Helenic Mathematics Euclid
Important notes:
- On the problems that require computations, the steps to find the answer must be included in your submission. An answer with little or no explanantion will receive little or no credit.
- The extra credit problems can be attempted if and only if you complete all the non-extra credit ones.
- In problems 2, 3 and 4 you are supposed to find, read and understand the appropriate propositions (Pythagorean theorem, construction of a square, infinitude of primes) and proofs in Euclid’s elements, copy the statements (not the proofs) and explain the proofs in your own words. You can use the Geogebra apps we worked on in class or make your own illustrations. Excellent explanations will be granted extra-credit points.
- propositon 1 read it
- Commensturablity look for it
- Learn and understand definition of commensurability, incommensurability, axiom (recall that Euclid divided axioms into “common notions” and “postulates”.)
- Write down a note stating that you read the “important notes” above.
- Euclid's proof of the Pythagoran theorem. Suggested Geogebra apps are here, here and here.
- Explain the proof of "To construct a square given a segment."
- Explain the proof of "there are infinitely many primes."
- In the proof of the propoposition stated in Problem 4, given three primes a new one is defined. Find four concrete examples of this procedure, assuming that one of the given primes is 41. Extra credit. Find an example where three or more primes can be defined in the same way. For instance, starting with the primes 7, 11, and 13 one can find three “new” primes: 2, 3 and 167.
- Euclid’s Elements influenced at least two US presidents. One was mentioned in class. Find both of these presidents and write a short paragraph describing how they were influenced by the Elements.
- Recall that in class we discussed the Euler characteristc of a solid with polygonal faces (You can also read about it here). Check that the Euler characteristic of a pyramid with pentagonal base is 2. (Note: The fact that Euler characteristic of a solid with polygonal faces is equal to 2 was mentioned as a theorem that is not Euclid’s elements)
- Choose one proposition of the Elements, not discussed in class or in this homework. Read it, understand it and write down the proof in your own words. You can assume everything that has been proved before that proposition.
Mayan Mathematics
- Given the Mayan date, say, (10,12,100) in the Calendar Round (the three wheels 13x20x365) find the date 2.2.5.11.15 days later. (We ar following the notation used in class).
- Given a certain number of days, say 35788, find the corresponding calendric date, (of the form a.b.c.d.e)
Helenic Mathematics Before Euclid
- zeno discuss a paradox defend attack
- Extra credit: Find a proof of the Pythagorean theorem not discussed in class.
Office hours are an important part in supporting you throughout this course.
Even if you don’t have specific questions, needs, and concerns, I would love to
meet up with you at least once during this semester. There are a couple of
options to meet up:
- Explain why the Egyptian multiplication algorithm works.
- https://www.storyofmathematics.com/egyptian.html
- Find a quotation that includes a definition of mathematics (for instance, one from the first week slides or one that you found) and explain what you think it means. Discuss whether you agree or not.
- Write down the idea you found most interesting, from those we discussed since the begining of the class.
- https://www.storyofmathematics.com/ find wrong info online
- http://www-math.ucdenver.edu/~wcherowi/courses/history2/hmqual.html
- http://www-math.ucdenver.edu/~wcherowi/courses/history2/hmqual.html
