Homework 1

Notes

• In order to get full credit in the open ended questions, you need to spend some time reflecting upon and writing your answers. In most of these problems, the probability that a one-sentence answer will get full credit is close to 0 (and so will be what you learn from the exercise)
• Make sure you show all your work on the problems that require it so. Otherwise, even if you give a correct answer, if you do not explain how you obtained it, you'll get very little or no credit.
• It would be great if you discussed ideas with your classmates. The write-up, however, must be done individually.
• These questions are designed to exercise your amazing human mind. Unless specifically allowed, please do not use AI (ChatGPT or similar) to answer them.
• Recall that the slides of the lectures can be found here.

Problems

1. Find a quote about mathematics that includes or implies a definition (of mathematics) and explain what you think it means. Discuss whether you agree or disagree and explain your reasons.
2. Let your imagination fly and then reflect on your findings.
   1. Create a hypothesis as to the meaning of the scratches on the Ishango bone or why they might have been made. Be creative. The only requirement is that your hypothesis matches the pattern exhibited in the bone.
   2. Discuss whether your hypothesis fits in with what we know of math history at that point.
3. Locate one primary and one secondary source for one of the following topics. Include a photo of the primary source, as well as a brief description and estimated date. For the secondary source, include a complete reference (a link is not enough; you must write down the author, title, journal, date, etc.). Explain how you know the secondary source is peer-reviewed. In both cases, provide a link. (Note that it is fine to use ChatGPT to find a partial answer to this question, but make sure you use your human mind to complete it).
   1. Mathematics in Ancient Egypt
   2. Mathematics in Ancient Mesopotamia
   3. Mathematics in Ancient China
4. Discuss when and by whom x was first used as an unknown in mathematics, and whether there is agreement among scholars about the answer to this question. Provide a complete reference (as indicated in the previous problem) for your answer. Explain why you think the source you cite is appropriate and your answer is plausible.
5. Units of time, such as a day, a month, and a year, have ratios. In fact, you probably know that a year is about 365 1/4 days long. Imagine that you were never taught this fact.
   1. How would you -how did people originally- determine how many days are are in a year?
   2. What do you think was the purpose of a calendar for very ancient societies? Write down two possible uses of calendars for them.
6. In what sense is it possible to know the exact value of a number such as √2. Obviously, if a number is known only by its whole infinite decimal expansion, nobody does know and nobody will ever know the exact value of this number. What immediate practical consequences, if any, does this fact have? Is there any other sense in which one could be said to know this value exactly? If there is no direct consequences of being ignorant of its exact value, is there any practical value in having the concept of an exact square root of 2? Why not simply replace it by a suitable approximation such as 1.41421? Consider also other "irrational" numbers, such as π and e. What is the value of having the concept of such numbers as opposed to approximate rational replacements for them.

Note: The last two problems are adapted from the Roger Cook's book History of Mathematics: A Brief Course, by Roger Cooke, 2005, Wiley-Interscience.