the 20th century. Discuss who discovered the
quantum theory and talk a little bit about the person’s background. Also discuss what the
quantum theory explains.
3. When Galileo was teaching in Padua, he used inductive reasoning to discover some of the
physical laws that govern the time of descent for a ball that rolls on a cyclodial path from point A
to point B and for a ball that rolls on a straight incline from point A to point B. Search for
information about Galileo’s experiment with a cycloid and discuss what Galileo Discovered.
4. Paul Erdos was a mathematician known for his elegant solutions of problems in number
theory, combinatorics, discrete mathematics, and graph theory. He loved to solve mathematical
problems and for those problems he could not solve, he offered financial rewards, up to $10,000,
to the person who could provide a solution. Discuss the life of Paul Erdos. Include information
about the type of problem that Erdos considered the most interesting. What did Erdos have to
say about the Collatz problem?
5. There are many unsolved problems in mathematics. One famous unsolved problem is known
as the Collatz problem, or the 3x + 1 problem. This problem was created by L. Collatz in 1937.
Although the procedures in the Collatz problem are easy to understand, the problem remains
unsolved. Discuss the Collatz problem and include the explanation of the meaning of a
“hailstone” sequence.
6. The Hilbert Hotel is an imaginary hotel created by the mathematician David Hilbert. The
hotel has an infinite number of rooms. Each room is numbered with a natural number; room 1,
room 2, room 3, and so on. Explain some of the interesting questions that arise when guests
arrive to stay at the hotel.
7. Raymond Smullyan is a logician, a philosopher, a professor, and an author of many books on
logic, and puzzles. Some of his fans rate his puzzle books as the best ever written. Discuss the
life of Smullyan and his work in the area of logic.
8. Lewis Carroll (Charles Dodgson) wrote many puzzles, many of which he recorded in his
diaries. Find puzzles created by Lewis Carroll. For the two puzzles that you think are the most
interesting, explain the puzzles and give their solution.
9. Lewis Carroll (Charles Dodgson) devised a bilateral diagram to analyze sullogisms. His
method has some advantages over Euler diagrams and Venn diagrams. Discuss Carroll’s method
of analyzing syllogisms. Explain his method and its advantages.
10. The Ionic Greek numeration system assigned numerical values to the letters of the Greek
alphabet. Research the Ionic Greek numeration system. Include information about some of the
advantages and disadvantages of this system compared with our present Hindu-Arabic
numeration system.
11. In 1977, Ron Rivest, Adi Shamir, and Leonard Adleman invented a method for encrypting
information. Their method is known as the RSA algorithm. Today the RSA algorithm is used
by both the Microsoft Internet Explorer and Netscape Navigator web browsers, as well as by
VISA and MasterCard to ensure secure electronic credit card transactions. The RSA algorithm
involves large prime numbers. Research the RSA algorithm and discuss the reasons why this
algorithm has become one of the most popular of all the encryption algorithms.
12. Discuss the life of John Venn. Include information about where he was educated as well as
which mathematician influenced him to create the Venn Diagrams. Discuss the difference
between his influence and his own diagrams.
13. Choose 1 famous mathematician of the 20th century and a mathematician from any other
century. Discuss why each mathematician is famous and find any similarities or differences
between the backgrounds of these mathematicians.
14. In section 1 of chapter 2 we discussed the concept of cardinal numbers. Find information
about ordinal numbers and nominal numbers. Explain the difference between these three types
of numbers. Also, give 3 examples of how each type of number is used in the “real world”.
15. Georg Cantor is recognized as the founder and leader in the development of set theory. Do
research and write a paper on his life and his contributions to set theory and the field of
mathematics.
16. Write a report on the life and accomplishments of George Boole, who was an important
contributor to the development of logic. Indicate how his work eventually led to development of
the computer.
17. Show how logic is used in advertising. Discuss several advertisements and show how logic
is used to persuade the reader. Also, find examples of valid or invalid arguments in printed
matter such as a newspaper or magazine articles. Explain why the arguments are valid or
invalid.
18. Investigate the development of the Hindu-Arabic system of numeration. Start with the
earliest records of this system in India.
19. Report on the history of pi. Indicate when the symbol  was first used and list the first 50
digits of .