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.
- Here and here you have two versions of Euclid’s Elements, and here is Oliver Byrne’s version used in class.
- 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 "there are infinitely many primes."
- In the proof of the propoposition ("there are infinitely many primes.") stated the previous problem, 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.
- Is it possible to construct a square with the same area of a given circle, only using straight-edge and compass? Give a justifcation to your answer.
- Euclid’s Element influenced at least two US presidents. One is mentioned here. Find both of these presidents and write a short paragraph describing how they were influenced by the Elements.